Lecture Notes on Analysis II MA131. Xue-Mei Li

Transcription

1 Lecture Notes on Analysis II MA131 Xue-Mei Li March 8, 2013

2 The lecture notes are based on this and previous years lectures by myself and by the previous lecturers. I would like to thank David Mond, who is the driving force for me to finish this work and for his important feed backs. In this edition I would like to thank Ruoran Huang and Chris Midgley for a list of typos. Xue-Mei Li Office C [email protected] Important note: Please tell Xue-Mei Li ([email protected]) or David Mond ([email protected]) if you find any mistakes in these lecture notes, so that future printed editions, and the online lecture notes, can be improved. If something seems like a mistake but you are not sure, write to us anyway if there is a mistake, your message will help correct it, and if there is not, you will learn something about Analysis. 1

3 Contents 0.1 Introduction Continuity of Functions of One Real Variable Functions Lecture Useful Inequalities and Identities Continuous Functions Lecture Continuity and Sequential Continuity Lecture Properties of Continuous Functions Lecture Continuity of Trigonometric Functions Measurement of angles Continuous Functions on closed Intervals: I The Intermediate Value Theorem Lectures 5 & Continuous Limits Continuous Limits Lecture Limit and continuity Negation Continuous limit and sequential limit The Sandwich Theorem The Algebra of Limits and limit of composition of functions Accumulation points and Limits* Limits to Limits at The Extreme Value Theorem Bounded Functions The Bolzano-Weierstrass Theorem

4 4.3 The Extreme Value Theorem Lecture The Inverse Function Theorem for continuous functions The Inverse of a Function Monotone Functions Continuous Injective and Monotone Surjective Functions Lecture The Inverse Function Theorem (Version 1) Lecture Differentiation Definition of Differentiability Lecture The Weierstrass-Carathéodory Formulation for Differentiability Properties of Differentiation Lecture The Inverse Function Theorem Lecture One Sided Derivatives The mean value theorem Local Extrema Lecture Global Maximum and Minimum Rolle s Theorem The Mean Value Theorem Lecture Monotonicity and Derivatives Inequalities From the MVT Lecture Asymptotics of f at Infinity Lecture An Example: Gaussian Envelopes Higher order derivatives Continuously Differentiable Functions Higher Order Derivatives Convexity Still higher derivatives* Power Series Functions Radius of Convergence The Limit Superior of a Sequence Hadamard s Test Continuity of Power Series Functions Term by Term Differentiation

5 10 Classical Functions The Exponential and the Natural Logarithm Function The Sine and Cosine Functions Taylor s Theorem and Taylor Series Non-centered Power Series Uniqueness of Power Series Expansion Taylor s Theorem with remainder in Lagrange form Taylor s Theorem in Integral Form* Cauchy s Mean Value Theorem A Table Techniques for Evaluating Limits Use of Taylor s Theorem L Hôpital s Rule Unfinished Business 160 4

6 The module 0.1 Introduction Analysis II, together with Analysis I, is a 24 CATS core module for first year students. The content of the module will be delivered in 29 lectures. Assessment 7.5%, Term 1 assignments, 7.5%, Term 2 assignments, 25%, January exam (on Analysis 1) 60%, final exam (3 hour in June, cover analysis 1+2) Fail the final ===> Fail the module. Assignments: Given out on Thursday and Fridays in lectures, to hand in the following Thursday in supervisor s pigeonhole by 14:00. Each assignment is divided into part A, part B and part C. Part B is for assignments. Learn Definitions and statements of theorems. Take the exercises very seriously. It is the real way you learn (and hence pass the summer exam) Books: G. H. Hardy: A Course of Pure Mathematics, Third Edition (First Edition 1908), Cambridge University Press. E. T. Whittaker &G. N. Watson A Course of Modern Analysis T. M. Apostol Calculus, Vol I&II, 2nd Edition, John Wiley & Sons E. Hairer & G. Wanner: Analysis by Its History, Springer. Objectives of the module: Study functions of one real variable. This module leads to analysis of functions of two or more variables (Analysis III), Ordinary differential equations, partial differential equations, Fourier Analysis, Functional analysis etc. What we cover: 5

7 Continuity of functions of one real variable (lectures 1-3) the Intermediate Value theorem, continuous functions on closed intervals Continuous limits (circa lecture 8-9) Extreme Value Theorem, Inverse Function Theorem (circa 10-11) Differentiation (circa lecture 12), Mean Value Theorem Power series (ca. lecture 23) Taylor s Theorem (ca. lecture 23) L Hôpital s rule (ca. lecture 26) Relation to Analysis I. Continuity and continuous limits of function will be formulated in terms of sequential limits. Power series are functions obtained by sums of infinite series. Relation to Analysis II. Uniform convergence and the Fundamental Theorem of Calculus from Analysis III would make a lot of proofs here easier. Relation to Complex Analysis The power series expansion. Relation to Ordinary Differential Equations Differentiability of functions. Solve e.g. f (x) = f(x), f(0) = 1. Feedbacks Ask questions in lectures. Talk to me after or before the lectures. Plan your study: Shortly after each hour of lecture, you need to spend one hour going over lecture notes and mulling over the assignments. It is a rare student who has understood everything during the lecture. There is plentiful evidence that putting in this effort shortly after the lecture pays ample dividends in terms of understanding. In particular, if you begin a lecture having understood the previous one, you learn much more from it, so the process is cumulative. Please plan two hours per week for the exercises. 6

8 Topics by Lecture (approximate dates) 1. Introduction. Continuity, ɛ δ formulation 2. Properties of continuous functions, sequential continuity 3. Algebra of continuity 4. Composition of continuous functions, examples 5. The Intermediate Value Theorem (IVT) 6. The Intermediate Value Theorem (continued) 7. Continuous Limits, ɛ δ formulation, relation with to sequential limits and continuity 8. One sided limits, left and right continuity 9. The Extreme Value Theorem 10. The Inverse Function Theorem (continuous version) 11. Differentiation, Weierstrass formulation 12. Algebraic rules, chain rule 13. The Inverse Function Theorem (Differentiable version) 14. Local Extrema, critical points, Rolle s Theorem 15. The Mean Value Theorem 16. Deduce Inequalities, asymptotics of f at infinity 17. Higher order derivatives, C k (a, b) functions, examples of functions, convexity and graphing 18. Formal power series, radius of convergence 19. Limit superior, 20. Hadamard Test for power series, Functions defined by power series 21. Term by Term Differentiation Theorem 22. Classical Functions of Analysis 7

9 23. Polynomial approximation, Taylor s series, Taylor s formula 24. Taylor s Theorem 25. Cauchy s Mean Value Theorem, Techniques for evaluating limits 26. L Hôpital s rule 27. L Hôpital s rule 28. Unfinished business, Summary, Matters related to exams 29. Question Time, etc 8

10 Chapter 1 Continuity of Functions of One Real Variable Let R be the set of real numbers. We will often use the letter E to denote a subset of R. Here are some examples of the kind of subsets we will be considering: E = R, E = (a, b) (open interval), E = [a, b] (closed interval), E = (a, b] (semi-closed interval), E = [a, b), E = (a, ), E = (, b), and E = (1, 2) (2, 3). The set of rational numbers Q is also a subset of R. 1.1 Functions Lecture 1 Definition By a function f : E R we mean a rule which to every number in E assigns a number from R. This correspondence is denoted by y = f(x), or x f(x). We say that y is the image of x and x is a pre-image of y. The set E is the domain of f. The range, or image, of f consists of the images of all points of E. It is often denoted f(e). We denote by N the set of natural numbers, Z the set of integers and Q the set of rational numbers: N = {1, 2,... } Z = {0, ±1, ±2,... } Q = { p q : p, q Z, q 0}. 9

11 Example E = [ 1, 3]. { 2x, 1 x 1 f(x) = 3 x, 1 < x 3. The range of f is [ 2, 2]. 2. E = R. Range f= {0, 1}. f(x) = { 1, if x Q 0, if x Q 3. E = R. { 1/q, if x = p f(x) = q where p Z, q N have no common divisor 0, if x Q Range f= {0} { 1 q, q N {0}}. 4. E = R. f(x) = { sin x x if x 0 1 if x = 0. Range f =? A little work is needed here. 5. E = ( 1, 1). f(x) = ( 1) n xn n. n=1 This function is a representation of log(1+x), see chapter on Taylor series. Range f = ( log 2, ). 1.2 Useful Inequalities and Identities a 2 + b 2 2ab a + b a + b b a max( b a, a b ). Proof The first follows from (a b) 2 > 0. The second inequality is proved by squaring a + b and noting that ab a b. The third inequality follows from the fact that b = a + (b a) a + b a 10

12 and by symmetry a b + b a. Pythagorean theorem : sin 2 x + cos 2 x = 1. cos 2 x sin 2 x = cos(2x), cos(x) = 1 2 sin 2 ( x 2 ). 1.3 Continuous Functions Lecture 2 What do we mean by saying that f(x) varies continuously with x? It is reasonable to say f is continuous if the graph of f is an unbroken continuous curve. The concept of an unbroken continuous curve seems easy to understand. However we may need to pay attention. For example we look at the graph of { x, x 1 f(x) = x + 1, if x > 1 It is easy to see that the curve is continuous everywhere except at x = 1. The function is not continuous at x = 1 since there is a gap of 1 between the values of f(1) and f(x) for x close to 1. It is continuous everywhere else. y 1 x Now take the function F (x) = { x, x 1 x , if x > 1 The function is not continuous at x = 1 since there is a gap of However can we see this gap on a graph with our naked eyes? No, unless you have exceptional eyesight! 11

13 Here is a theorem we will prove, once we have the definition of continuous function. Theorem (Intermediate value theorem): Let f : [a, b] R be a continuous function. Suppose that f(a) f(b), and that the real number v lies between f(a) and f(b). Then there is a point c [a, b] such that f(c) = v. This looks obvious, no? In the picture shown here, it says that if the graph of the continuous function y = f(x) starts, at (a, f(a)), below the straight line y = v and ends, at (b, f(b)), above it, then at some point between these two points it must cross this line. y f(b) v f(a) y=f(x) a c b x But how can we prove this? Notice that its truth uses some subtle facts about the real numbers. If, instead of the domain of f being an interval in R, it is an interval in Q, then the statement is no longer true. For example, we would probably agree that the function F (x) = x 2 is continuous (soon we will see that it is). If we now consider the function f : Q Q defined by the same formula, then the rational number v = 2 lies between 0 = f(0) and 9 = f(3), but even so there is no c between 0 and 3 (or anywhere else, for that matter) such that f(c) = 2. Sometimes what seems obvious becomes a little less obvious if you widen your perspective. 12

14 These examples call for a proper definition for the notion of continuous function. In Analysis, the letter ε is often used to denote a distance, and generally we want to find way to make some quantity smaller than ε : The sequence (x n ) n N converges to l if for every ε > 0, there exists N N such that if n > N, then x n l < ε. The sequence (x n ) n N is Cauchy if for every ε > 0, there exists N N such that if m, n > N then x m x n < ε. In each case, by taking n, or n and m, sufficiently big, we make the distance between x n and l, or the distance between x n and x m, smaller than ε. Important: The distance between a and b is a b. The set of x such that x a < δ is the same as the set of x with a δ < x < a δ. The definition of continuity is similar in spirit to the definition of convergence. It too begins with an ε, but instead of meeting the challenge by finding a suitable N N, we have to find a positive real number δ, as follows: Definition Let E be subset of R and c a point of E. 1. A function f : E R is continuous at c if for every ε > 0 there exists a δ > 0 such that if x c < δ and x E then f(x) f(c) < ε. 2. If f is continuous at every point c of E, we say f is continuous on E or simply that f is continuous. The reason that we require x E is that f(x) is only defined if x E! If f is a function with domain R, we generally drop E in the formulation above. Example The function f : R R given by f(x) = 3x is continuous at every point x 0. It is very easy to see this. Let ε > 0. If δ = ε/3 then x x 0 < δ = x x 0 < ε/3 = 3x 3x 0 < ε = f(x) f(x 0 ) < ε as required. Exercise For each of the following cases, show that the function given is continuous at every point x 0 R. 13

15 1. f(x) = 3x f(x) = 5x 3. f(x) = f(x) = λ with λ a non-zero constant. Example Prove that the function f : R R given by f(x) = x 2 + x is continuous at x = 2. Take ε > 0. We wish to have f(x) f(2) < ɛ. Let us simplify and bound the left hand side quantity, f(x) f(2) = x 2 + x (4 + 2) x x 2 = x + 2 x 2 + x 2 = ( x ) x 2. Now we should give an upper bound for x + 2. If x 2 1, then This shows that x + 2 = (x 2) + 4 x f(x) f(2) ( x ) x 2 6 x 2. We can take δ = min( ε 6, 1). If x 2 < δ, the above reasoning gives f(x) f(2) < ɛ. There are two things which make the proof easier than it looks at first sight: 1. If δ works, then so does any δ > 0 which is smaller than δ. There is no single right answer. Sometimes you can make a guess which may not be the biggest possible δ which works, but still works. In the last example, we could have taken δ = min{ ε 10, 1} instead of min{ ε 6, 1}, just to be on the safe side. It would still have been right. 14

16 2. Fortunately, we don t need to find δ very often. In fact, it turns out that sometimes it is easier to use general theorems than to prove continuity directly from the definition. Sometimes it s easier to prove general theorems than to prove continuity directly from the definition in an example. My preferred proof of the continuity of the function f(x) = x 2 + x goes like this: (a) First, prove a general theorem showing that if f and g are functions which are continuous at c, then so are the functions f + g and fg, defined by { (f + g)(x) = f(x) + g(x) (fg)(x) = f(x) g(x) (b) Second, show how to assemble the function h(x) = x 2 + x out of simpler functions, whose continuity we can prove effortlessly. In this case, h is the sum of the functions x x 2 and x x. The first of these is the product of the functions x x and x x. Continuity of the function f(x) = x is very easy to show! As you can imagine, the general theorem in the first step will be used on many occasions, as will the continuity of the simple functions in the second step. So general theorems save work. But that will come later. For now, you have to learn how to apply the definition directly in some simple examples. First, a visual example with no calculation at all: 15

17 f(c) + ε f(c) f(c) ε a c b Here, as you can see, for all x in (a, b), we have f(c) ε < f(x) < f(c)+ε. What could be the value of δ? The largest possible interval centred on c is shown in the picture; so the biggest possible value of δ here is b c. If we took as δ the larger value c a, there would be points within a distance δ of c, but for which f(x) is not within a distance ε of f(c). Exercise Suppose that f(x) = x 2, x 0 = 7 and ε = 5. What is the largest value of δ such that if x x 0 < δ then f(x) f(x 0 ) < ε? What if ε = 50? What if ε = 0.1? Drawing a picture like the last one may help. Discontinuity Let us now do a logic problem. We gave a definition of what is meant by f is continuous at c : ε > 0 δ > 0 such that if x c < δ then f(x) f(c) < ε. How to express the meaning of f is not continuous at c in these same terms? there exists an ε > 0 such that for all δ > 0, there exists an x satisfying x c < δ but f(x) f(c) ε. 16

18 Example Consider f : R R, { 0, if x Q f(x) = 1, if x Q. Then f is discontinuous at every point. The key point is that between any two numbers there is an irrational number and a rational number. More precisely, take ε = 0.5. Let δ > 0 be any positive number. If c Q, then f(c) = 0. There is an x Q with x c < δ; because x Q, f(x) = 1. Thus f(x) f(c) = 1 0 > 0.5. So f is not continuous at c. If c Q then f(c) = 1. Take ε = 0.5. For any δ > 0, take x Q with x c < δ; then f(x) f(c) = 0 1 > 0.5 = ε. Example Consider f : R R, { 0, if x Q f(x) = x, if x Q Claim: This function is continuous at c = 0 and discontinuous everywhere else. Let c = 0. For any ε > 0 take δ = ε. If x c < δ, then f(x) f(0) = x in case c Q, and f(x) f(0) = 0 0 = 0 in case c Q. In both cases, f(x) f(0) x < δ = ε. Hence f is continuous at 0. Exercise Show that the function f of the last example is not continuous at c 0. Hint: take ε = c /2. Example Suppose that g : R R and h : R R are continuous, and let c be some fixed real number. Define a new function f : R R by { g(x), x < c f(x) = h(x), x c. Then the function f is continuous at c if and only if g(c) = h(c). Proof: Since g and h are continuous at c, for any ε > 0 there are δ 1 > 0 and δ 2 > 0 such that g(x) g(c) < ε when x c < δ 1 and such that h(x) h(c) < ε when x c < δ 2. 17

19 Define δ = min(δ 1, δ 2 ). Then if x c < δ, g(x) g(c) < ε and h(x) h(c) < ε. Case 1: Suppose g(c) = h(c). x c < δ, For any ε > 0 and the above δ, if { g(x) g(c) < ε if x < c f(x) f(c) = h(x) h(c) < ε if x c So f is continuous at c. Case 2: Suppose g(c) h(c). Take ε = 1 g(c) h(c). 2 By the continuity of g at c, there exists δ 0 such that if x c < δ 0, then g(x) g(c) < ε. If x < c, then moreover f(c) f(x) = h(c) g(x) ; h(c) g(x) = h(c) g(c) + g(c) g(x) h(c) g(c) g(c) g(x). We have h(c) g(c) = 2ε, and if c x < δ 0 then g(c) g(x) < ε. It follows that if c δ 0 < x < c, f(c) f(x) = h(c) h(x) > ε. So if c δ 0 < x < c, then no matter how close x is to c we have f(x) f(c) > ε. This shows f is not continuous at c. Example Let E = [ 1, 3]. Consider f : [ 1, 3] R, { 2x, 1 x 1 f(x) = 3 x, 1 < x 3. The function is continuous everywhere. 18

20 Example Let f : R R. { 1/q, if x = p f(x) =, q > 0 q 0, if x Q p, q coprime integers Then f is continuous at all irrational points, and discontinuous at all rational points. Proof Every rational number p/q can be written with positive denominator q, and in this proof we will always use this choice. Case 1. Let c = p/q Q. We show that f is not continuous at c. Take ε = 1 2q. No matter how small is δ there is an irrational number x such that x c < δ. And f(x) f(c) = 1 q > ε. Case 2. Let c Q. We show that f is continuous at c. Let ε > 0. If x = p Q, q f(x) f(c) = 1. So the only rational numbers p/q for which f(x) f(p/q) ε q are those with denominator q less than or equal to 1/ε. Let A = {x Q (c 1, c + 1) : q 1 }. Clearly c / A. The crucial point is ε that if it is not empty, A contains only finitely many elements. To see this, observe that its members have only finitely many possible denominators q, since q must be a natural number 1/ε. For each possible q, we have p/q (c 1, c + 1) p (qc q, qc + q), so no more than 2q different values of p are possible. It now follows that the set B := { x c : x A}, if not empty, has finitely many members, all strictly positive. Therefore if B is not empty, it has a least element, and this element is strictly positive. Take δ to be this element. If B is empty, take δ =. In either case, (c δ, c + δ) does not contain any number from A. Suppose x c < δ. If x / Q, then f(x) = f(x) = 0, so f(x) f(c) = 0 < ε. If x = p/q Q then since x / A, f(x) f(c) = 1 q < ε. Example f : ( 1, 1) R. f(x) = ( 1) n xn n n=1 Is f continuous at 0? This is a difficult problem for us at the moment. We cannot even graph this function and so we have little intuition. But we can answer this question after the study of power series. Exercise Is there a value a such that the function 1, if x > 0 f a (x) = a, if x = 0 1, if x < 0. is continuous at x = 0? Justify your answer. 19

21 1.4 Continuity and Sequential Continuity Lecture 3 Definition We say f : E R is sequentially continuous at c E if for every sequence {x n } E such that lim n x n = c we have lim f(x n) = f(c). n Theorem Let E be a subset of R and c E. Let f : E R. The following are equivalent: 1. f is continuous at c. 2. f is sequentially continuous at c. Proof Suppose that f is continuous at c. For any ε > 0 there is a δ > 0 so that f(x) f(c) < ε, whenever x c < δ. Let {x n } be a sequence with lim n x n = c. There is an integer N such that x n c < δ, n > N, Hence if n > N, f(x n ) f(c) < ε. We have proved that f(x n ) f(c). Thus f is sequentially continuous at c. Suppose that f is not continuous at c. Then ε > 0 such that for every δ > 0, there is a point x E with x c < δ but f(x) f(c) ε. In particular, this holds for δ = 1 n : there exists a point x n E with x n c < 1 n, but f(x n ) f(c) ε. In this way we have constructed a sequence {x n } which tends to c, but for which f(x n ) f(c). This shows that f is not sequentially continuous at c. 20

22 Sequential continuity is handy when it comes to showing a function is discontinuous. Example Let f : R R be defined by { sin( 1 f(x) = x ) x 0 0 x = 0 is not continuous at 0. Proof Observe that f(0) = 0. Take x n = 1 2nπ+ π ; then x n 0 as n, 2 but f(x n ) = 1 0. Hence f is not sequentially continuous at 0 and it is therefore not continuous at 0. Example Let f(x) = { x + 1 x 2 12 x = 2 The function f is not continuous at x = 2. Proof Take x n = n ; then x n 2 as n, but f(x n ) = n 3 f(2). Hence f is not sequentially continuous at 2 and it is therefore not continuous at 2. We end this section with a small result which will be very useful later. The following lemma says that if f is continuous at c and f(c) 0 then f does not vanish close to c. In fact f has the same sign as f(c) in a neighbourhood of c. We introduce a new notation: if c is a point in R and r is also a real number, then B r (c) = {x R : x c < r}. It is sometimes called the ball of radius r centred on c. The same definition makes sense in higher dimensions (B r (c) is the set of points whose distance from c is less than r); in R 2 and R 3, B r (c) looks more round than it does in R. Lemma Non-vanishing lemma Suppose that f : E R is continuous at c. 21

23 1. If f(c) > 0, then there exists r > 0 such that f(x) > 0 for x B r (c). 2. If f(c) < 0, then there exists r > 0 such that f(x) < 0 for x B r (c). In both cases there is a neighbourhood of c on which f does not vanish. Proof Suppose that f(c) > 0. Take ε = f(c) 2 > 0. Let r > 0 be a number such that if x c < r and x E then f(x) f(c) < ε. For such x, f(x) > f(c) ε = f(c) 2 > 0. If f(c) < 0, take ε = f(c) 2 > 0. There is r > 0 such that on B r (c), f(x) < f(c) + ε = f(c) 2 < 0. Corollary Let f : (a, b) R be continuous at c (a, b). If f(c) 0 then 1 f is defined in a neighbourhood of c. Exercise Let f be continuous at c. Show that if v R and f(x) < v then there exists δ > 0 s.t. for all x (c δ, c + δ), f(x) < v. 1.5 Properties of Continuous Functions Lecture 4 In this section we learn how to build continuous functions from known continuous functions. Notation: for r > 0, let B r (c) = {x E c r < x < c + r}. This is the subset of E whose elements satisfy c r < x < c + r. We recall the following properties of limits of sequences: Proposition (Algebra of Limits) Suppose a n and b n are sequences with lim n a n = a and lim n b n = b. Then 1. lim n (a n + b n ) = a + b; 2. lim n (a n b n ) = ab; 3. lim n a n bn = a b, provided that b n 0 for all n and b 0. 22

24 What can we deduce about the continuity of f + g, fg and f/g if f and g are continuous? Proposition The algebra of continuous functions Suppose f and g are both defined on a subset E of R and are both continuous at c, then 1. af + bg is continuous at c where a and b are any constants. 2. fg is continuous at c 3. f/g is continuous at c if g(c) 0. Note: If g(c) 0 then there is a neighbourhood of c on which g(x) 0, by the non-vanishing lemma The function f/g referred to in part (3) of the proposition is well defined on this neighbourhood, and it is this function, on this neighbourhood, that we claim is continuous. Proof Let {x n } be any sequence in E converging to c. Then lim f(x n) = f(c), n lim g(x n) = g(c). n By the algebra of convergent sequences we see that lim [af(x n) + bg(x n )] = [af(c) + bg(c)]. n Thus lim (af + bg)(x n) = (af + bg)(c). n Hence af + bg is sequentially continuous at c. Since lim n f(x n )g(x n ) = f(c)g(c), lim (fg)(x n) = (fg)(c) n and fg is sequentially continuous at c. Suppose that g(c) 0. By Lemma 1.4.5, there is a neighbourhood B r (c) such that g(x) 0. We may now assume that E = B r (c) and hence assume that x n B r (c). Any sequences in B r (c) satisfies g(x n ) 0. By sequential continuity of g, lim n g(x n ) = g(c) 0. Consequently f(x n ) lim n g(x n ) = f(c) g(c) and f g is sequentially continuous at c. 23

25 The continuity of the function f(x) = x 2 + x, which we proved from first principles in Example 1.3.5, can be proved more easily using Proposition For the function g(x) = x is obviously continuous (take δ = ε), the function h(x) = x 2 is equal to g g and is therefore continuous by 1.5.2(2), and finally f = h + g and is therefore continuous by 1.5.2(1). Example The function f given by f(x) = 1 x is continuous on R {0}. It cannot be extended to a continuous function on R since if x n = 1 n, f(x n ). Proposition composition of continuous functions Suppose f : E R is continuous at c, g is defined on the range of f, and g is continuous at f(c). Then g f is continuous at c. Proof Let {x n } be any sequence in E converging to c. By the continuity of f at c, lim n f(x n ) = f(c). By the continuity of g at f(c), lim g(f(x n)) = g(f(c)). n This shows that g f is continuous at c. Example A polynomial of degree n is a function of the form P n (x) = a n x n + a n 1 x n a 0. Polynomials are continuous, by repeated application of Proposition 1.5.2(1) and (2). 2. A rational function is a function of the form: P (x) Q(x) where P and Q are polynomials. The rational function P Q is continuous at x 0 provided that Q(x 0 ) 0, by 1.5.2(3). Example The exponential function exp(x) = e x is continuous. This we will prove later in the course. 2. Given that exp is continuous, the function g defined by g(x) = exp(x 2n+1 + x) is continuous (use continuity of exp, 1.5.5(1) and 1.5.4). Example The function x sin x is continuous. This will be proved later using the theory of power series. An alternative proof is given in the next section. From the continuity of sin x we can deduce that cos x, tan x and cot x are continuous: 24

26 Example The function x cos x is continuous by 1.5.4, since cos x = sin(x + π 2 ) and is thus the composite of the continuous function cos and the continuous function x x + π/2 Example The function x tan x is continuous. Use tan x = sin x cos x and 1.5.2(3). Discussion on tan x. If we restrict the domain of tan x to the region ( π 2, π 2 ), its graph is a continuous curve and we believe that tan x is continuous. On a larger range, tan x is made of disjoint pieces of continuous curves. How could it be a continuous function? Surely the graph looks discontinuous at π 2!!! The trick is that the domain of the function does not contain these points where the graph looks broken. By the definition of continuity we only consider x with values in the domain. The largest domain for tan x is R { π 2 + kπ, k Z} = k Z(kπ π 2, kπ + π 2 ). For each c in the domain, we locate the piece of continuous curve where it belongs. We can find a small neighbourhood on which the graph is is part of this single piece of continuous curve. For example if c ( π 2, π 2 ), make sure that δ is smaller than min( π 2 c, c + π 2 ). 1.6 Continuity of Trigonometric Functions Measurement of angles Two different units are commonly used for measuring angles: Babylonian degrees and radians. We will use radians. The radian measure of an angle x is the length of the arc of a unit circle subtended by the angle x. x 1 x 25

27 The following explains how to measure the length of an arc. Take a polygon of n segments of equal length inscribed in the unit circle. Let l n be its length. The length increases with n. As n, it has a limit. The limit is called the circumference of the unit circle. The circumference of the unit circle was measured by Archimedes, Euler, Liu Hui etc.. It can now be shown to be an irrational number. Historically sin x and cos x are defined in the following way. Later we define them by power series. The two definitions agree. Take a right-angled triangle, with the hypotenuse of length 1 and an angle x. Define sin x to be the length of the side facing the angle and cos x the length of the side adjacent to it. Extend this definition to [0, 2π]. For example, if x [ π 2, π], define sin x = cos(x π 2 ). Extend to the rest of R by decreeing that sin(x + 2π) = sin x, cos(x + 2π) = cos x. Lemma If 0 < x < π 2, then sin x < x < tan x. Proof Take a unit disk centred at 0, and consider a sector of the disk of angle x which we denote by OBA. y B O F A E x x 2π The area of the sector OBA is: times the area of the disk which is π. hence Area(Sector OBA) = 1 2 x. Let a = OF and b = F A then a + b = 1. The area of the triangle OBA is the sum of the areas of OBF and BF A: Area(triangle OBA) = 1 2 a sin(x) b sin x = 1 2 sin(x). 26

28 Since the triangle is strictly smaller than the sector, sin x < x. Consider the right-angled triangle OBE, with one side tangent to the circle at B. Because BE = tan x, the are of the triangle is, area(obe) = 1 tan x. 2 This triangle is strictly bigger than the sector OBA. So and therefore x < tan x. Area(Sector OBA) < Area(Triangle OBE), A slightly different formulation for f is continuous at c: for ɛ > 0, there is δ > 0 such that if h < δ, f(c + h) f(c) < ɛ. To see this for any x write x = x c + c and let h = x c (the difference). Theorem The function x sin x is continuous. Proof Let x R, We show that f is continuous at x. Let h R be such that h < π 2. We use the relation cos(h) = 1 2 ( sin( h 2 )) 2. sin(x + h) sin x = sin x cos h + cos x sin h sin x = sin x(cos h 1) + cos x sin h = 2 sin x sin 2 ( h ) cos x sin h 2 2 sin x sin 2 ( h ) + cos x sin h 2 h2 2 + h = h ( h 2 For any ε > 0, choose δ = min(2ε, π 2 ). If h < δ then sin(x + h) sin x 2 h < ε. + 1) 2 h. Exercises Show that the following functions are continuous at all points of their domains. 27

29 (a) f : R {x R : cos x = 0} R, f(x) = tan x. (b) f : R {x R : sin x = 0} R, f(x) = cot x. (c) f : R R, f(x) = sinh x (you may assume here that exp is continuous). (d) f : R R, f(x) = cos(x 2 ). (e) f : R R, f(x) = cos(sin x) 2. Show that the function f : [0, ) R, f(x) = x is continuous. Unlike in the previous exercise, here you may have to use the definition of continuity directly. Drawing a picture may help you to guess a suitable value δ for each ε > Can you see how to prove that f : (0, ) R, f(x) = ln x is continuous? What about the function f : [ 1, 1] R defined by f(x) = arcsin x? 28

30 Chapter 2 Continuous Functions on closed Intervals: I Definition A number c is an upper bound of a set A if for all x S we have x c. A number c is the least upper bound of a set A if c is an upper bound for A. if U is an upper bound for A then c U. The least upper bound of the set A is denoted by sup A. The completeness Axiom If S R is a non empty set bounded above it has a least upper bound in R. In particular there is a sequence x n S such that c 1 n x n < c. This sequence x n converges to c. 2.1 The Intermediate Value Theorem Lectures 5 & 6 Recall Theorem 1.3.1, which we state again: Theorem Intermediate Value Theorem (IVT) Let f : [a, b] R be continuous. Suppose that f(a) f(b). Then for any v strictly between f(a) and f(b), there exists c (a, b) such that f(c) = v. A rigorous proof was first given by Bolzano in Proof We may assume f(a) < f(b) (in case f(b) < f(a) define g = f.) Consider the set A = {x [a, b] : f(x) v}. Note that a A and A is 29

31 bounded above by b, so it has a least upper bound, which we denote by c. We will show that f(c) = v. y f v a c b x Since c = sup A, there exists x n A with x n c. Then f(x n ) f(c) by continuity of f. Since f(x n ) v then f(c) v. Suppose f(c) < v. Then by the Non-vanishing Lemma applied to the function x v f(x), there exists r > 0 such that for all x B r (c), f(x) < v. But then c + r/2 A. This contradicts the fact that c is an upper bound for A. The idea of the proof is to identify the greatest number in (a, b) such that f(c) = v. The existence of the least upper bound (i.e. the completeness axiom) is crucial here, as it is on practically every occasion on which one wants to prove that there exists a point with a certain property, without knowing at the start where this point is. Exercise To test your understanding of this proof, write out the (very similar) proof for the case f(a) > v > f(b). Example Let f : [0, 1] R be given by f(x) = x 7 +6x+1, x [0, 1]. Does there exist x [0, 1] such that f(x) = 2? Proof The answer is yes. Since f(0) = 1, f(1) = 8 and 2 (1, 8), there is a number c [0, 1] such that f(c) = 2 by the IVT. 30

32 Remark*: There is an alternative proof for the IVT, using the bisection method. It is not covered in lectures. Suppose that f(a) < 0, f(b) > 0. We construct nested intervals [a n, b n] with length decreasing to zero. We then show that a n and b n have common limit c which satisfy f(c) = 0. Divide the interval [a, b] into two equal halves: [a, c 1], [c 1, b]. If f(c 1) = 0, done. Otherwise on (at least) one of the two sub-intervals, the value of f must change from negative to positive. Call this subinterval [a 1, b 1]. More precisely, if f(c 1) > 0 write a = a 1, c 1 = b 1; if f(c 1) < 0, let a 1 = c 1, b 1 = b. Iterate this process. Either at the k th stage f(c k ) = 0, or we obtain a sequence of intervals [a k, b k ] with [a k+1, b k+1 ] [a k, b k ], a k increasing, b k decreasing, and f(a k ) < 0, f(b k ) > 0 for all k. Because the sequences a k and b k are monotone and bounded, both converge, say to c and c respectively. We must have c c, and f(c ) 0 f(c ). If we can show that c = c then since f(c ) 0, f(c ) 0 then we must have f(c ) = 0, and we have won. It therefore rermains only to show that c = c. I leave this as an (easy!) exercise. Example The polynomial p(x) = 3x 5 + 5x + 7 = 0 has a real root. Proof Let f(x) = 3x 5 + 5x + 7. Consider f as a function on [ 1, 0]. Then f is continuous and f( 1) = = 1 < 0 and f(0) = 7 > 0. By the IVT there exists c ( 1, 0) such that f(c) = 0. Exercise Show that every polynomial of odd degree has a real root. Discussion on assumptions. In the following examples the statement fails. For each one, which condition required in the IVT is not satisfied? Example Let f : [ 1, 1] R, { x + 1, x > 0 f(x) = x, x < 0 Then f( 1) = 1 < f(1) = 2. Can we solve f(x) = 1/2 ( 1, 2)? No: that the function is not continuous on [ 1, 1]. 2. Define f : Q [0, 2] R by f(x) = x 2. Then f(0) = 0 and f(2) = 4. Is it true that for each v with 0 < v < 4, there exists c Q [0, 2] such that f(c) = v? No! If, for example, v2, then there is no rational number c such that f(c) = v. Here, the domain of f is Q [0, 2], not an interval in the reals, as required by the IVT. 31

33 Example The Intermediate Value Theorem may hold for some functions which are not continuous. For example it holds for the following function: { sin(1/x), x 0 f(x) = 0 x = 0, 0 x 1. Imagine the graphs of two continuous functions f and g. If they cross each other where x = c then f(c) = g(c). If the graph f is higher at a and the graph of g is higher at b then they must cross, as is shown below. Example Let f, g : [a, b] R be continuous functions. Suppose that f(a) < g(a) and f(b) > g(b). Then there exists c (a, b) such that f(c) = g(c). Proof Define h = f g. Then h(a) < 0 and h(b) > 0 and h is continuous. Apply the IVT to h: there exists c (a, b) with h(c) = 0, which means f(c) = g(c). Let f : E R be a function. Any point x E such that f(x) = x is called a fixed point of f. Theorem (Fixed Point Theorem) Suppose g : [a, b] [a, b] is a continuous function. Then there exists c [a, b] such that g(c) = c. Proof The notation that g : [a, b] [a, b] implies that the range of f is contained in [a, b]. Set f(x) = g(x) x. Then f(a) = g(a) a a a = 0, f(b) = g(b) b b b = 0. If f(a) = 0 then a is the sought after point. If f(b) = 0 then b is the sought after point. If f(a) > 0 and f(b) < 0, apply the intermediate value theorem to f to see that there is a point c (a, b) such that f(c) = 0. This means g(c) = c. Remark This theorem has a remarkable generalisation to higher dimensions, known as Brouwer s Fixed Point Theorem. Its statement requires the notion of continuity for functions whose domain and range are of higher dimension - in this case a function from a product of intervals [a, b] n to itself. Note that [a, b] 2 is a square, and [a, b] 3 is a cube. I invite you to adapt the definition of continuity we have given, to this higher-dimensional case. 32

34 Theorem (Brouwer, 1912) Let f : [a, b] n [a, b] n be a continuous function. Then f has a fixed point. The proof of Brouwer s theorem when n > 1 is rather harder than when n = 1. It uses the techniques of Algebraic Topology. Exercises Suppose that a sequence (x n ) is defined by a starting with some initial value x 1, choosing a function f, and for n 2 taking x n = f(x n 1 ) (Question 2 contains some examples). Show that if lim n x n = l, and if f is continuous at l, then f(l) = l. 2. Find the limit of the sequences defined by (a) x 1 = 1, x n+1 = 1 1+x n for n 0. (b) x 1 = 2, x n+1 = 1 2+x n for n A sequence of rectangles (R n ) is defined as follows: beginning with some rectangle R 1, at each stage we construct R n+1 by by adding to R n a square drawn on its longer side. H G A D F B C E R1=ABCD R2=ABEF R3=HBEG To do: (a) Suppose that the sequence has a limiting shape (i.e. sequence length of long side x n := length of short side that the tends to a limit). Show that this limit has only one possible value, and find it. (b) Show that the sequence does have a limiting shape. 33

35 Chapter 3 Continuous Limits 3.1 Continuous Limits Lecture 7 We wish to give a precise meaning to the statement f approaches l as x approaches c which is denoted by If f(x) = lim f(x) = l. x c { 2x + 1 x 1 0, x = 1 Could we say something about f(x) as x approaches 1? Let us agree that there is a limit. Should the limit be 3 or 0? It should be 3, and the value f(1) = 0 is not relevant. We do not care the value of f at c, and indeed for that matter f is not required to be defined at c. Definition Let c (a, b). Let l be a real number. Let f : E R. (1) Suppose that E (a, c) (c, b). We say that f tends to l as x approaches c, and write lim f(x) = l, x c if for any ε > 0 there exists δ > 0, such that for all x E satisfying we have 0 < x c < δ (3.1.1) f(x) l < ε. In lim x c f(x) = l for some l we say that f has a( finite) limit at c. 34

36 (2) Suppose that E (a, c). We say that f has left limit at c, if for any ε > 0 there exists δ > 0, such that for all x E satisfying c δ < x < c, we have f(x) l < ε. This is denoted by lim f(x) = l. x c (3) Suppose that E (c, b). We say that f has right limit at c, if for any ε > 0 there exists δ > 0, such that for all x E satisfying c < x < c+δ, we have f(x) l < ε. This is denoted by We observe that lim f(x) = l. x c+ Proposition Let f : (a, b) R and c (a, b). It is clear that lim x c f(x) = l is equivalent to both lim x c+ f(x) = l and lim x c f(x) = l. The condition that E contains an interval around c can be replaced by c is an accumulation point of E, see the Appendix, Chapter 3.2. This assumption will ensure that if there is a limit, the limit is unique. Proposition If f : R has a left limit (or has a right limit) at c, this limit is unique. Proof We prove only the case when lim x c f(x) exists. Suppose f has two limits l 1 and l 2 with l 1 l 2. Take ε = 1 4 l 1 l 2 > 0. By definition, there exists δ > 0 such that if 0 < x c < δ, f(x) l 1 < ε, f(x) l 2 < ε. (such x does exists from the assumption that either (a, c) (c, b) E!!) By the triangle inequality, l 1 l 2 f(x) l 1 + f(x) l 2 < 2ε = 1 2 l 1 l 2. This is a contradiction. Example The statement that lim x 0 f(x) = 0 is equivalent to the statement that lim x 0 f(x) = 0. Define g(x) = f(x), then g(x) 0 = f(x). 35

37 3.1.1 Limit and continuity Theorem Let c (a, b). Let f : (a, b) R. The following are equivalent: 1. f is continuous at c. 2. lim x c f(x) = f(c). Proof Just compare the ɛ δ definitions for both the statements. Example Let f(x) = x2 +3x+2 sin(πx)+2. We observe that sin(πx) + 2 is never zero and f is continuous everywhere, especially at x = 1. Then Example Does x 2 + 3x + 2 lim x 1 sin(πx) + 2 = f(1) = 3 lim x x 1 x exist? Find its value if it does exist. Let 1 + x 1 x f(x) =. x The natural domain of f contains {x : 0 < x < 1/2}. We first simplify the algebraic expression of the function, 1 + x 1 x 2x lim = lim x 0 x x 0 x( 1 + x + 1 x) = lim 2. x x + 1 x We have used x 0 in the last step. Let g(x) = 2 1+x+ 1 x. Then g is defined everywhere and is continuous. Furthermore f(x) = g(x) when x 0. Hence lim f(x) = lim g(x) = g(0) = 1. x 0 x 0 To see g is continuous: First x is continuous at x = 1, (prove it by ε δ argument!). The functions x 1 + x and x 1 x are compositions of continuous functions: they are both continuous at x = 0; Their sum is non-zero when x = 0, it follows, by the algebra of continuous functions, that the function g is is also continuous at x = 0. x 36

38 Definition A function f : (a, c] R is said to be left continuous at c if for any ε > 0 there exists a δ > 0 such that if x (c δ, c] (a, c], we have f(x) f(c) < ε. 2. A function f : [c, b) R is said to be right continuous at c if for any ε > 0 there exists a δ > 0 such that if x [c, c + δ) [c, b) f(x) f(c) < ε. Proposition ) The statement that f is right continuous at c is equivalent to the statement that lim x c+ f(x) = f(c). 2) The statement that f is left continuous at c is equivalent to the statement that lim x c f(x) = f(c). Theorem Let f : (a, b) R and for c (a, b). The following are equivalent: (a) f is continuous at c (b) f is right continuous at c and f is left continuous at c. (c) Both lim x c+ f(x) and lim x c f(x) exist and are equal to f(c). (d) lim x c f(x) = f(c) Example Let y { x f 1 (x) = 2, 0 x < π 2 π cos(x), 2 x π. f 1 π 2 π x 37

39 The function is continuous everywhere except to c = π 2. Furthermore it has left and right limit everywhere, this we only need to check to c = π 2 : lim f 1(x) = lim x π 2 x π 2 x2 = ( π 2 )2 lim f 1(x) = lim cos(x) = cos(π x π 2 + x π ) = 0. Since lim x π 2 + f 1 (x) = f 1 ( π 2 ), f 1 is right continuous at π 2. It is not continuous, nor does it have a limit at π 2. Example We change the value of f 1 at c = π 2 functions: { x f 2 (x) = 2, 0 x π 2 π cos(x), 2 < x π. y y to form two new f 2 f 3 π x π x x 2, 0 x < π 2 π f 3 (x) = cos(x), 2 < x π 1, x = π 2. Then both functions have left and right limits everywhere, and are continuous everywhere except at c. Here, lim f 2(x) = x π 2 lim f 2(x) = x π 2 + lim f 3(x) = x π 2 lim f 3(x) = x π 2 + lim x π 2 f 1(x) = ( π 2 )2 lim f 1(x) = x π 2 + lim cos(x) = cos(π x π ) = 0. Neither f 2 nor f 3 has limit at c = π 2. Furthermore f 2 is left continuous at c, not right continuous at c. The function f 3 is neither left continuous nor right continuous at c. 38

40 Example Denote by [x] the integer part of x. Let f(x) = [x]. Show that for k Z, lim x k+ f(x) and lim x k f(x) exist. Show that lim x k f(x) does not exist. Show that f is discontinuous at all points k Z. Proof Let c = k, we consider f near k, say on the open interval (k 1, k+1). Note that { k, if k x < k + 1 f(x) = k 1, if k 1 x < k. It follows that lim f(x) = lim k = k x k+ x 1+ lim x k f(x) = lim x 1 (k 1) = k 1. Since the left limit does not agree with the right limit, lim x k f(x) does not exist. By the limit formulation of continuity, ( A function continuous at k must have a limit as x approaches k ), f is not continuous at k. Example For which number a is f defined below continuous at x = 0? { a, x 0 f(x) = 2, x = 0. Since lim x 0 f(x) = 2, f is continuous if and only if a = Negation The statement it is not true that lim x c f(x) = l means precisely that there exists a number ε > 0 such that for all δ > 0 there exists x (a, b) with 0 < x c < δ and f(x) l ε. That lim x c f(x) does not exist can occur in two ways: 1. the limit does not exist, or 2. the limit exists, but differs from l. In this case we write lim f(x) l. x c 1 Example Show that lim x 0 x does not exist. 1 Let l R. Let ɛ = 1. For any δ, take x with x 0 < min( l +3, δ), then 1 x > l + 3 and 1 x l 1 x l = ( l + 3) l = 3 > ɛ. 39

41 3.1.3 Continuous limit and sequential limit Theorem Let c (a, b) and f : (a, b) {c} R. The following are equivalent. 1. lim x c f(x) = l 2. For every sequence x n (a, b) {c} with lim n x n = c we have lim f(x n) = l. n Proof Step 1. Suppose that lim x c f(x) = l. Let ɛ > 0. If 0 < x c < δ, f(x) l < ɛ. Let x n (a, b) {c} with lim n x n = c. There exists N > 0 such that if n > N, x n c < δ and hence f(x n ) l < ɛ. This proves that lim n f(x n ) = l. Step 2. Suppose that lim x c f(x) = l does not hold. There is a number ε > 0 such that for all δ > 0, there is a number x (a, b) {c} with 0 < x c < δ, but f(x) l ε. Taking δ = 1 n we obtain a sequence x n (a, b) {c} with x n c < 1 n with f(x n ) l ε for all n. Thus the statement lim n f(x n ) = l cannot hold. But lim n x n = c and statement 2 fails. Example Let f : R {0} R, f(x) = sin( 1 x ). Then does not exist. lim x 0 sin( 1 x ) Proof Take two sequences of points, x n = z n = π nπ 1 π 2 + 2nπ. Both sequences tend to 0 as n. But f(x n ) = 1 and f(z n ) = 1 so lim n f(x n ) lim n f(z n ). By the sequential formulation for limits, Theorem , lim x 0 f(x) cannot exist. 40

42 In[22]:= ê xd, 8x, 0, 1.5<D Out[22]= In[21]:= ê xd, 8x, 0, 0.01<D Out[21]= In[23]:= ê H1 + xl, 8x, -5, 5<D 3 2 Out[23]=

43 Example Let f : R {0} R, f(x) = 1 number l R s.t. lim x 0 f(x) = l. x. Then there is no Proof Suppose that there exists l such that lim x 0 f(x) = l. Take x n = 1 n. Then f(x n ) = n and lim n f(x n ) does not converges to any finite number! This contradicts that lim n f(x n ) = l. Example Denote by [x] the integer part of x. Let f(x) = [x]. Then lim x 1 f(x) does not exist. Proof We only need to consider the function f near 1. Let us consider f on (0, 2). Then { 1, if 1 x < 2 f(x) = 0, if 0 x < 1. Let us take a sequence x n = n converging to 1 from the right and a sequence y n = 1 1 n converging to 1 from the left. Then lim x n = 1, n lim y n = 1. n Since f(x n ) = 1 and f(y n ) = 0 for all n, the two sequences f(x n ) and f(y n ) have different limits and hence lim x 1 f(x) does not exist The Sandwich Theorem The following follows from the sandwich theorem for sequential limits. Proposition (Sandwich Theorem/Squeezing Theorem) Let c (a, b) and f, g, h : (a, b) {c} R. If h(x) f(x) g(x) on (a, b) {c}, and lim h(x) = lim g(x) = l x c x c then lim f(x) = l. x c Proof Let x n (a, b) {c} be a sequence converging to c. Then by Theorem , lim n g(x n ) = lim n h(x n ) = l. Since h(x n ) f(x n ) g(x n ), lim f(x n) = l. n By Theorem again, we see that lim x c f(x) = l. 42

44 Example Let is continuous at 0. f(x) = { x sin(1/x) x 0 0, x = 0 Proof Note that 0 x sin( 1 x ) x. Now lim x 0 x = 0 by the the continuity of the function f(x) = x. By the Sandwich theorem the conclusion lim x 0 x sin( 1 x ) = 0. Since f is continuous at 0. lim f(x) = lim x sin( 1 x 0 x 0 x ) = 0 = f(0), The following example will be revisited, when we learn L Hôpital s rule (Example ). Example ( An Important Limit ) sin x lim x 0 x = 1. Proof Recall if 0 < x < π 2, then sin x x tan x and sin x tan x sin x x 1. cos x sin x x 1 (3.1.2) The relation (3.1.2) also holds if π 2 < x < 0: Letting y = x then 0 < y < π 2. All three terms in (3.1.2) are even functions: sin x x = sin y y and cos y = cos( x). Since lim cos x = 1 x 0 by the Sandwich Theorem and (3.1.2), lim x 0 sin x x = 1. 43

45 3.1.5 The Algebra of Limits and limit of composition of functions Proposition (Algebra of limits) Let c (a, b) and f, g : (a, b) {c} R. Suppose that Then lim f(x) = l 1, x c lim g(x) = l 2. x c If l 2 0, lim(f + g)(x) = lim f(x) + lim g(x) = l 1 + l 2. x c x c x c lim(fg)(x) = lim f(x) lim g(x) = l 1l 2. x c x xc x c f(x) lim x c g(x) = lim x c f(x) lim x c g(x) = l 1. l 2 This can be proved by the corresponding theorems on sequential limits, or by modifying the functions f, g so that the new functions are continuous and use the corresponding theorem for continuity. Proposition Limit of the composition of two functions Let c (a, b), f : (a, b) {c} R. Suppose lim f(x) = l. (3.1.3) x c Let g : (a 1, b 1 ) {l} R. (a 1, b 1 ) {l}. Suppose that Suppose that the range of f is contained in Then lim g(y) = l (3.1.4) y l lim g(f(x)) = l. (3.1.5) x c Proof Given ε > 0, since lim y l g(y) = l, we can choose δ 1 > 0 such that 0 < y l < δ 1 = g(y) l < ε. (3.1.6) By (3.1.3) we can choose δ 2 > 0 such that 0 < x c < δ 2 = f(x) l < δ 1. (3.1.7) 44

46 Since f(x) l for all x in the domain of f, it follows from (3.1.7) that if 0 < x c < δ 2 then 0 < f(x) l < δ 1 ; it therefore follows, by (3.1.6), that g(f(x)) l < ε. Example (Important limit to remember, c.f. Example ) Proof 1 cos x lim = 0. x 0 x 1 cos x lim x 0 x 2 sin 2 ( x 2 = lim ) x 0 x = lim sin( x x x 0 2 )sin( 2 ) x = lim x 0 sin( x 2 ) lim x 0 = 0 1 = 0. 2 sin( x 2 ) x 2 The following lemma should be compared to Lemma The existence of the lim x c f(x) and its value depend only on the behaviour of f near to c. It is in this sense we say that limit is a local property. Recall that B r (c) = (c r, c + r); let B r (c) = (c r, c + r) {c}. If a property holds on B r (c) for some r > 0, we often say that the property holds close to c. Lemma Let g : (a, b) {c} R, and c (a, b). Suppose that lim g(x) = l. x c 1. If l > 0, then g(x) > 0 on B r (c) for some r > If l < 0, then g(x) < 0 on B r (c) for some r > 0. In both cases g(x) 0 for x B r (c). Proof Define g(x) = { g(x), x c l, x = c. Then g is continuous at c and so by Lemma 1.4.5, g is not zero on (c r, c+r) for some r. The conclusion for g follows. 45

47 3.2 Accumulation points and Limits* This section is not delivered in the lectures. Recall that a point x 0 is called an accumulation point (or limit point) of a set E if for any δ > 0, there is a point x A such that 0 < x x 0 < δ. Example If E = (1, 2) [3, 4) its set of accumulation point is [1, 2] [3, 4]. {1, 2,..., } has no accumulation points. Let E = {a 1, a 2,... }. If a is the limit of a sub-sequence of {a n}, then a is an accumulation point of E. Every real number is an accumulation point of Q. Every real number is an accumulation point of R Q. Definition Let c be an accumulation point of E. A function f : E R has a limit l as x approaches c, where l R, if for every ε > 0 there exists δ > 0 such that for all x E satisfying 0 < x c < δ, f(x) l < ε. In this case we write lim f(x) = l. x c Example Why do we need c to be an accumulation point of E? Take E = { 1 }, n define f( 1 ) = n. Let c = 1, not an accumulation point. Then any number l could be n 2 the limit of f(x) as x c = 1. In fact if we take δ = 1, then no element of E satisfies < x 1 < δ. Hence if 0 < x 1 < δ = 1 and x E, f(x) l < ε for any number l! The concept of limit in this case is absurd. Proposition If lim x c f(x) exists, then the limit must be unique. Proof If f has two limits u and v, take ε = 1 u v. Then there is a δ > 0 such that if 4 0 < x c < δ, u v f(x) u + f(x) v < ε. Such x exists as c is an accumulation point of A, giving a contradiction. Proposition Let f : E R be a function and let c be an accumulation point of E. The following statements are equivalent: Proof 1. f has a limit at c. 2. For any ε > 0 there is a δ > 0, such that for all x, y E satisfying 0 < x c < δ and 0 < y c < δ we have f(x) f(y) < ε. 1. Suppose that Statement 1 holds. Let l = lim x c f(x). Then for any ε > 0 there is a δ > 0 such that if x, y E satisfy 0 < x c < δ and 0 < y c < δ, then By the triangle inequality, f(x) l < ε 2, f(y) l < ε 2. f(x) f(y) f(x) l + f(x) l < ε. 46

48 2. We assume that Statement 2 holds and prove that f has a limit as x tends to c. Let (x n) be any sequence in E {c} tending to c. I claim that from Statement 2 it follows that (f(x n)) is a Cauchy sequence. For, given ε > 0, we first choose δ > 0 such that if 0 < x c < δ and 0 < y c < δ then f(x) f(y) < ε, and then choose N such that if n N then x n c < δ. It follows that if m, n N then f(x m) f(x n) < ε. Now, because {f(x n)} is a Cauchy sequence, it converges, say to l R. In fact, this l must be the same for any two sequences (x n) and (y n) in E {c} both tending to c. Suppose f(x n) l 1 and f(y n) l 2. From the two sequences we can form a third, (z n), by alternating x n and y n: x 1, y 1, x 2, y 2, x 3, y 3,.... (3.2.1) This sequence too tends to c, so its image under f converges. Since f(x n) l 1 and f(y n) l 2, and (f(x n)) and (f(y n)) are subsequences of the sequence f(z n), we must have l 1 = l 2. We have shown that there is an l R such that for every sequence (x n) in E {c} converging to c, we have f(x n) l. By , this means that lim x c f(x) = l. Exercise Can you find a proof that (2) implies (1) in Proposition 3.2.5, that does not make use of Cauchy sequences? Definition Consider f : E R and c an accumulation point of E. 1. We say lim x c f(x) =, if M > 0, δ > 0, such that f(x) > M whenever x E satisfies 0 < x c < δ. 2. We say lim x c f(x) =, if M > 0, δ > 0, such that f(x) < M whenever x E satisfies 0 < x c < δ. 3.3 Limits to What do we mean by lim x c f(x) =? Since is not a real number, we formulate what it means to approach. Definition Let c (a, b) and f : (a, b) {c} R. (a) We say that lim f(x) =, x c if M > 0, δ > 0, such that for all x (a, b) with 0 < x c < δ we have f(x) > M. 47

49 (b) We say that lim f(x) =, x c if M > 0, δ > 0, such that for all x (a, b) with 0 < x c < δ we have f(x) < M. 2. Let f : (c, b) R. We say that lim f(x) = x c+ (resp. ), if M > 0, δ > 0, s.t. for all x (c, b) (c, c + δ), f(x) > M (resp. f(x) < M)). 3. Let f : (a, c) R. We say that lim f(x) = x c (resp. ), if M > 0, δ > 0, such that for all x (a, c) with c δ < x < c f(x) > M (resp. f(x) < M). Example Show that lim x 0 1 x = Proof Let M > 0. Define δ = 1 M. Then if 0 < x 0 < δ, we have f(x) = 1 x > 1 δ = M. Exercise Show that lim x 0 1 sin x =. Proof Let M > 0. Since lim x 0 sin x = 0, there is a δ > 0 such that if 0 < x < δ, then sin x < 1 M. Since we are interested in the left limit, we may consider x ( π 2, 0). In this case sin x = sin x. So we have sin x < 1 M which is the same as sin x > 1 M. In conclusion lim x 0 1 sin x =. 48

50 3.4 Limits at What do we mean by lim f(x) = l, or lim f(x) =? x x Definition Consider f : (a, ) R. We say that lim f(x) = l, x if for any ε > 0 there is an M such that if x > M we have f(x) l < ε. 2. Consider f : (, b) R. We say that lim f(x) = l, x if for any ε > 0 there is an M > 0 such that for all x < M we have f(x) l < ε. Definition Consider f : (a, ) R. We say lim f(x) =, x if M > 0, a number X > 0, such that f(x) > M, for all x > X. lim x f(x) = can also be defined. Example Show that lim x + (x 2 + 1) = +. Proof For any M > 0, we look for x with the property that x > M. Take A = M then if x > A, x 2 +1 > M +1. Hence lim x + (x 2 +1) = +. Remark In all cases, 1. There is a unique limit if it exists. 49

51 2. There is a sequential formulation. For example, lim x f(x) = l if and only if lim n f(x n ) = l for all sequences {x n } E with lim n x n =. Here E is the domain of f. 3. Algebra of limits hold. 4. The Sandwich Theorem holds. Exercise Let f(x) = x 1+x. It is defined on R { 1}. 1. Show that the one sided limits at c = 1 are infinite: lim x 1+ x =, lim 1 + x x 1 2. Show that the limit at infinity exists: lim x x = 1, lim 1 + x x x 1 + x =. x 1 + x = Plot y = 1 and x = 1 in dotted lines. Plot on the same graph (0, f(0)), (2, f(2)), (5, f(5)), and (8, f(8)). 4. Plot the graph of y = f(x) on the right hand side of x > Plot the points on the graph of y = f(x) corresponding to the points x = 1.1, 1.01, 2, Now complete the graph of y = f(x) to the left of f = 1. (see the second appended graph). 7. N.B. f(x) = x. Its graph can be obtained from shifting and rotating the graph of y = 1 x. Exercise Show that lim x c f(x) = + implies that lim x c 1 f(x) = 0. Does the converse hold? Give examples to illustrate your point. Exercises Find the following limits: lim x e 1/x2, lim x x 2, lim x x 2. 50

52 In[43]:= ê H1 + xl<, 8x, -3, 2<, PlotRange Ø 8-20, 20<D Out[43]= In[45]:= Plot@8x ê H1 + xl<, 8x, -3, 2<, PlotRange Ø 8-20, 20<D Out[45]= In[60]:= Plot@Sin@xD ê x, 8x, -Pi, Pi<D Out[60]= 0.4

53 In[57]:= 8x, -10, 10<D In[58]:= Plot@Tan@xD, 8x, -3 Pi, 3 Pi<D Out[58]= In[64]:= Plot@ArcCot@xD, 8x, -3 Pi, 2 Pi<D Out[64]=

54 Chapter 4 The Extreme Value Theorem By now we understand quite well what is a continuous function. Let us look at the landscape drawn by a continuous function. There are peaks and valleys. Is there a highest peak or a lowest valley? 4.1 Bounded Functions Consider the range of f : E R: f(e) := {f(x) x E}. Is f(e) bounded from above and from below? If so it has a greatest lower bound m 0 and a least upper bound M 0. And m 0 f(x) M 0, x E. Do m 0 and M 0 belong to f(e)? That they belong to f(e) means that there exist x E and x E such that f(x) = m 0 and f( x) = M 0. Definition We say that f : E R is bounded above if there is a number M such that for all x E, f(x) M. We say that f attains its maximum if there is a number c E such that for all x E, f(x) f(c) That a function f is bounded above means that its range f(e) is bounded above. Definition We say that f : E R is bounded below if there is a number m such that for all x E, f(x) m. We say that f attains its minimum if there is a number c E such that for all x E, f(x) f(c). 53

55 That a function f is bounded below means that its range f(e) is bounded below. Definition A function which is bounded above and below is bounded. 4.2 The Bolzano-Weierstrass Theorem This theorem was introduced and proved in Analysis I. Lemma ( Bolzano-Weierstrass Theorem) A bounded sequence has at least one convergent sub-sequence. 4.3 The Extreme Value Theorem Lecture 9 Theorem ( The Extreme Value theorem) Let f : [a, b] R be a continuous function. Then 1. f is bounded above and below, i.e. there exist numbers m and M such that m f(x) M, x [a, b]. 2. There exist x, x [a, b] such that Remark: f(x) f(x) f(x), a x b. f(x) = inf{f(x) x [a, b]}, f(x) = sup{f(x) x [a, b]}. Proof We first prove that f is bounded above and sup f([a, b]) is attained. 1. Suppose that f is not bounded above. Then for any M > 0 there is a point x [a, b] such that f(x) > M. In particular, for every n there is a point x n [a, b] such that f(x n ) n. In particular lim n f(x n ) =. The sequence (x n ) n N [a, b] is bounded. By Bolzano-Weierstrass Theorem there is a convergent subsequence x nk whose limit is in [a, b] and denoted by x 0. By sequential continuity of f at x 0, lim f(x n k ) = f(x 0 ) k 54

56 However as a sub-sequence of a convergent subsequence, lim f(x n k ) =. k This gives a contradiction. We conclude that f must be bounded from above. 2. Next we show that f attains its maximum value. Let A = {f(x) x [a, b]}. Since A is not empty and bounded above, it has a least upper bound, M 0. Then there is a sequence x n [a, b] with lim f(x n) = M 0. n (By the definition of the least upper bound, M 0 1 n is not an upper bound of A. There is hence f(x n ) A such that M 0 1 n f(x) M 0. By the Sandwich theorem, lim f(x n) = M 0. ) n Since a x n b, it has a convergent subsequence x nk, whose limit will be denoted by x [a, b]. By the sequential continuity of f at x 0, f( x) = lim k f(x n k ) = M 0. That is, f attains its maximum at x. 3. Let g(x) = f(x). By the above argument the continuous function g is bounded above and there is x [a, b] such that But g(x) = f(x) and g(x) = sup{g(x) : a x b}. sup{g(x) : a x b} = sup{ f(x) : a x b} = inf{f(x) : a x b}. This means that f(x) = inf{f(x) x [a, b]}. 55

57 Remark By the extreme value theorem, if f : [a, b] R is a continuous function then the range of f is the closed interval [f(x), f( x)]. By the Intermediate Value Theorem (IVT), f : [a, b] [f(x), f( x)] is surjective. Discussion on the assumptions: Example Let f : (0, 1] R, f(x) = 1 x. Note that f is not bounded. The condition that the domain of f be a closed interval is violated. Let g : [1, 2) R, g(x) = x. It is bounded above and below. But the value sup{g(x) x [1, 2)} = 2 is not attained on [1, 2). violated. Again, the condition closed interval is Example Let f : [0, π] Q, f(x) = x. Let A = {f(x) x [0, π] Q}. Then sup A = π. But π is not attained on [0, π] Q. The condition which is violated here is that the domain of f must be an interval. Example Let f : R R, f(x) = x. Then f is not bounded. The condition bounded interval is violated. Example Let f : [ 1, 1] R, { 1 f(x) = x, x 0 2, x = 0 Then f is not bounded. The condition f is continuous is violated. 56

58 Chapter 5 The Inverse Function Theorem for continuous functions In this chapter we answer the following question: If f has an inverse and f is continuous, is f 1 continuous? 5.1 The Inverse of a Function Definition Let E and B be subsets of R. 1. We say f : E B is injective f(x) f(y) whenever x y. Here x, y E. 2. We say f : E B is surjective, if for any y B there is a point x E such that f(x) = y. 3. We say f is bijective if it is surjective and injective. A bijection f : E B has an inverse f 1 : B E: Definition If f : E B is a bijection, we define f 1 : B E by N.B. f 1 (y) = x if f(x) = y. 1) If f : E B is a bijection, then so is its inverse function f 1 : B E. 57

59 2) If f : E R is injective it is a bijection from its domain E to its range B. So it has an inverse: f 1 : B E. 3) If f 1 is the inverse of f, then f is the inverse of f Monotone Functions The simplest injective function is an increasing function, or a decreasing function. They are called monotone functions. Definition Consider the function f : E R. 1. We say f is increasing, if f(x) < f(y) whenever x < y and x, y E. 2. We say f is decreasing, if f(x) > f(y) whenever x < y and x, y E. 3. It is monotone or strictly monotone if it is either increasing or decreasing. Compare this with the following definition: Definition Consider the function f : E R. 1. We say f is non-decreasing if for any pair of points x, y with x < y, we have f(x) f(y). 2. We say f is non-increasing if for any pair of points x, y with x < y, we have f(x) f(y). Some authors use the term strictly increasing for increasing, and use the term increasing where we use non-decreasing. We will always use increasing and decreasing in the strict sense defined in If f : [a, b] Range(f) is an increasing function, its inverse f 1 is also increasing. If f is decreasing, f 1 is also decreasing (Prove this!). 5.3 Continuous Injective and Monotone Surjective Functions Lecture 10 Increasing functions and decreasing functions are injective. Are there any other injective functions? 58

60 Yes: the function indicated in Graph A below is injective but not montone. Note that f is not continuous. Surprisingly, if f is continuous and injective then it must be monotone. y y Graph A x Graph B x If f : [a, b] is increasing, is f : [a, b] [f(a), f(b)] surjective? Is it necessarily continuous? The answer to both questions is No, see Graph B. Again, continuity plays a role here. We show below that for an increasing function, being surjective is equivalent to being continuous! Theorem Let a < b and let f : [a, b] R be a continuous injective function. Then f is either strictly increasing or strictly decreasing. Proof First note that f(a) f(b) by injectivity. Case 1: Assume that f(a) < f(b). f(a) f(b) a x b Step 1: We first show that if a < x < b, then f(a) < f(x) < f(b). (5.3.1) Note that f(x) f(a) and f(x) f(b) by injectivity. If f(a) < f(x) < f(b) fails, then either (i) f(x) < f(a); or (ii) f(b) < f(x). 59

61 f(b) f(a) f(x) (i) Suppose that f(x) < f(a). Then a < x < b, a x b f(x) < f(a) < f(b). Take v = f(a). Apply IVT to f on [x, b]: there exists c (x, b) with f(c) = v = f(a). Since c a, this violates injectivity. (ii) Suppose f(b) < f(x). Then a < x < b, f(a) < f(b) < f(x). Take v = f(b). By the IVT for f on [a, x], there exists c (a, x) with f(c) = v = f(b). This again violates injectivity. f(x) f(a) a x b f(b) We have therefore proved (5.3.1). We next show that f is strictly increasing on [a, b]. Take y < x from [a, b]. Since a < x b, by (5.3.1), f(a) < f(x) f(b). Apply (5.3.1) to a < y < x we see that f(y) < f(x). This completes the proof that if f(a) < f(b), f is strictly increasing. Case 2: If f(a) > f(b) we show f is decreasing. Let g = f. Then g(a) < g(b). By Case 1 g is increasing and so f is decreasing. 60

62 Theorem If f : [a, b] [f(a), f(b)] is strictly increasing and surjective, it is continuous. 2. If f : [a, b] [f(b), f(a)] is strictly decreasing and surjective, it is continuous. Proof We prove the increasing case. Fix c (a, b). Take ε > 0. We wish to find the set of x such that f(x) f(c) < ε, or Let ε = min{ε, f(b) f(c), f(c) f(a)}. Then f(c) ε < f(x) < f(c) + ε. (5.3.2) f(a) < f(c) ε < f(c), f(c) < f(c) + ε < f(b). Since f : [a, b] [f(a), f(b)] is surjective we may define Since f is increasing, As ε ε, we conclude a 1 = f 1 (f(c) ε ) b 1 = f 1 (f(c) + ε ). a 1 < x < b 1 = f(c) ε < f(x) < f(c) + ε. a 1 < x < b 1 = f(c) ε < f(x) < f(c) + ε. We have proved that f is continuous at c (a, b). The continuity of f at a and b can be proved similarly. 5.4 The Inverse Function Theorem (Version 1) Lecture 10 If f : [a, b] R is strictly increasing and continuous, then f : [a, b] [f(a), f(b)] is surjective by the IVT. It has inverse f 1 : [f(a), f(b)] [a, b]. 61

63 Theorem [The Inverse Function Theorem: continuous version] If f : [a, b] R is continuous and injective, its inverse f 1 : range(f) [a, b] is continuous. Proof Since f : [a, b] R is continuous and injective, f is either strictly increasing or strictly decreasing by Theorem We consider the case when f is strictly increasing. Its inverse f 1 : [f(a), f(b)] R is then increasing and surjective. By Theorem it is continuous. Example For n N, the function f : [0, ) [0, ) defined by f(x) = x 1 n : [0, 1] R is continuous. Proof Let g(x) = x n. Then g : [0, a] [0, a n ] is increasing and continuous, and therefore has continuous inverse. Every point c (0, ) lies in the interior of some interval [0, a 2 ]. Thus f is continuous at c. Example Consider sin : [ π 2, π 2 ] [ 1, 1]. It is increasing and continuous. Define its inverse arcsin : [ 1, 1] [ π 2, π 2 ]. Then arcsin is continuous. 62

64 Chapter 6 Differentiation Speaking intuitively, a function is smooth if its graph is smooth you cannot cut your finger by running it along the graph. The graphs of sin x or of a polynomial are smooth; the graph of f(x) = x is not smooth at the point (0, 0). These curves are not smooth. What about a mathematical definition? The key is the idea of the tangent line. At a smooth point p on a curve, there is a unique straight line that approximates the curve, in the neighbourhood of p, better than any other straight line. We call this the tangent line. At a smooth point p, the tangent line is close to the line joining two nearby point on the graph. We are still speaking intuitively here; let us now make some formal definitions. Let f : R R be a function. We would like to decide when there is a tangent line to the graph at the point (x, f(x)). We select a nearby point by changing the value of the x-coordinate slightly, adding a small number 63

65 h. So the nearby point is ((x + h), f(x + h)). We measure the slope of the line joining the points (x, f(x)) and ((x + h), f(x + h)) on the graph of f: it is f(x + h) f(x) (6.0.1) (x + h) x f(x + h) f(x) = h tangent f(x+h) f(x) h As h gets smaller, the chord tends towards the tangent x x+h x What happens as h gets smaller and smaller? Let us contrast two simple calculations. First, we calculate for f(x) = x 2. We have f(x + h) f(x) = (x + h)2 x 2 = h h As h gets smaller, this tends to 2x: 2xh + h2 h = 2x + h. (6.0.2) f(x + h) f(x) lim = 2x. (6.0.3) h 0 h Second, let f(x) = x and let x = 0. Then f(0 + h) f(0) = h { h (0 + h) 0 h = h if h > 0 h h if h < 0 { 1 if h > 0 = 1 if h < 0. It follows that lim x 0 f(0 + h) f(0) h f(0 + h) f(0) = 1, lim = 1, h 0+ h and, since the limit from the left and the limit from the right do not agree, does not exist. lim h 0 f(0+h) f(0) h 64

66 6.1 Definition of Differentiability Lecture 11 Definition Let f : (a, b) R be a function and let x (a, b). We say f is differentiable at x if f(x + h) f(x) lim h 0 h exists and is a real number (not ± ). If this is the case, the value of the limit is the derivative of f at x, which is usually denoted f (x). Thus, the function f(x) = x 2 is differentiable at every point x, with derivative f (x) = 2x. The function f(x) = x is not differentiable at x = 0. In fact it is differentiable everywhere else: at any point x 0, there is a neighbourhood in which f is equal to the function g(x) = x, or the function h(x) = x. Both of these are differentiable, with g (x) = 1 for all x,and h (x) = 1 for all x. The differentiability of f at x is determined by its behaviour in the neighbourhood of the point x, so if f coincides with a differentiable function near x then it too is differentiable there. We list some of the motivations for studying derivatives: Calculate the angle between two curves where they intersect. (Descartes) Find local minima and maxima (Fermat 1638) Describe velocity and acceleration of movement (Galileo 1638, Newton 1686). Express some physical laws (Kepler, Newton). Determine the shape of the curve given by y = f(x). Example f : R {0} R, f(x) = 1 x is differentiable at x 0 0. Proof. f(x) f(x 0 ) lim = lim x x 0 x x 0 x x 0 1 x 1 x 0 x x 0 x 0 x x = lim 0 x x x 0 x x 0 = lim 1 x x 0 x 0 x = 1 x 2, 0 in the last step we use that 1 x is continuous at x

67 Example sin x : R R is differentiable everywhere and (sin x) = cos x. Proof Let x 0 R. sin(x 0 + h) sin x 0 lim h 0 h sin(x 0 ) cos h + cos(x 0 ) sin h sin x 0 = lim h 0 h sin(x 0 )(cos h 1) + cos(x 0 ) sin h = lim h 0 = ( h cos h 1 sin(x 0 ) lim h 0 h sin h + cos(x 0 ) lim h 0 h = sin(x 0 ) 0 + cos(x 0 ) 1 = cos(x 0 ). ) We used the previously calculated limits: (cos h 1) sin h lim = 0, lim h 0 h h 0 h = 1. The graph of a differentiable function. The graph of a differentiable function should be smooth: no corners, no cusps. The graph below is of f(x) = { x 2 sin(1/x) if x 0 0 if x = 0 (6.1.1) We will refer to this function as x 2 sin(1/x) even though its value at 0 requires a separate line of definition. 66

68 Example The function f(x) = x 2 sin(1/x) is differentiable at 0, with derivative f (0) = 0. Example The function { x sin(1/x), x 0 f(x) = 0, x = 0 is continuous everywhere. But it is not differentiable at x 0 = Proof The function is continuous at x 0 by an easy application of the algebra of continuity. That f(x) is continuous at x = 0 is discussed in Example If x 0 = 0, let g(x) = f(x) f(0) x 0 = sin(1/x), x 0. Note that g is defined on R {0}. We have previously proved that lim x 0 g(x) does not exist. So f is not differentiable at 0. (Take x n = 1 2πn 0, y n = 1 2πn+ π 0. Then g(x n ) = 0, g(y n ) = 1.) 2 This function is differentiable at every point x 0, but we will not be able to show this until we have proved the chain rule for differentiation. Exercise Show that f(x) = is differentiable at x = 0. { x 2 sin(1/x) x 0 0, x = 0 67

69 6.2 The Weierstrass-Carathéodory Formulation for Differentiability Below we give the formulation for the differentiability of f at x 0 given by Carathéodory in Theorem [ Weierstrass-Carathéodory Formulation] Consider f : (a, b) R and x 0 (a, b). The following statements are equivalent: 1. f is differentiable at x 0 2. There is a function φ continuous at x 0 such that Furthermore f (x 0 ) = φ(x 0 ). Proof f(x) = f(x 0 ) + φ(x)(x x 0 ). (6.2.1) 1) 2) Suppose that f is differentiable at x 0. Set { f(x) f(x0 ) φ(x) = x x 0, x x 0 f. (x 0 ), x = x 0 Then f(x) = f(x 0 ) + φ(x)(x x 0 ). Since f is differentiable at x 0, φ is continuous at x 0. 2) 1). Assume that there is a function φ continuous at x 0 such that Then f(x) = f(x 0 ) + φ(x)(x x 0 ). f(x) f(x 0 ) lim = lim φ(x) = φ(x 0 ). x x 0 x x 0 x x 0 The last step follows from the continuity of φ at x 0. Thus f (x 0 ) exists and is equal to φ(x 0 ). Remark: Compare the above formulation with the geometric interpretation of the derivative. The tangent line at x 0 is y = f(x 0 ) + f (x 0 )(x x 0 ). 68

70 If f is differentiable at x 0, f(x) = f(x 0 ) + φ(x)(x x 0 ) = f(x 0 ) + φ(x 0 )(x x 0 ) + [φ(x) φ(x 0 )](x x 0 ) = f(x 0 ) + f (x 0 )(x x 0 ) + [φ(x) φ(x 0 )](x x 0 ). The last step follows from φ(x 0 ) = f (x 0 ). Observe that lim [φ(x) φ(x 0 )] = 0 x x 0 and so [φ(x) φ(x 0 )](x x 0 ) is insignificant compared to the the first two terms. We may conclude that the tangent line y = f(x 0 ) + f (x 0 )(x x 0 ) is indeed a linear approximation of f(x). Lecture 12 Corollary If f is differentiable at x 0 then it is continuous at x 0. Proof If f is differentiable at x 0, f(x) = f(x 0 ) + φ(x)(x x 0 ) where φ is a function continuous at x 0. By algebra of continuity f is continuous at x 0. Example The converse to the Corollary does not hold. Let f(x) = x. It is continuous at 0, but fails to be differentiable at 0. Example Consider f : R R given by { x f(x) = 2, x Q 0, x Q. Claim: f is differentiable only at the point 0. Proof Take x 0 0. Then f is not continuous at x 0, as we learnt earlier, and so not differentiable at x 0. Take x 0 = 0, let g(x) = f(x) f(0) x = { x, x Q 0, x Q. We learnt earlier that g is continuous at 0 and hence has limit g(0) = 0. Thus f (0) exists and equals 0. 69

71 Example f 1 (x) = is differentiable at x = 0. { x 2 sin(1/x) x 0 0, x = 0 Proof Then Let φ(x) = { x sin(1/x) x 0 0, x = 0 f 1 (x) = φ(x)x = f 1 (0) + φ(x)(x 0). Since φ is continuous at x = 0, f 1 is differentiable at x = 0. [We proved that φ is continuous in Example??. ] 6.3 Properties of Differentiation Lecture 12 Rules of differentiation can be easily deduced from properties of continuous limits. They can be proved by the sequential limit method or even directly from the definition in terms of ε and δ, if we have the patience. Theorem (Algebra of Differentiability ) Suppose that x 0 (a, b) and f, g : (a, b) R are differentiable at x Then f + g is differentiable at x 0 and (f + g) (x 0 ) = f (x 0 ) + g (x 0 ); 2. (product rule) fg is differentiable at x 0 and (fg) (x 0 ) = f (x 0 )g(x 0 ) + f(x 0 )g (x 0 ) = (f g + fg )(x 0 ). Proof If f and g are differentiable at x 0 there are two functions φ and ψ continuous at x 0 such that f(x) = f(x 0 ) + φ(x)(x x 0 ) (6.3.1) g(x) = g(x 0 ) + ψ(x)(x x 0 ). (6.3.2) Furthermore f (x 0 ) = φ(x 0 ) g (x 0 ) = ψ(x 0 ). 70

72 1. Adding the two equations (6.3.1) and (6.3.2), we obtain (f + g)(x) = f(x 0 ) + φ(x)(x x 0 ) + g(x 0 ) + ψ(x)(x x 0 ) = (f + g)(x 0 ) + [φ(x) + ψ(x)](x x 0 ). Since φ + ψ is continuous at x 0, f + g is differentiable at x 0. And (f + g) (x 0 ) = (φ + ψ)(x 0 ) = f (x 0 ) + g (x 0 ). 2. Multiplying the two equations (6.3.1) and (6.3.2), we obtain ( )( ) (fg)(x) = f(x 0 ) + φ(x)(x x 0 ) g(x 0 ) + ψ(x)(x x 0 ) Let = (fg)(x 0 ) ( ) + g(x 0 )φ(x) + f(x 0 )ψ(x) + φ(x 0 )ψ(x 0 )(x x 0 ) (x x 0 ). θ(x) = g(x 0 )φ(x) + f(x 0 )ψ(x) + φ(x 0 )ψ(x 0 )(x x 0 ). Since φ, ψ are continuous at x 0, θ is continuous at x 0 by the algebra of continuity. It follows that fg is differentiable at x 0. Furthermore (fg) (x 0 ) = θ(x 0 ) = g(x 0 )φ(x 0 )+f(x 0 )ψ(x 0 ) = g(x 0 )f (x 0 )+f(x 0 )g (x 0 ). In Lemma 1.4.5, we showed that if g is continuous at a point x 0 and g(x 0 ) 0, then there is a neighbourhood of x 0, U r x 0 = (x 0 r, x 0 + r) on which f(x) 0. Here r is a positive number. This means f/g is well defined on U r x 0 and below in the theorem we only need to consider f restricted to U r x 0. Theorem (Quotient Rule) Suppose that x 0 (a, b) and f, g : (a, b) R are differentiable at x 0. Suppose that g(x 0 ) 0, then f g is differentiable at x 0 and ( ) f (x 0 ) = f (x 0 )g(x 0 ) f(x 0 )g (x 0 ) g g 2 = f g g f (x 0 ) g 2 (x 0 ). 71

73 Proof There are two functions φ and ψ continuous at x 0 such that f(x) = f(x 0 ) + φ(x)(x x 0 ) f (x 0 ) = φ(x 0 ) g(x) = g(x 0 ) + ψ(x)(x x 0 ) g (x 0 ) = ψ(x 0 ). Since g is differentiable at x 0, it is continuous at x 0. By Lemma 1.4.5, g(x) 0, x (x 0 r, x 0 + r) for some r > 0 such that (x 0 r, x 0 + r) (a, b). We may divide f by g to see that ( f g )(x) = f(x 0) + φ(x)(x x 0 ) g(x 0 ) + ψ(x)(x x 0 ) Define = f(x 0) g(x 0 ) f(x 0)[g(x 0 ) + ψ(x)(x x 0 )] g(x 0 )[g(x 0 ) + ψ(x)(x x 0 )] + g(x 0)[f(x 0 ) + φ(x)(x x 0 )] g(x 0 )[g(x 0 ) + ψ(x)(x x 0 )] = f(x 0) g(x 0 ) + g(x 0)φ(x) f(x 0 )ψ(x) g(x 0 )[g(x 0 ) + ψ(x)(x x 0 )] (x x 0). θ(x) = g(x 0)φ(x) f(x 0 )ψ(x) g(x 0 )[g(x 0 ) + ψ(x)(x x 0 )]. Since g(x 0 ) 0, θ is continuous at x 0 and (f/g) is differentiable at x 0. And ( f g ) (x 0 ) = θ(x 0 ) = g(x 0)f (x 0 ) + f(x 0 )g (x 0 ) [g(x 0 )] 2. Exercise Prove the sum, product rule and quotient rules for the derivative directly from the definition of the derivative as a limit, without using the Carathéodory formulation. Theorem (chain rule) Let x 0 (a, b). Suppose that f : (a, b) (c, d) is differentiable at x 0 and g : (c, d) R is differentiable at f(x 0 ). Then g f is differentiable at x 0 and (g f) (x 0 ) = g (f(x 0 )) f (x 0 ). 72

74 f g x 0 y 0 = f(x 0 ) g(y 0 ) Proof There is a function φ continuous at x 0 such that f(x) = f(x 0 ) + φ(x)(x x 0 ). Let y 0 = f(x 0 ). There is a function ψ continuous at y 0 such that g(y) = g(y 0 ) + ψ(y)(y y 0 ). (6.3.3) Substituting y and y 0 in (6.3.3) by f(x) and f(x 0 ), we have ( ) ( )( ) g f(x) = g(f(x 0 )) + ψ f(x) f(x) f(x 0 ) ( ) = g(f(x 0 )) + ψ f(x) φ(x)(x x 0 ). Let θ(x) = ( ψ f(x) ) φ(x). Then (6.3.4) gives g f(x) = g f(x 0 ) + θ(x)(x x 0 ). (6.3.4) Since f is continuous at x 0 and ψ is continuous at f(x 0 ), the composition ψ f is continuous at x 0. Then θ, as a product of continuous functions, is continuous at x 0. Using the Carathéodory formulation of differentiability, 6.2.1, it therefore follows from (6.3.4) that g f is differentiable at x 0, with ( ) (g f) (x 0 ) = θ(x 0 ) = ψ f(x 0 ) φ(x 0 ) = g ( ) f(x 0 ) f (x 0 ). Exercise The following proof of the Chain Rule works under one additional assumption. What is it? Assume f differentiable at x 0 and g differentiable at f(x 0 ). We have ( ) ( ) g(f(x 0 + h)) g(f(x 0 )) g(f(x0 + h)) g(f(x 0 ) f(x0 + h) f(x 0 ) = h f(x 0 + h) f(x 0 ) h (6.3.5) 73

75 Write y 0 = f(x 0 ) and t = f(x 0 + h) f(x 0 ). Then f(x 0 + h) = y 0 + t and g(f(x 0 + h)) g(f(x 0 )) f(x 0 + h) f(x 0 ) = g(y 0 + t) g(y 0 ). t If f is differentiable at x 0, then it is continuous there. Hence when h 0, t 0 also. It follows that as h 0, (g(y 0 + t) g(y 0 ))/t tends to g (f(x 0 )). Therefore by (6.3.5), g(f(x 0 + h)) g(f(x 0 )) lim h 0 h g(f(x 0 + h)) g(f(x 0 ) f(x 0 + h) f(x 0 ) = lim lim h 0 f(x 0 + h) f(x 0 ) h 0 h = g (f(x 0 ))f (x 0 ). Example We can use the product and chain rules to prove the quotient rule. Since g(x 0 ) 0, it does not vanish on some interval (x 0 δ, x 0 + δ). The function f/g is defined everywhere on this interval. Let h(x) = 1 x. Then f g = f h g. Since g does not vanish anywhere on (x 0 δ, x 0 + δ), h g is differentiable at x 0 and therefore so is f h g. For the value of the derivative, note that h (y) = 1/y 2, so ( ) f (x 0 ) = f (x 0 )h(g(x 0 )) + f(x 0 )h (g(x 0 ))g (x 0 ) g = f (x 0 ) g(x 0 ) + f(x 1 0) g 2 (x 0 ) g (x 0 ) = f (x 0 )g(x 0 ) f(x 0 )g (x 0 ) g 2. (x 0 ) 6.4 The Inverse Function Theorem Lecture 13 Is the inverse of a differentiable function differentiable? 74

76 y f y = x f(c) c f 1 c f(c) x The graph of f is the graph of f 1, reflected by the line y = x. We might therefore guess that f 1 is as smooth as f. This is almost right, but there is one important exception: where f (x 0 ) = 0. Conisder the following graph, of f(x) = x 3. Its tangent line at (0, 0) is the line {y = 0}. y x 3 y = x x If we reflect the graph in the line y = x, the tangent line at (0, 0) becomes the vertical line {x = 0}, which has infinite slope. Indeed, the inverse of f 75

77 is the function f 1 (x) = x 1/3 0, which is not differentiable at 0: (x + h) 1/3 x 1/3 lim h 0 h =. Exercise Prove this formula for the limit. Recall that if f is continuous and injective, the inverse function theorem for continuous functions, 5.4.1, states that f is either increasing or decreasing, and that if f is increasing then f : [a, b] [f(a), f(b)] is a bijection, if f is decreasing then f : [a, b] [f(b), f(a)] is a bijection, and in both cases f 1 is continuous. Theorem (The inverse Function Theorem, II) Let f : [a, b] [c, d] be a continuous bijection. Let x 0 (a, b) and suppose that f is differentiable at x 0 and f (x 0 ) 0. Then f 1 is differentiable at y 0 = f(x 0 ). Furthermore, (f 1 ) (y 0 ) = 1 f (x 0 ). f x 0 y 0 f 1 76

78 Proof Since f is differentiable at x 0, f(x) = f(x 0 ) + φ(x)(x x 0 ), where φ is continuous at x 0. Letting x = f 1 (y), So f(f 1 (y)) = f(f 1 (y 0 )) + φ(f 1 (y))(f 1 (y) f 1 (y 0 )). y y 0 = φ(f 1 (y))(f 1 (y) f 1 (y 0 )). Since f is a continuous injective map its inverse f 1 is continuous. The composition φ f 1 is continuous at y 0. By the assumption φ f 1 (y 0 ) = φ(x 0 ) = f (x 0 ) 0, φ(x) 0 for x close to x 0. Define θ(y) = It follows that θ is continuous at y 0 and 1 φ(f 1 (y)). f 1 (y) = f 1 (y 0 ) + θ(y)(y y 0 ). Consequently, f 1 is differentiable at y 0 and (f 1 ) (y 0 ) = θ(y 0 ) = 1 f (f 1 (y 0 )) = 1 f (x 0 ). Proof [A second proof for the inverse function theorem] A function f is differentiable at x 0 if and only if for all x n (a, b) {x 0 } with lim n x n = x 0, f f(x n ) f(x 0 ) (x 0 ) = lim. n x n x 0 By injectivity f(x n ) f(x 0 ). Since f (x 0 ) 0, lim n x n x 0 f(x n ) f(x 0 ) 1 f (x 0 ). Take y n (c, d) with lim n y n = y 0. Let x n = f 1 (y n ). Since f 1 is continuous, lim x n = lim f 1 (y n ) = f 1 (y 0 ) = x 0 n n Thus f 1 (y n ) f 1 (y 0 ) x n x 0 lim = lim n y n y 0 n f(x n ) f(x 0 ) = 1 f (x 0 ). 77

79 In conclusion, f 1 is differentiable at f 0 and (f 1 ) (y 0 ) = 1 f (x 0 ). Comment: The kind of continuous bijective functions we see in everyday life are differentiable everywhere except perhaps at a discrete set of points. However there exist continuous bijective functions f, differentiable at x 0 with f (x 0 ) 0, but such that there exists a sequence x n tending to x 0 such that f is not differentiable at x n. Such functions are quite contrived. Exercise (i) Draw the graph of a function with this behaviour at some point x 0. (ii) Give a formula defining a function with this behaviour at some point x 0. Recall the following lemma (Lemma ). Lemma Let g : (a, b) {c} R, and c (a, b). Suppose that lim g(x) = l. x c 1. If l > 0, then g(x) > 0 for x close to c. 2. If l < 0, then g(x) < 0 for x close to c. If f (x 0 ) 0, by the non-vanishing Lemma above, applied to g(x) = f(x) f(x 0 ) x x 0, we may assume that on (x 0 r, x 0 + r) f(x) f(x 0 ) x x 0 > 0. Remark If f > 0 on an interval (x 0 r, x 0 + r), by Corollary to the Mean Value Theorem in the next chapter, f is an increasing function. Thus f : [x 0 r 2, x 0 + r 2 ] : [f(x 0 r 2 ), f(x 0 + r 2 )] is a bijection. In summary if f (x) > 0 for all x (a, b), then f is invertible. 2. Suppose that f is differentiable on (a, b) and f is continuous on (a, b). If x 0 (a, b) and f (x 0 ) > 0 then f is positive on an interval near x 0 and on which the function is invertible. 78

80 6.5 One Sided Derivatives Definition A function f : (a, x 0 ] R is said to have left derivative at x 0 if f(x) f(x 0 ) lim x x 0 x x 0 exists. The limit will be denoted by f (x 0 ), and called the left derivative of f at x A function f : (x 0, b) R is said to have right derivative at x 0 if f(x) f(x 0 ) lim x x 0 + x x 0 exists. The limit will be denoted by f (x 0 +), and called the right derivative of f at x 0. Theorem A function f has a derivative at x 0 if and only if both f (x 0 +) and f (x 0 ) exist and are equal. Example f(x) = { x sin(1/x) x > 0 0, x 0 Claim: f (0 ) = 0 but f (0+) does not exist. Proof f(x) f(0) 0 0 lim = lim = 0. x 0 x 0 x 0 x But f(x) 0 lim = lim x 0+ x 0 sin(1/x) x 0+ does not exist, as can be seen below. Take x n = 1 2nπ 0, y n = Both sequences converge to 0 from the right. But π nπ sin(1/x n ) = 0, sin(1/y n ) = 1. They have different limits so lim x 0+ sin(1/x) does not exist, and so f (0+) does not exist. 79

81 Example g(x) = Claim: g (0+) does not exist. Proof Note that Take But { x sin(1/x) x > 0 0, x 0 g(x) g(0) 1 lim = lim sin(1/x). x 0+ x 0 x 0+ x y n = π nπ 1 π sin(1/y n ) = + 2nπ +. yn 2 1 Hence x sin(1/x) cannot converge to a finite number as x approaches 0 from the right. We conclude that g (0+) does not exist. Exercises Find, directly from the definition of derivative as a limit, the derivatives of (a) f(x) = x n for n N (b) f(x) = x n for n N. (c) f(x) = ln(x) (d) f(x) = cos x (e) f(x) = tan x. 2. A function f : R R is called even if f( x) = f(x) for all x, and odd if f( x) = f(x) for all x. (a) Suppose that f is differentiable. Show that f is even f is odd. (b) Suppose that f is odd, and a bijection. Show that f 1 is odd. Is there a comparable result for the case where f is even? 80

82 Chapter 7 The mean value theorem If we have some information on the derivative of a function what can we say about the function itself? What do we think when our minister of the economy tells us that The rate of inflation is slowing down? This means that if f(t) is the price of a commodity at time t, its derivative is decreasing, but not necessarily the price itself (in fact most surely not, otherwise we would be told so directly)! I heard the following on the radio: the current trend of increasing unemployment is slowing down. What does it really mean? Exercise Consider the following graph..1 To do: (i) Suppose that this is the graph of some function f, and make a sketch of the graph of f. (ii) Suppose instead that this is the graph of the derivative f, and make a sketch of the graph of f. Hint for (ii): First find the critical points (where f = 0). For each 81

83 one, decide whether it is a local maximum or a local minimum: if x is a critical point and f is increasing at x, then x it is a local minimum, and if f is decreasing, then x is a local maximum. Identify inflection points (where f reaches local minimum or local maximum). At these points the graph changes from convex to concave or vice versa. Try to decide whether the graph between two consecutive critical points is increasing or decreasing, convex or concave. 7.1 Local Extrema Lecture 14 Definition Consider f : [a, b] R and x 0 [a, b]. 1. We say that f has a local maximum at x 0, if for all x in some neighbourhood (x 0 δ, x 0 + δ) (where δ > 0) of x 0, we have f(x 0 ) f(x). If f(x 0 ) > f(x) for x x 0 in this neighbourhood, we say that f has a strict local maximum at x We say that f has a local minimum at x 0, if for all x in some neighbourhood (x 0 δ, x 0 + δ) (where δ > 0) of x 0, we have f(x 0 ) f(x). If f(x 0 ) < f(x) for x x 0 in this neighbourhood, we say that f has a strict local minimum at x We say that f has a local extremum at x 0 if it either has a local minimum or a local maximum at x 0. Lemma Consider f : (a, b) R. Suppose that x 0 (a, b) is either a local minimum or a local maximum of f. If f is differentiable at x 0 then f (x 0 ) = 0. Proof Case 1. We first assume that f has a local maximum at x 0 : for some δ 1 > 0. Then f(x 0 ) f(x), x (x 0 δ 1, x 0 + δ 1 ), f f(x 0 + h) f(x 0 ) (x 0 +) = lim 0 h 0+ h since h (0, δ 1 ) eventually. Similarly f f(x 0 + h) f(x 0 ) (x 0 ) = lim 0. h 0 h 82

84 Since the derivative exists we must have f (x 0 ) = f +(x 0 ) = f (x 0 ). Finally we deduce that f (x 0 ) = 0. Case 2. If x 0 is a local minimum of f, take g = f. Then x 0 is a local maximum of g and g (x 0 ) = f (x 0 ) exists. It follows that g (x 0 ) = 0 and consequently f (x 0 ) = 0. Example Let f(x) = 1 4 x4 1 2 x2. Then f(x) = 1 4 x2 (x 2 2). It has a local maximum at x = 0, since f(0) = 0 and near x = 0, f(x) < 0. It has critical points at x = ±1. We will prove that they are local minima later. f Example f(x) = x 3. This function is strictly increasing, and there is no local minimum or local maximum at 0, even though f (0) = 0. 83

85 f(x) = x 3 0 Example { 1, x [ π, 0], f(x) = cos(x), x [0, 3π] π f Then x = 2π is a strict local maximum, and x = π is a strict local minimum; the function is constant on [ π, 0], so each point in the interior of this interval is both a local minimum and a local maximum! The point x = 0 is a local maximum but not a local minimum. The point π is both a local minimum and a local maximum. 7.2 Global Maximum and Minimum If f : [a, b] R is continuous by the extreme value theorem, it has a minimum and a maximum. How do we find them? What we learnt in the last section suggested that we find all critical points. Definition A point c is a critical point of f if either f (c) = 0 or f (c) does not exist. 84

86 To find the (global) maximum and minimum of f, evaluate the values of f at a, b and at all the critical points. Select the largest value. 7.3 Rolle s Theorem y x 0 f x Theorem (Rolle s Theorem) Suppose that 1. f is continuous on [a, b]. 2. f is differentiable on (a, b). 3. f(a) = f(b). Then there is a point x 0 (a, b) such that f (x 0 ) = 0. Proof If f is constant, Rolle s Theorem holds. Otherwise by the Extreme Value Theorem, there are points x, x [a, b] with f( x) f(x) f( x), x [a, b]. Since f is not constant f(x) f( x). Since f(a) = f(b), one of the point x or x is in the open interval (a, b). Denote this point by x 0. By the previous lemma f (x 0 ) = 0. Example Discussions on Assumptions: 85

87 1. Continuity on the closed interval [a, b] is necessary. For example consider f : [1, 2] R. { f(x) = 2x 1, x (1, 2] f(x) = f(1) = 3, x = 1. The function f is continuous on (1, 2] and is differentiable on (a, b). Also f(2) = 3 = f(1). But there is no point in x 0 (1, 2) such that f (x 0 ) = 0. y 3 f y x 1 2 x 1 1 x 2. Differentiability is necessary. e.g. take f : [ 1, 1] R, f(x) = x. Then f is continuous on [ 1, 1], f( 1) = f(1). But there is no point on x 0 ( 1, 1) with f (x 0 ) = 0. This point ought to be x = 0 but the function fails to be differentiable at x 0. Exercise Let f(x) = x sin x 1 2 x sin x Does f have a critical point on ( π 6, 1)? 7.4 The Mean Value Theorem Lecture 15 Theorem (The Mean Value Theorem) Suppose that 1. f is continuous on [a, b]. 2. f is differentiable on (a, b). Then there is a point c (a, b) such that f (c) = f(b) f(a). b a 86

88 f(b) f(a) Note: b a is the slope of the chord joining the points (a, f(a)) and (b, f(b)) on the graph of f. The Mean Value Theorem says that there is a point c (a, b) such that the tangent to the graph at (c, f(c)) is parallel to this chord. y f (c) = f(b) f(a) b a f a c b x Proof The equation of the chord joining (a, f(a)) and (b, f(b)) is Let y = g(x) = f(x) f(b) f(a) (x a) + f(a). b a f(b) f(a) (x a) + f(a) b a Then g(b) = g(a) = 0, and g is continuous on [a, b] and differentiable on (a, b). By Rolle s Theorem applied to g on [a, b], there exists c (a, b) with g (c) = 0. But g (c) = f f(b) f(a) (c). b a Corollary Suppose that f is continuous on [a, b] and is differentiable on (a, b). Suppose that f (c) = 0, for all c (a, b). Then f is constant on [a, b]. Proof Let x (a, b]. By the Mean Value Theorem on [a, x], there exists c (a, x) with f(x) f(a) = f (c). x a Since f (c) = 0, this means that f(x) = f(a). 87

89 Exercise Show that if f, g are continuous on [a, b] and differentiable on (a, b), and f = g on (a, b), then f = g + C where C = f(a) g(a). Example If f (x) = 1 for all x then f(x) = x + c, where c is some constant. To see this, let g(x) = f(x) x. Then g (x) = 0 for all x. Hence g(x) is a constant: g(x) = g(a) = f(a) a. Consequently f(x) = x + g(x) = f(a) + x a. Exercise If f (x) = x for all x, what does f look like? Remark When f is continuous, we can deduce the Mean Value Theorem from the Intermediate Value Theorem, together with some facts about integration which will be proved in Analysis III. If f is continuous on [a, b], it is integrable. Since f is continuous, by the Extreme Value Theorem, there exist x 1, x 2 [a, b] such that f (x 1 ) and f (x 2 ) are respectively the minimum and the maximum values of f on the interval [a, b]. Hence we have f (x 1 )(b a) Divide by b a to see that f (x 1 ) b a 1 b a f (x)dx f (x 2 )(b a). b a f (x)dx f (x 2 ). Now the Intermediate Value Theorem, applied to f on the interval between x 1 and x 2, says there exists c between x 1 and x 2 such that f (c) = 1 b a b a f (x)dx. Finally, the fundamental theorem of calculus, again borrowed from Analysis III, tells us that b a f (x) = f(b) f(a). 7.5 Monotonicity and Derivatives If f : [a, b] R is increasing and differentiable at x 0, then f (x 0 ) 0. This is because f f(x) f(x 0 ) (x 0 ) = lim. x x 0 + x x 0 Both numerator and denominator in this expression are non-negative, so the quotient is non-negative also, and hence so is its limit. 88

90 Corollary Suppose that f : R R is continuous on [a, b] and differentiable on (a, b). 1. If f (x) 0 for all x (a, b), then f is non-decreasing on [a, b]. 2. If f (x) > 0 for all x (a, b), then f is increasing on [a, b]. 3. If f (x) 0 for all x (a, b), then f is non-increasing on [a, b]. 4. If f (x) < 0 for all x (a, b), then f is decreasing on [a, b]. Proof Part 1. Suppose f (x) 0 on (a, b). Take x, y [a, b] with x < y. By the mean value theorem applied on [x, y], there exists c (x, y) such that f(y) f(x) = f (c) y x and f(y) f(x) = f (c)(y x) 0. Thus f(y) f(x). Part 2. If f (x) > 0 everywhere then as in part 1, for x < y, f(y) f(x) = f (c)(y x) > 0. Proposition If f : (a, b) R and f (x 0 ) > 0 then there exists δ > 0 such that if x (x 0 δ, x 0 ), then f(x) < f(x 0 ) Proof Because if x (x 0, x 0 + δ), then f(x 0 ) < f(x). f(x 0 + h) f(x 0 ) lim > 0, h 0 h there exists δ > 0 such that if 0 < h < δ then f(x 0 + h) f(x 0 ) h > 0. Writing x in place of x 0 + h, so that h is replaced by x x 0, this says if 0 < x x 0 < δ then f(x) f(x 0 ) x x 0 > 0. 89

91 When x 0 δ < x < x 0, the denominator in the last quotient is negative. Since the quotient is positive, the numerator must be negative too. When x 0 < x < x 0 + δ, the denominator in the last quotient is positive. Since the quotient is positive, the numerator must be positive too. In this proposition we compare the value of f at the variable point x with its value at the fixed point x 0. A statement that f is increasing would have to compare the values of f at variable points x 1, x 2. Suppose that f : (a, b) R is differentiable at x 0 and that f (x 0 ) > 0. Does this imply that f is increasing on some neighbourhood of x 0? If you attempt to answer this question by drawing pictures and exploring the possibilities, it is quite likely that you will come to the conclusion that the answer is Yes. But it is not true! We will see this shortly, by means of an example (Example 7.5.3). First, notice that if we assume not only that f (x 0 ) > 0, but also that f is differentiable at all points x in some neighbourhood of x 0, and that f is continuous at x 0, then in this case it is true that f is increasing on a neighbourhood of x 0. This is simply because the continuity of f at x 0 implies that f (x) > 0 for all x in some neighbourhood of x 0, and then Part 2 of Corollary implies that f is increasing on this neighbourhood. To see that it is NOT true without this extra assumption, consider the following example. Example The function { x + x f(x) = 2 sin( 1 ), x 0, x [ 1, 1] x 2 0, x = 0. is differentiable everywhere and f (0) = 1. 0 Proof Since the sum, product, composition and quotient of differentiable 90

92 functions are differentiable, provided the denominator is not zero, f is differentiable at x 0, and f (x) = 1 + 2x sin( 1 x 2 ) 2 x cos( 1 x 2 ). At x = 0, f (0) = f(x) f(0) f(x) lim = lim x 0 x 0 x 0 x = x + x 2 sin( 1 ) x lim x 0 x = lim(1 + x sin( 1 )) = 1. x 0 x2 91

93 Notice, however, that although f is everywhere differentiable, the function f (x) is not continuous at x = 0. This is clear: f (x) does not tend to a limit as x tends to 0. And in fact, Claim: Even though f (0) = 1 > 0, there is no interval ( δ, δ) on which f is increasing. It shows that, although f (x 0 ) > 0 we cannot conclude that f is increasing on some neighbourhood of x 0. Proof { f 1 + 2x sin( 1 (x) = x ) 2 2 x cos( 1 x ), x 0 2 1, x = 0. Consider the intervals: I n = If x I n, 2πn 1 x 2πn + π 4 and [ ] 1 1,. 2πn + π 4 2πn cos 1 x 2 cos π 4 = 1 2. Thus if x I n f 1 (x) πn 1 = 2πn 2 πn + 2 2πn πn < 0. Consequently f is decreasing on I n. As every open neighbourhood of 0 contains an interval I n, the claim is proved. Example shows that f need not be continuous. However, it turns out that the kind of discontinuities it may have are rather different from the discontinuities of a function like g(x) = [x]. To begin with, even though it may not be continuous, rather surprisingly f has the intermediate value property : Exercise Let f be differentiable on (a, b). 1. Suppose that a < c < d < b, and that f (c) < v < f (d). Show that there exists x 0 (c, d) such that f (x 0 ) = v. Hint: let g(x) = f(x) vx. Then g (c) < 0 < g (d). By Proposition 7.5.2, g(x) < g(c) for x slightly greater than c, and g(x) < g(d) for x slightly less than d. It follows that g achieves its minimum value on [c, d] at some point in the interior (c, d). 2. Suppose that lim x x0 f (x 0 ) exists. Use the intermediate value property to show that it is equal to f (x 0 ). Prove also the analogous statement for lim x x0+ f (x). Deduce that if both limits exist then f is continuous at x 0. 92

94 7.6 Inequalities From the MVT Lecture 16 We wish to use information on f to deduce information on f. Suppose we have a bound for f on the interval [a, x]. Applying the mean value theorem to the interval [a, x], we have f(x) f(a) x a = f (c) for some c (a, x). From the bound on f we can deduce something about f(x). We will prove later that ln : (0, ) R is differentiable everywhere and ln (x) = 1 x. We already know for any b > 0, ln : [1, b] R is continuous, as it is the inverse of the continuous function e x : [0, ln b] [1, b]. Example For any x > 1, 1 1 x < ln(x) < x 1. Proof Fix x > 1 and consider the function ln on [1, x]. Apply the Mean Value Theorem to the function f(y) = ln y on the interval [1, x]: there exists c (1, x) such that ln(x) ln 1 = f (c) = 1 x 1 c. Since 1 x < 1 c < 1, 1 x < ln x x 1 < 1. Multiplying through by x 1 > 0, we see x 1 x < ln x < x 1. Example Let f(x) = 1 + 2x + 2 x. Show that f(x) x on [0.1, ). Proof First f (x) = 2 2. Since f (x) < 0 on [0.1, 1), by Corollary x 2 of the MVT, f decreases on [0.1, 1]. So the maximum value of f on [0.1, 1] is f(0.1) = On [1, ), f(x) 1 + 2x + 2 = 3 + 2x. 93

95 7.7 Asymptotics of f at Infinity Lecture 16 Example Let f : [1, ) R be continuous. Suppose that it is differentiable on (1, ). If lim x + f (x) = 1 there is a constant K such that f(x) 2x + K. Proof 1. Since lim x + f (x) = 1, taking ε = 1, there is a real number A > 1 such that if x > A, f (x) 1 < 1. Hence f (x) < 2 if x > A. We now split the interval [1, ) into [1, A] (A, ) and consider f on each interval separately. 2. Case 1: x [1, A]. By the Extreme Value Theorem, f has an upper bound K on [1, A]. If x [1, A], f(x) K. Since x > 0, f(x) K + 2x, x [1, A]. 3. Case 2. x > A. By the Mean Value Theorem, applied to f on [A, x], there exists c (A, x) such that f(x) f(a) x A = f (c) < 2, by part 1). So if x > A, f(x) f(a) + 2(x A) f(a) + 2x K + 2x. The required conclusion follows. 94

96 3 2 x + 4 y 1 2 x A The graph of f lies in the wedge-shaped region with the yellow boundaries. Exercise Let f(x) = x + sin( x). Find lim x + f (x). Exercise In Example 7.7.1, can you find a lower bound for f? Example Let f : [1, ) R be continuous. Suppose that it is differentiable on (1, ). If lim x + f (x) = 1 then there are real numbers m and M such that for all x [1, ), Proof m x f(x) M x. 1. By assumption lim x + f (x) = 1. For ε = 1 2, there exists A > 1 such that if x > A, 1 2 = 1 ε < f (x) < 1 + ε = Let x > A. By the Mean Value Theorem, applied to f on [A, x], there exists c (A, x) such that x f (c) = f(x) f(a) x A. 95

97 Hence So if x > A, 1 f(x) f(a) < < 3 2 x A 2. f(a) (x A) f(x) f(a) + 3 (x A) 2 f(a) 1 2 A x f(x) f(a) x. 3. On the finite interval [1, A], we may apply the Extreme Value Theorem to f(x) 1 2x and to f(x) so there are m, M such that Then if x [1, A], m f(x) 1 x, f(x) M. 2 f(x) M M x f(x) = f(x) 1 2 x x m x. 4. Since m f(a) 1 2 A, the required identity also holds if x > A. Exercise In Example can you show that there exist real numbers m and M such that for all x [0, ), m x f(x) 1.01x + M? Exercise Let f : [1, ) R be continuous. Suppose that it is differentiable on (1, ). Suppose that lim x + f (x) = k. Show that for any ε > 0 there are numbers m, M such that m + (k ε)x f(x) M + (k + ε)x. Comment: If lim x + f (x) = k, the graph of f lies in the wedge formed by the lines y = M + (k + ε)x and y = m + (k ε)x. This wedge contains and closely follows the line with slope k. But we have to allow for some oscillation and hence the ε, m and M. In the same way if f (x) Cx α when x is sufficiently large then f(x) is controlled by x 1+α with allowance for the oscillation. 96

98 Exercise What can we say if lim x f (x) = 0? Let us first consider the monotone functions f(x) = x α. Example Let f : (0, ) R, f(x) = x α, x > 0. Then f (x) = αx α 1 and lim f(x) = x + lim f (x) = x + +, if α > 0 1, if α = 0 0, if α < 0. +, if α > 1 1, if α = 1 0, if α < 1. If α < 1 the function x α grows sub-linearly at x. Let us now add some oscillation. How do we do this? Example Consider f(x) = x α + sin x. The function f is dominated by x α when α > 0 and x goes to +. For α < 0, as x goes to + it behaves like sin x and oscillates between 1 and 1. 97

99 The graph of sin Hsqrt HxLLL and sin HxL 81 Pi^2D Hzoomed outl. The pink graph is that of sin x. Note that how it oscillates compare to that of sin Hsqrt HxLL. In[47]:= ê 2LD, 8x, 0, H9 PiL^2<D Out[47]= The graph looking like a straight line is that of x + sin Hsqrt HxLL In[44]:= Plot@8x + Sin@x^H1 ê 2LD, 10 * Sin@xD<, 8x, 0, H2 PiL^2<D Out[44]= The fast oscillation graph is that of sin Hx^2L. In[43]:= Plot@8Sin@x^H2LD, 10 * Sin@xD<, 8x, 0, 4 Pi<D 10 5 Out[43]=

100 Example Let β > 0 and f : (0, ) R, f(x) = sin( 1 x β ). When x is large, we do not see oscillation from the sine function. Indeed if x > ( 2 π ) 1 β, then f (x) < 0 (check this!) and sin 1 is strictly decreasing. x β Example Let β > 0 and f : (0, ) R, f(x) = x α sin( 1 x β ). We know that lim x sin( 1 x ) = 0; how about lim x x sin 1 x? Indeed for any β > 0, let y = 1. We see x β Hence lim x + xβ sin( 1 x β ) = lim sin(y) = 1. y 0+ y lim x + xα sin( 1 [ x β ) = lim x + xα β x β sin( 1 ] x β ). As x goes to, the function f behaves like x α β. Example Let f(x) = x sin( 1 x 1 4 ). Then f(x) does not have a limit as x goes to, and is not even bounded. Nevertheless ( 1 lim f (x) = lim x + x + 2 x sin( 1 ) 1 1 cos( 1 ) ) x x 3 4 x 1 4 Let us now look at a more interesting oscillating function. Example Let β > 0. Let f : (0, ) R, f(x) = x α sin(x β ). = The bigger is β, the more quickly x β grows (once x > 1). The more quickly x β grows, the more quickly sin(x β ) oscillates. So large β means faster oscillation. 2. If α < 0, lim x + f(x) = 0; if α 0, lim x + f(x) does not exist. To see the conclusion for α 0 take x n = (2πn) 1 β lim f(x n) = lim n + n + xα n 0 = 0. 99, then

101 Take y n = (2πn + π 2 ) 1 β, then lim f(y n) = lim n + n yα n 1 = (2πn + π { 2 ) α β, if α > 0; = 1, if α = 0. So lim n f(y n ) lim n + f(x n ). 3. f (x) = αx α 1 sin(x β ) + βx α+β 1 cos(x β ). Note that β > 0 and so α 1 < α + β 1 and x α+β 1 dominates x α 1 (a) Suppose that α + β 1, then lim x f (x) does not exist. (b) Suppose that α + β < 1. Since α < 1, lim f (x) = 0. x + Example In each of the following three cases, in which we assign values to α and β in the previous example, evaluate lim x + f(x) and lim x + f (x). Take α = 0, β = 3 4, f(x) = sin(x 3 4 ). Take α = 1 2, β = 1 4, f(x) = x sin(x 1 4 ). Take α = 1, β = 1 4, f(x) = x sin(x 1 4 ). Example Consider g(x) = x sin x. lim g(x) = 1 x + lim x + g (x) = lim x + ( 1x 2 sin(x) + 1x ) cos(x) = An Example: Gaussian Envelopes We do not cover this section in class. However it is a fun example. Do read on! Let f σ (x), σ I be a family of continuous functions with index σ in a set I. We say that f is parametrized by σ. We seek a function g(x) such that for each x, g(x) is the maximal value of f σ (x) when σ runs through all values in I. That is g(x) = sup{f σ (x)}. σ I 100

102 We say that g is the upper envelope of f σ. To work out the upper envelope of f σ, we fix x and consider f σ (x) as a function of the variable σ. Then we seek its maximal value. If I = [a, b] and f σ (x) is continuous in σ, then by the extreme value theorem, the maximal value of f σ (x) is attained at a point ˆσ. For different values of x, we may have a different maximal point ˆσ. Example We find the upper envelope for the family of Gaussian curves: f σ (x) = 1 2πσ e x2 2σ 2. Graph for σ = {0.15, 0.2, 0.22, 0.25, 0.3, 0.35, 0.4, 0.5, 0.7, 0.8, 0.9, 1.0, 1.2, 1.5, 2.4, 3, 4}. Fix the value x. We compute f σ (x) for the critical value ˆσ for which fˆσ (x) is a maximum. Set fσ σ f σ σ = 1 ( 1 2π σ 2 + x2 ) x 2 e σ 4 (σ, x) = 0 and obtain the solution σ = x. Now Consequently 2σ 2. f σ (x) = 1 ( 2 2π σ 3 4x2 σ 5 (( 1 ) σ 2 + x2 σ 4 )2 e x2 2σ 2. f ˆσ (x) = <

103 So F σ (x) achieves its maximum when σ = x. The curve y = f(ˆσ, x) is now given by 1 y = e π x It is called the upper envelope of the family of curves f σ. For each value of σ, the graph of f σ touches the upper envelope at precisely x = σ. 102

104 Chapter 8 Higher order derivatives 8.1 Continuously Differentiable Functions When f is differentiable at every point of an interval, we investigate the properties of its derivative f (x). Definition We say that f : [a, b] R is continuously differentiable if it is differentiable on [a, b] and f is continuous on [a, b]. Instead of continuously differentiable we also say that f is C 1 on [a, b]. Note that by f (a) we mean f (a+) and by f (b) we mean f (b ). A similar notion exists for f defined on (a, b), or on (a, ), on (, b), or on (, ). Example The function g defined by g(x) = x 2 sin(1/x 2 ) for x 0, g(0) = 0, is differentiable on all of R, but not C 1 its derivative is not continuous at 0. See Example (which deals with the function f(x) = x + g(x)) for details. Example Let 0 < α and { x h(x) = 2+α sin(1/x) x > 0 0, x = 0 Claim: The function h is C 1 on [0, ). Proof We have h x 2+α sin(1/x) (0) = lim = lim x 0+ x 0 x 0+ x1+α sin(1/x) = 0, 103

105 and h (x) = (2 + α)x 1+α sin(1/x) x α cos(1/x) for x > 0. Since (2 + α)x 1+α sin(1/x) x α cos(1/x) (2 + α) x 1+α + x α, by the sandwich theorem, lim x 0+ h (x) = 0 = h (0). Hence h is C 1. Show that h is not differentiable at x = 0, if α (0, 1). Definition The set of all continuously differentiable functions on [a, b] is denoted by C 1 ([a, b]; R) or by C 1 ([a, b]) for short. Similarly we may define C 1 ((a, b); R), C 1 ((a, b]; R), C 1 ([a, b); R), C 1 ([a, ); R), C 1 ((, b]; R), C 1 ((, ); R) etc. 8.2 Higher Order Derivatives Example Let g(x) = { x 3 sin(1/x) x 0 0, x = 0 Claim: The function g is twice differentiable on all of R. Proof It is clear that g is differentiable on R {0}. It follows directly from the defintion of derivative that g is differentiable at 0 and g (0) = 0. So g is differentiable everywhere and its derivative is { g 3x (x) = 2 sin(1/x) x cos(1/x) x 0 0, x = 0 By the sandwich theorem, lim x 0 3x 2 sin(1/x) x cos(1/x) = 0 = g (0). Hence g is C 1. f. If the derivative of f is also differentiable, we consider the derivative of Definition Let f : (a, b) R be differentiable. We say f is twice differentiable at x 0 if f : (a, b) R is differentiable at x 0. We write: f (x 0 ) = (f ) (x 0 ). It is called the second derivative of f at x 0. It is also denoted by f (2) (x 0 ). 104

106 Following from these Definition We say that f is n times differentiable at x 0 if f (n 1) is differentiable at x 0. The the nth derivative of f at x 0 is: f (n) (x 0 ) = (f (n 1) ) (x 0 ). Definition We say that f is C, or smooth on (a, b) if it is n times differentiable for every n N. Definition We say that f is n times continuously differentiable on [a, b] if f (n) exists and is continuous on [a, b]. This means all the lower order derivatives and f itself are continuous, of course - otherwise f (n) would not be defined.. Definition The set of all functions which are n times differentiable on (a, b) and such that the derivative f (0), f (1),... f (n) are continuous functions on (a, b) is denoted by C n (a, b). There is a similar definition with [a, b], or other types of intervals, in place of (a, b). 8.3 Convexity We give a theorem for deciding whether a critical point is a local minimum or a local maximum. Consider the two functions: f(x) = x 2 and g(x) = 1 x 2 with critical point x = 0. We have f > 0 and g < 0. What do their graphs look like? Theorem Let f : (a, b) R be differentiable in a neighbourhood of the point x 0. Suppose that f (x 0 ) = 0, and that f also is differentiable at x 0. Then if f (x 0 ) > 0, f has a local minimum at x 0, and if f (x 0 ) < 0, f has a local maximum at x 0. Proof Suppose that f (x 0 ) > 0. That is, since f (x 0 ) = 0, f (x 0 + h) lim > 0. h 0 h Hence there exists δ > 0 such that for 0 < h < δ, f (x 0 + h) h > 0. (8.3.1) 105

107 This means that if 0 < h < δ, then f (x 0 + h) > 0, while if δ < h < 0, then f (x 0 + h) < 0. (We are repeating the argument from the proof of Proposition ) It follows by the MVT that f is decreasing in (x 0 δ, x 0 ] and increasing in [x 0, x 0 + δ). Hence x 0 is a strict local minimum. The case where f (x 0 ) < 0 follows from this by taking g = f. Corollary If f : [a, b] R is such that f (x) > 0 for all x (a, b) and f is continuous on [a, b] then for all x (a, b), f(x) f(a) + f(b) f(a) (x a). b a In other words, the graph of f between (a, f(a)) and (b, f(b)) lies below the line segment joining these two points. y g f a b x Proof Let g(x) = f(a) + f(b) f(a) (x a). b a The graph of g is the straight line joining the points (a, f(a)) and (b, f(b)). Let h(x) = f(x) g(x). Then 1. h(a) = h(b) = 0; 2. h is continuous on [a, b] and twice differentiable on (a, b); 3. h (x) = f (x) g (x) = f (x) > 0 for all x (a, b). By the extreme value theorem there are points x 1, x 2 [a, b] such that h(x 1 ) h(x) h(x 2 ), x [a, b]. 106

108 If x 2 (a, b), it is a local maximum and hence h (x 2 ) = 0 (see Lemma 7.1.2). But then by 8.3.1, since h (x 2 ) > 0, x 2 would also be a strict local minimum! This is absurd. So x 2 cannot be in the interior of the interval [a, b] it must be an end-point, a or b. In either case h(x 2 ) = 0. We have proved that 0 is the maximum value of h and so h(x) 0 and f(x) g(x). Exercise Let f be a function for which f (x) > 0 for all x. Show that the region above the graph of f is convex that is, for any two points in this region, the line segment joining them is completely contained in the region. For this reason, such functions are called convex functions Still higher derivatives* What if f (x 0 ) = f (x 0 ) = 0 and f (x 0 ) 0? What can we conclude about the behaviour of f in the neighbourhood of x 0? Let us reason as we did in the proof of Suppose f (x 0 ) > 0. Then there exists δ > 0 such that and if x 0 δ < x < x 0 then f (x) < f (x 0 ) and therefore f (x) < 0 if x 0 < x < x 0 + δ then f (x 0 ) < f (x) and therefore 0 < f (x). It follows that f is decreasing in [x 0 δ, x 0 ] and increasing in [x 0, x 0 + δ]. Since f (x 0 ) = 0, we conclude that f is positive in [x 0 δ, x 0 ) and also positive in (0, x 0 + δ]. We have proved Proposition If f (x 0 ) = f (x 0 ) = 0 and f (x 0 ) > 0 then there exists δ > 0 such that f is increasing on [x 0 δ, x 0 + δ]. Observe that this is exactly the behaviour we see in f(x) = x 3 in the neighbourhood of 0. Exercise Prove that if f (x 0 ) = f (x 0 ) = 0 and f (x 0 ) < 0 then there exists δ > 0 such that f is decreasing on [x 0 δ, x 0 + δ]. 2. What happens if f (x 0 ) = f (x 0 ) = f (x 0 ) = 0 and f (4) (x 0 ) > 0? 3. What happens if f (x 0 ) = = f (k) (x 0 ) = 0 and f (k+1) (x 0 ) 0? 107

109 Exercise Sketch the graph of f if f is the function whose graph is y f x shown and f(0) =

110 Chapter 9 Power Series Functions Definition By a power series we mean an expression of the type n=n 0 a n x n or n=n 0 a n (x x 0 ) n where each a n is in R (or, later on in the subject, in the complex domain C). Usually the initial index n 0 is 0 or 1. By convention, x 0 = 1 for all x (including 0) and 0 n = 0 for all n N with n > 0, and 0! = 1. Example n=1 ( 1)n xn n, n 0 = 0, 3. a n = ( 1) n 1 n, n 0 = n=0 xn n!, a n = 1 n!, n 0 = n=0 ( x 3 )n, a n = 1 3 n n 0 = 0 5. n=0 n!xn, a n = n!, n 0 = 0. If we substitute a real number in place of x, we obtain an infinite series. For which values of x does the infinite series converge? Lemma (The Ratio test) The series n=n 0 b n 1. converges absolutely if there is a number r < 1 such that b n+1 b n < r eventually. 2. diverges if there is a number r > 1 such that > r eventually. 109 b n+1 b n

111 Recall by a statement holding eventually we mean that there exists N 0 such that the statement holds for all n > N 0. Lemma (The Limiting Ratio test) The series n=0 b n 1. converges absolutely if lim n b n+1 b n < 1; 2. diverges if lim n b n+1 b n > 1. Example Consider ( x ) n. n=0 3 Let bn = ( x b n+1 b n = ( x 3 )n+1 ( x 3 )n = x 3, 3 ) n. Then { < 1, if x < 3 > 1, if x > 3. By the ratio test, the power series is absolutely convergent if x < 3, and divergent if x > 3. If x = 3, the power series is equal to , which is divergent. If x = 3 the power series is equal to ( 1) ( 1) , and again is divergent. Example Consider b n+1 b n = ( 1)n+1 n+1 x n+1 = x ( 1)n n xn n=0 ( 1) n n ( ) n n x, n + 1 xn. Let b n = ( 1)n n xn. Then The series is convergent for x < 1, and divergent for x > 1. For x = 1, we have ( 1) n n which is convergent. For x = 1 we have 1 n which is divergent. Example Consider the power series x (n+1) (n+1)! x n n! = n=0 xn n! x n < 1 The power series converges for all x (, ). n=0 n!xn converges only at x = 0. Indeed if x 0, { < 1, if x < 1 1, if x > 1.. We have as n. (n + 1)x (n+1) n!x n = (n + 1) x 110

112 If for some x R, the infinite series n=0 a nx n converges, we define f(x) = a n x n. n=0 The domain of f consists of all x such that n=1 a nx n is convergent. If x = 0, a n 0 n = a 0 n=0 is convergent, and so f(0) = a 0. Problem: Identify the set of points for which the power series is convergent. At which points is the function continuous? differentiable? 9.1 Radius of Convergence The following Lemma provides insight into the radius of convergence of power series. Lemma If for some number x 0, n=0 a nx n 0 converges, then n=0 a nc n converges absolutely for any C with C < x 0. Proof As a n x n 0 is convergent, lim a n x 0 n = 0. n Convergent sequences are bounded. So there is a number M such that for all n, a n x n 0 M. Suppose C < x 0. Let r = C x 0. Then r < 1, and ( ) a n C n = C a nx n n 0 Mr n. By the comparison theorem, n=0 a nc n converges absolutely. Theorem Consider a power series n=0 a nx n. Then one of the following holds: (1) The series only converges at x = 0. (2) The series converges for all x (, ). x n 0 111

113 (3) There is a positive number 0 < R <, such that n=0 a nx n converges for all x with x < R; n=0 a nx n diverges for all x with x > R. This number R is called the radius of convergence of the power series. In cases (1) and (2) we define the radius of convergence to be 0 and respectively. Proof Define S = { x } a n x n is convergent, n=0 Note that S is not empty : 0 S. Suppose that S is not bounded. By Lemma 9.1.1, n=0 a nx n converges for all x. This is case (2). If x = 0 is the only value for which the power series n=0 a nx n converges, we are in case (1). Suppose that S contains a point that is not {0}. Suppose that S is bounded. Let A = { x : x S}, and let R = sup(a). Let x be such that x < R. Then there is a number b A such that b x, otherwise x would be an upper bound of A. By definition of A, either b or b in S. Because n=0 a nb n converges absolutely, by Lemma it follows that n=0 a nx n converges. Suppose that x > R. Then x S hence n=0 a nx n diverges. Corollary The radius of convergence of n=0 a nx n agrees with sup{x R : a n x n converges}. n=0 When x = R and x = R, we cannot say anything about the convergence of the series n=0 a nx n without further examination. Lemma Suppose that n=n 0 a n x n has radius of convergence R and n=n 0 b n x n has radius of convergence S, let c R {0}, and let T and U be the radii of convergence of n=n 0 (a n +b n )x n and n=n 0 ca n x n respectively. Then T min{r, S} and U = R. 112

114 Exercise Give an example where T > min{r, S}. Remark Complex Analysis studies power series in which both the variable and the coefficients may be complex numbers. We will refer to them as complex power series. The proof of Lemma works unchanged for complex power series, and in consequence there is a version of Theorem for complex power series: every complex power series n=0 a n(z z 0 ) n has a radius of convergence, a number R R { } such that if z z 0 < R then the series converges absolutely. The set of z for which the series converges is now a disc centred on z 0, rather than an interval as in the real case. Also different from the real case is the fact (which we do not prove here) that a complex power series with radius of convergence 0 < R < necessarily diverges for some z with z = R. 9.2 The Limit Superior of a Sequence In this section we develop a method to find the radius of convergence of a power series. The method is based on the root test for the convergence of a series. This says that the series n=0 a n converges if lim n a n 1/n < 1 and diverges if lim n a n 1/n > 1. The problem with this test is that in many cases, such as where a n = (1 + ( 1) n ), the sequence a n 1/n does not have a limit as n. The way round this problem is to replace (a n ) by a related sequence which always does have a limit. Recall that if A is a set then { the least upper bound of A, if A is bounded above sup A = +, if A is not bounded above { the greatest lower bound of A, if A is bounded below inf A =, if A is not bounded below Given a sequence (a n ), we define a new sequence (s n ) by s 1 = sup{a 1, a 2, a 3,...} s 2 = sup{a 2, a 3, a 4,...} = s n = sup{a n, a n+1,...} = 113

115 Lemma Let (a n ) be a sequence of real numbers, and let s n be defined as above. Then one of the following occurs: 1. (a n ) is not bounded above. In this case s n = for all n, and so lim s n =. n 2. (a n ) is bounded above and (s n ) is bounded below. In this case lim s n R. n 3. (a n ) is bounded above, but (s n ) is not bounded below. In this case lim s n =. n Proof (1) If {a 1, a 2,...} is not bounded above, then nor is the set {a n, a n+1,...} for any n. Thus s n = for all n. (2) If (a n ) is bounded above, then s n R for all n. Moreover, the sequence (s n ) is non-increasing. If (s n ) is bounded from below, by the monotone convergence theorem, it must converge to a real limit in fact it converges to its infimum: lim s n = inf{s n : n N}. n (3) If (a n ) is bounded above and if the sequence (s n ) is not bounded below. The the non-increasing sequence (s n ) converges to. Note that if (a n ) is bounded above and below, then so is (s n ), for L a n U for all n implies L s n U for all n. Definition Let (a n ) be a sequence. With (s n ) as above, we define This is also denoted by lim n a n. lim sup a n = lim s n. n n 114

116 Example Hence 1. Let b n = 1 1 n. Then { sup 1 1 n, 1 1 } n + 1,... = 1. lim sup(1 1 n n ) = lim 1 = 1. n 2. Let b n = n. Then sup{1 + 1 n, n+1,... } = n. Hence lim sup(1+ 1 { n n ) = lim sup n n, } n + 1,... = lim (1+ 1 n n ) = 1. Proposition Let l be a real number or. If lim n a n = l then lim n sup a n = l also. Proof (1) If l = +, {a n } is not bounded from above and lim sup n a n =. (2) If l =, a n is eventually smaller than any given number M, and so is s n giving lim sup n a n =. (3) Suppose that l is a real number. Let ε > 0. By the convergence of (a n ), there exists N such that if n N then l ε < a n < l + ε. For n N, l ε sup({a n, a n+1,...}) l + ε. Hence lim n sup({a n, a n+1,...}) = l. i Exercises The limit inferior of the sequence (a n ) is defined as follows: we define a new sequence i n by i n = inf{a k : k n}, and set lim inf a n = lim i n. n n (a) Show that lim inf a n always exists (in R {± }). (b) Show that if a n l as n then lim n inf a n = l. 2. Show that if b R 0 then lim sup ba n = b lim sup a n. What can you say if b < 0? 3. Suppose l = lim sup n b n is finite, and let ε > 0. Show 115

117 (a) There is a natural number N ε such that if n > N then b n < l + ε. That is, b n is bounded above by l + ε, eventually. (b) For any natural number N, there exists n > N s.t. b n > l ε. That is, there is a sub-sequence b nk with b nk > l ε. Deduce that if l = lim sup n b n there is a sub-sequence b nk such that lim k b nk = l. 4. We say that l is a limit point of the sequence (x n ) if there is a subsequence (x nk ) tending to l. Let L((x n )) be the set of limit points of (x n ). Show that lim sup b n = max L((x n )) n lim inf b n = min L((x n )). n 5. Suppose that (b n ) is a sequence tending to b and b 0. Let (x n ) be another sequence, not necessarily convergent. Show that lim sup (b n x n ) = b lim sup x n. n n What happens if b n = 1, a n = 2 when n is odd and a n = 3 when n is odd? 9.3 Hadamard s Test Theorem (Cauchy s root test) The infinite sum n=1 z n is 1. convergent if lim sup n z n 1 n < divergent if lim sup n z n 1 n > 1. Proof Let r = lim sup z n 1 n. n 1. Suppose that r < 1. Choose ε > 0 so that r + ε < 1. Then there exists N 0 such that whenever n > N 0, z n < (r + ε) n. The geometric series n=1 (r + ɛ)n is convergent. By comparison test, n=n 0 z n is convergent. And is convergent. z n = n=1 N 0 1 n=1 116 z n + n=n 0 z n

118 2. Suppose that r > 1. Choose ε so that r ε > 1. Then for infinitely many n, z n > (r ε) n > 1. Consequently z n 0 and n=1 z n cannot be convergent. Theorem (Hadamard s Test) Let n=0 a nx n be a power series. 1. If lim sup n a n 1 n =, the power series converges only at x = If lim sup n a n 1 n = 0, the power series converges for all x (, ). 3. Let r = lim sup n a n 1 n. If 0 < r <, the radius of convergence R of the power series is equal to 1/r. Proof We only prove case 3. If x < R, lim sup n a n x n 1 n = r x < 1 and by Cauchy s root test the series n=0 a nx n converges. Suppose that x > R, then lim sup n a n x n 1 n = r x > 1, and by Cauchy s root test, the series n=0 a nx n diverges. Example Consider ( 3) n x n n=0 n+1. We have ( ( 3) n ) 1 n lim n n + 1 = 3 lim n 1 (n + 1) 1 2n = 3. So R = 1 3. If x = 1/3, we have a divergent sequence. The interval of convergence is x = ( 1/3, 1/3]. Example Consider n=1 a nx n where a 2k = 2 2k and a 2k+1 = 5(3 2k+1 ). This means { 2 a n = n, n is even 5(3 n ), n is odd Thus which has two limit points: 2 and 3. a n 1 n = { 2, n is even 3(5 1 n ), n is odd lim sup a n 1 n = 3. and R = 1/3. For x = 1 3, x = 1 3, a nx n 0 and does not converge to 0 (check). So the interval of convergence is ( 1/3, 1/3)., 117

119 Example Consider n=1 a nx n where a 2k = ( 1 5 )2k and a 2k+1 = ( 1 3) 2k+1. lim sup a n 1 n = 1/3. So R = 3. For x = 3, x = 3, a n x n 1 and does not tend to 0. So the interval of convergence is ( 3, 3). Example Consider the series k=1 k2 x k2. We have { n, if n is the square of a natural number a n = 0, otherwise. Thus lim sup a n 1 n = 1 and R = 1. Does the series converge when x = ±1? Example Prove that the following power series is convergent on (, ). exp(x) = sinx = = cos x = n=0 n=0 n=0 x n n! = 1 + x + x2 2! + x3 3! +... ( 1) n (2n + 1)! x2n+1 = x x3 3! + x5 5!... ( 1) n (2n)! x 2n = 1 x2 2! + x4 4! x6 6! The use of the names exp, sin, cos will be justified later when we leaner Taylor series expansions of smooth functions. 9.4 Continuity of Power Series Functions Define f : ( R, R) R by f(x) = a n x n. n=0 The maximum domain of f contains all numbers x such that the infinite sum is a real number. For such x, let f N (x) = N n=0 a nx n. Then f(x) = lim N f N(x). 118

120 Each function f N is continuous and differentiable; can we say the same about f? In Analysis 3 we ll introduce uniform convergence and show that the limit of a sequence of continuous functions is continuous provided that the convergence is uniform. In this section we will consider continuity. Proposition Let f : ( R, R) R by f(x) = a n x n, n=0 where R is the radius of convergence of the power series. Then f is continuous at every x in ( R, R). Proof Fix x 0 ( R, R). Let r = 1 min(r x0, x0 + R), then [x0 r, x0 + r] ( R, R). 2 Let ρ = max{ x 0 r, x 0 + r }. Let x be such that x x 0 r, then x ρ < R. For ε > 0, there exists N(ɛ) such that if n > N(ɛ), the tail of the convergent series n=1 an ρn satisfies: a n ρ n < (1/3)ε Consequently For such x, n=n+1 f(x) f(x 0) = a n x n n=0 n=n(ɛ)+1 n=n(ɛ)+1 a nx n a nx n 0 a n ρ n < (1/3)ε x [x 0 r, x 0 + r]. n=0 (N(ɛ) N(ɛ) a nx n a nx n 0 + n=0 n=0 n=n(ɛ)+1 N (ɛ)a n(x n x n 0 ) + ε/3 + ε/3 n=0 a nx n + n=n(ɛ)+1 a nx n 0 The polynomial N(ɛ) n=0 anxn is continuous at x 0. There is δ > 0 such that if x x 0 < δ, N(ɛ) n=0 an(xn x n 0 ) < ɛ/3. Choose δ = min(δ, r). If x x 0 < δ, f(x) f(x 0) ε/3 + ε/3 + ε/3 = ε. The required continuity now follows. 119

121 9.5 Term by Term Differentiation If f N (x) = N n=0 a nx n = a 0 + a 1 x + + a N x N then f N(x) = N n=1 na n x n 1, f N(x) = N n(n 1)a n x n 2. n=2 Does it hold, if f(x) = n=0 a nx n, that ( f (x) d ) a n x n = na n x n 1 dx n=0 n=1 ( ) f (x) d2 dx 2 a n x n = n(n 1)a n x n 2? n=0 We are asking whether it is correct to differentiate a power series term by term. As preliminary question, we ask: if R is the radius of convergence of n=0 a nx n, what are the radii of convergence of the power series n=1 na nx n 1 and n=2 n(n 1)a nx n 2? Lemma The power series n=0 a nx n, n=1 na nx n 1 and n=2 n(n 1)a n x n 2 have the same radius of convergence. Proof Let R 1, R 2, R 3 be respectively the radii of convergence of the first, the second and the third series. Recall Hadamard s Theorem: if n=2 l = lim sup a n 1 n, n then R 1 = 1 l. We interpret 1 0 as and 1 as 0 to cover all three cases for the radius of convergence. Observe that ( ) x na n x n 1 = na n x n n=1 so n=1 na nx n 1 and n=1 na nx n have the same radius of convergence R 2. Note that lim sup na n 1 n = lim n 1 n lim sup a n 1 n = l. n n n By Hadamard s Theorem n=1 1 R 2 = = 1 lim sup n na n 1 n l. 120

122 Exercise: show that R 3 = 1 r. You may use the fact that lim (n 1) 1 n = 1. n Below we prove that a function defined by a power series can be differentiated term by term, and in the process show that f is differentiable on ( R, R). From this theorem it follows that f is continuous. This means that strictly speaking we did not need the separate proof of continuity that we gave in Theorem We gave this other proof because the idea of the proof will be used again in Analysis III, and therefore it is useful to become familiar with it. Theorem [Term by Term Differentiation] Let R be the radius of convergence of the power series n=0 a nx n. Let f : ( R, R) R be the function defined by this power series, f(x) = a n x n. n=0 Then f is differentiable at every point of ( R, R) and f (x) = na n x n 1. n=1 Proof Let x 0 ( R, R). We show that f is differentiable at x 0 and f (x 0) = n=1 nanxn 1 0. We wish to show that for any ε > 0 there is δ > 0 such that if x x 0 < δ, Since the expression simplifies: f(x) f(x 0) x x 0 na nx n 1 0 < ε. n=1 f(x) = a 0 + a 1x + a 2x f(x 0) = a 0 + a 1x 0 + a 2(x 0) 2, f(x) f(x 0) x x 0 na nx n 1 0 = a n(x n x n 0 ) na nx n 1 0 x x 0 n=1 n=1 ( ) x n x n 0 = a n nx n 1 0 x x 0 n=2 a n xn x n 0 nx n 1 0 x x 0. n=1 n=1. 121

123 By Lemma below, there is a δ 0 > 0 such that if x x 0 < δ 0, xn x n 0 nx n 1 0 x x 0 n(n 1)ρn 2 x x 0. Here ρ > 0 is a number such that ρ < R. By Lemma 9.5.1, the radius of convergence of n(n 1)x n is also R. Since ρ < R, n=2 n(n 1)ρn 2 is convergent and let ε A = n(n 1)ρ n 2 n=2 Finally, let δ = min{, δ0}. Then if 0 < x x0 < δ, A+1 f(x) f(x 0) na nx n 1 0 A x x0. x x 0 n=1 We proved that f (x 0) exists and is n=1 nanxn 1 0. Lemma Let x 0 R and let n N. If δ > 0 and 0 < x x 0 < δ, then xn x n 0 nx n 1 0 x x 0 n(n 1)ρn 2 x x 0, where ρ = max( x 0 δ, x 0 + δ ) can be chosen to be small than R. Proof Assume that x > x 0. Apply the Mean Value Theorem to f n(x) = x n to see that x n x n 0 x x 0 = nc n 1 n. for some c n (x 0, x). (Ifx < x 0 we apply the Mean Value Theorem on [x, x 0].) We next apply the mean value theorem again, this time to f n 1(x) = x n 1 on [x 0, c n] to obtain c n 1 n x n 1 0 = (n 1)(ξ n) n 2, c n x 0 for some ξ n (x 0, c n). We see that xn x n 0 nx n 1 0 x x 0 = n(n 1) ξ n n 2 c n x 0. Since ξ n (x 0, c n) (x 0, x), ξ n ρ and c n x 0 x x 0 we finally obtain that xn x n 0 nx n 1 0 x x 0 n(n 1)ρn 2 x x 0. How do we choose δ > 0 so that if ρ = max{ x : x [x 0 δ, x 0 + δ]}? We use the midpoint y 1 of [ R, x 0] and the midpoint y 2 of [x 0, R]. We then choose a sub-interval (x 0 δ 0, x 0 + δ 0) (y 1, y 2). 122

124 and x 0 R 2 x 0 +R 2 R y 1 x 0 y 2 R Let δ 0 := min{ 1 2 R x0, 1 x0 + R } 2 ρ := max{ 1 2 R x0, 1 x0 + R }. 2 Then ρ < R, as required. Since the derivative of f is a power series with the same radius of convergence, we apply Theorem to f to see that f is twice differentiable with the second derivative again a power series with the same radius of convergence. Iterating this procedure we obtain the following corollary. Corollary Let f : ( R, R) R, f(x) = a n x n, n=0 where R is the radius of convergence of the power series. C ( R, R). Then f is in Example The exponential function exp : R R is C. Example If x ( 1, 1), evaluate n=0 nxn. Solution. The power series n=0 xn has R = 1. But if x < 1 the geometric series has a limit and x n = 1 1 x. Hence n=0 By term by term differentiation, for x ( 1, 1), ( ) d x n = nx n 1. dx n=0 n=0 ( nx n = x d dx = x d dx n=0 n=1 ( 1 1 x x n ) ) = x (1 x)

125 Chapter 10 Classical Functions 10.1 The Exponential and the Natural Logarithm Function Consider the exponential function exp : R R, exp(x) = n=0 x n n! = 1 + x + x2 2! + x3 +..., x R. 3! Note that exp(0) = 1. By the term-by-term differentiation theorem 9.5.2, exp(x) = exp(x), and so exp is infinitely differentiable. d dx Proposition For all x, y R, exp(x + y) = exp(x) exp(y), exp( x) = 1 exp(x). Proof For any y R considered as a fixed number, let f(x) = exp(x + y) exp( x). Then f (x) = d d exp(x + y) exp( x) + exp(x + y) dx dx exp( x) = exp(x + y) exp( x) + exp(x + y)[ exp( x)] =

126 By the corollary to the Mean Value Theorem, f(x) is a constant. f(0) = exp(y), f(x) = exp(y), i.e. Since exp(x + y) exp( x) = exp(y). Take y = 0, we have exp(x + 0) exp( x) = exp(0) = 1. So and exp( x) = 1 exp(x) 1 exp(x + y) = exp(y) = exp(y) exp(x). exp( x) A neat argument, no? Proposition exp is the only solution to the ordinary differential equation (ODE) { f (x) = f(x) f(0) = 1. Proof Since d dx exp(x) = exp(x) and exp(0) = 1, the exponential function is one solution of the ODE. Let f(x) be any solution. Define g(x) = f(x) exp( x).then g (x) = f (x) exp( x) + f(x) d dx [exp( x)] = f(x) exp( x) f(x) exp( x) = 0. Hence for all x, g(x) = g(0) = f(0) exp(0) = 1 1 = 1. Thus f(x) exp( x) = 1 and any solution f(x) must be equal to exp(x). It is obvious from the power series that exp(x) > 0 for all x 0. Since exp( x) = 1/ exp(x), it follows that exp(x) > 0 for all x R. Exercise Prove that the range of exp is all of R >0. Hint: If exp(x) > 1 for some x then exp(x n ) can be made as large, or as small, as you wish, by suitable choice of n Z. 125

127 2. Show that exp : R R >0 is a bijection. We would like to say next: Proposition For all x R, where e = n=0 1 n!. exp(x) = e x But what does e x mean? We all know that e 2 means e e, but what about e π? Operations of this kind are not part of nature - we need to define them. It is easy to extend the simplest definition, that of raising a number to an integer power, to define what it means to raise a number to a rational power: first, for each n N {0} we define an n th root function n : R 0 R as the inverse function of the (strictly increasing, infinitely differentiable) function x x n. Then for any m/n Q, we set x m/n = ( n x ) m, where we assume, as we may, that n > 0. But this does not tell us what e π is. We could try approximation: choose a sequence of rational numbers x n tending to π, and define e π = lim n exn. We would have to show that the result is independent of the choice of sequence x n (i.e. depends only on its limit). This can be done. But then proving that the function f(x) = a x is differentiable, and finding its derivative, are rather hard. There is a much more elegant approach. First, we define ln : R >0 R as the inverse to the function exp : R R >0, which we know to be injective since its derivative is everywhere strictly positive, and surjective by Exercise Then we make the following definition: Definition For any x R and a R >0, a x := exp(x ln a). 126

128 Before anything else we should check that this agrees with the original definition of a x where it applies, i.e. where x Q. This is easy: because ln is the inverse of exp, and (by Proposition ) exp turns addition into multiplication, it follows that ln turns multiplication into addition: ln(a b) = ln a + ln b, from which we easily deduce that ln(a m ) = m ln a (for m N) and then that ln(a m/n ) = m/n ln a for m, n Z. Thus exp( m n ln a) = exp(ln(am/n ) = a m/n, the last equality because exp and ln are mutually inverse. We have given a meaning to e x, and we have shown that when x Q this new meaning coincides with the old meaning. Now that Proposition is meaningful, we will prove it. Proof In the light of Definition , Proposition reduces to exp(x) = exp(x ln e). (10.1.1) But ln e = 1, since exp(1) = e; thus (10.1.1) is obvious! Exercise Show that Definition , and the definition of a x sketched in the paragraph preceding Definition agree with one another. Proposition The natural logarithm function ln : (0, ) R is differentiable and ln (x) = 1 x. Proof Since the differentiable bijective map exp(x) has exp (x) 0 for all x, the differentiability of its inverse follows from the Inverse Function Theorem. And ln 1 (x) = exp = (y) y=ln(x) e x ln x x 1 exp(ln(x)) = 1 x. ln x 127

129 Remark: In some computer programs, eg. gnuplot, x 1 n is defined as following, x 1 n = exp(ln(x 1 n )) = exp( 1 n ln x). Note that ln(x) is defined only for x > 0. This is the reason that typing in ( 2) 1 3 returns an error message: it is not defined! 10.2 The Sine and Cosine Functions We have defined sin x = x x3 3! + x5 5! = ( 1) n (2n + 1)! x2n+1 n=0 cos x = 1 x2 2! + x4 4! x6 6! + = n=0 ( 1) n (2n)! Term by term differentiation shows that sin (x) = cos(x) and cos (x) = sin(x). Exercise For a fixed y R, put f(x) = sin(x + y) sin x cos y cos x sin y. Compute f (x) and f (x). Then let E(x) = (f(x)) 2 + (f (x)) 2. What is E? Apply the Mean Value Theorem to E, and hence prove the addition formulae sin(x + y) = sin x cos y + cos x sin y cos(x + y) = cos x cos y sin x sin y. The sine and cosine functions defined by means of power series behave very well, and it is gratifying to be able to prove things about them so easily. But what relation do they bear to the sine and cosine functions defined in elementary geometry? x 2n elementary trigonometry sin θ= o/h h o cosin θ= a/h coordinate geometry. θ (x,y) 1 sin θ= y cosin θ= x θ a 128

130 Chapter 11 Taylor s Theorem and Taylor Series We have seen that some important functions can be defined by means of power series, and that any such function is infinitely differentiable. If we are given an infinitely differentiable function, does there always exist a power series which converges to it? And if such a power series exists, how can we determine its coefficients? We approach an answer to the question by first looking at polynomials, i.e. at finite power series, and proving Taylor s Theorem. This is concerned with the following problem. Suppose that the function f is n times differentiable in the interval (a, b), and that x 0 (a, b). If we know the values of f and its first n derivatives at x 0, how much can we say about the behaviour of f in the rest of (a, b)? It turns out (and is easy to prove) that there is a unique polynomial P of degree n such that f(x 0 ) = P (x 0 ), f (x 0 ) = P (x 0 ),..., f (n) (x 0 ) = P (n) (x 0 ). Taylor s theorem gives a way of estimating the difference between f(x) and P (x) at other points x (a, b). Proposition Let f : (a, b) R be n times differentiable and let x 0 (a, b). Then there is a unique polynomial of degree n, P n (x), such that f(x 0 ) = P n (x 0 ), f (x 0 ) = P n(x 0 ),..., f (n) (x 0 ) = P (n) n (x 0 ), (11.0.1) 129

131 namely P n (x) = f(x 0 ) + f (x 0 )(x x 0 ) + f (2) (x 0 ) 2! n = k=0 (x x 0 ) f (n) (x 0 ) (x x 0 ) n n! f (k) (x 0 ) (x x 0 ) k. (11.0.2) k! Proof Evaluate the polynomial P n at x 0 ; it is clear that it satisfies P n (x 0 ) = f(x 0 ). Now differentiate and again evaluate at x 0. It is clear that P n(x 0 ) = f (x 0 ). By repeatedly differentiating and evaluating at x 0, we see that P n satisfies (11.0.1). The polynomial P n (x) = n f (k) (x 0 )(x x 0 ) k k=0 is called the degree n Taylor polynomial of f about x 0. Warning P n (x) depends on the choice of point x 0. Consider the function f(x) = sin x. Let us determine the polynomial P 2 (x) for two different values of x Taking x 0 = 0 we get 2. Taking x 0 = π/2 we get P 2 (x) = sin(0) + sin (0) x + sin (0) x 2 = x. 2 k! P 2 (x) = sin(π/2)+sin (π/2)(x π/2)+ sin (π/2) (x π/2)x 2 = (x π/2)2. Questions: 1. Let R n (x) = f(x) P n (x) be the error term. It measures how well the polynomial P n approximates the value of f. How large is the error term? Taylor s Theorem is concerned with estimating the value of the error R n (x). 130

132 2. Is it true that lim n R n (x) = 0? If so, then n f(x) = lim P f (k) (x 0 ) n(x) = lim (x x 0 ) k = n n k! k=0 and we have a power series expansion for f. k=0 f (k) (x 0 ) (x x 0 ) k. k! 3. Does every infinitely differentiable function have a power series expansion? Definition If f is C on its domain, the infinite series f (k) (x 0 )(x x 0 ) k k=0 is called the Taylor series for f about x 0, or around x 0. Does the Taylor series of f about x 0 converge on some neighbourhood of x 0? If so, it defines an infinitely differentiable function on this neighbourhood. Is this function equal to f? The answers to both questions are yes for some functions and no for some others. Definition If f is C on its domain (a, b) and x 0 (a, b), we say Taylor s formula holds for f near x 0 if for all x in some neighbourhood of x 0, f (k) (x 0 )(x x 0 ) k f(x) =. k! k=0 The following is an example of a C function whose Taylor series converges, but does not converge to f: Example (Cauchy s Example (1823)) { exp( 1 ), x 0 f(x) = x 2 0 x = 0. It is easy to see that if x 0 then f(x) 0. Below we sketch a proof that f (k) (0) = 0 for all k. The Taylor series for f about 0 is therefore: k=0 k! 0 k! xk = This Taylor series converges everywhere, obviously, and the function it defines is the zero function. So it does not converge to f(x) unless x =

133 How do we show that f (k) (0) = 0 for all k? Let y = 1 x, f +(0) exp( 1 ) x = lim 2 y = lim x 0+ x y + exp(y 2 ). y exp(y 2 ) = 0 and lim y Since exp(y 2 ) y 2, lim y + that f +(0) = f (0) = 0. A similar argument gives the conclusion for k > 1. needed. y = 0. It follows exp(y 2 ) An induction is Exercise If Q = a 0 +a 1 + +a m x m is a polynomial of degree m, m = 0, 1, 2,..., show that lim y + Q(y) exp(y 2 ) = Show that Q( 1 x lim ) exp( 1 ) x 2 = 0. x 0 x 3. Compute the derivative f (k) (x) for x 0, for k = 1, 2, 3. For example, f (x) = 2x 3 exp( 1 x 2 ) f (x) = [ 2x 3 ] 2 exp( 1 x 2 ) + 3!x 4 exp( 1 x 2 ) f (x) = [ 2x 3 ] 3 exp( 1 x 2 ) 4!x 5 exp( 1 x 2 ) 2(3!)x 6 exp( 1 x 2 ). By induction show that for x 0, f (k) (x) = P (x) Q(x) exp( 1 x 2 ) for some polynomials P (x), Q(x). Note that f (k+1) f (k) (x) Q k ( 1 x (0) = lim = lim ) exp( 1 ) x 2. x 0 x x 0 x 4. Show that f (k+1) (0) = 0 for all k. 132

134 Remark By Taylor s Theorem below, if R n (x) = xn+1 (n + 1)! f (n+1) (ξ n ) 0 where ξ n is between 0 and x, Taylor s formula holds. It follows that, for the function above, either lim n R n (x) does not exist, or, if it exists, it cannot be 0. Convince yourself of this (without rigorous proof) by observing that Q n+1 (y) contains a term of the form ( 1) n+1 (n + 2)!y (n+2). And indeed may not converge to 0 as n. x n+1 (n + 1)! ( 1)n+1 (n + 2)!ξn n 2 Definition * A C function f : (a, b) R is said to be real analytic if for each x 0 (a, b), its Taylor series about x 0 converges, and converges to f, in some neighbourhood of x 0. The preceding example shows that the function e 1/x2 is not real analytic in any interval containing 0, even though it is infinitely differentiable. Complex analytic functions are studied in the third year Complex Analysis course. Surprisingly, every complex differentiable function is complex analytic. Example Find the Taylor series for exp(x) = k=0 xk k! about x 0 = 1. For which x does the Taylor series converge? For which x does Taylor s formula hold? Solution. For all k, exp (k) (1) = e 1 = e. So the Taylor series for exp about 1 is exp (k) (1)(x 1) k e(x 1) k (x 1) k = = e. k! k! k! k=0 k=0 The radius of convergence of this series is +. Hence the Taylor series converges everywhere. Furthermore, since exp(x) = k=0 xk k!, it follows that (x 1) k = exp(x 1) k! k=0 k=0 and thus (x 1) k e = e exp(x 1) = exp(1) exp(x 1) = exp(x) k! and Taylor s formula holds. 133 k=0

135 Exercise Show that exp is real analytic on all of R Non-centered Power Series Definition If x 0 R, a formal power series centred at x 0 is an expression of the form: a n (x x 0 ) n = a 0 + a 1 (x x 0 ) + a 2 (x x 0 ) 2 + a 3 (x x 0 ) n=0 Example Take n=0 1 3 (x 2) n. If x = 1, n ( 1) n 1 n=0 is a convergent series. 1 3 n (x 2)n = n=0 1 3 n (1 2)n = What we learnt about power series n=0 a nx n can be applied here. For example, 1. There is a radius of convergence R such that the series n=0 a n(x x 0 ) n converges when x x 0 < R, and diverges when x x 0 > R. n=0 2. R can be determined by the Hadamard Theorem: 3 n R = 1 r, r = lim sup a n 1 n. n 3. The open interval of convergence for n=0 a n(x x 0 ) n is The functions f(x) = are related by (x 0 R, x 0 + R). a n (x x 0 ) n and g(x) = n=0 f(x) = g(x x 0 ). a n x n 4. The term by term differentiation theorem holds for x in (x 0 R, x 0 +R) and f (x) = na n (x x 0 ) n 1 = a 1 + 2a 2 (x x 0 ) + 3a 3 (x x 0 ) n=0 n=0 134

136 Example Consider n=0 1 3 n (x 2) n. Then lim sup( 1 3 n ) 1 1 n =, so R = 3. 3 The power series converges for all x with x 2 < 3. For x ( 1, 5), f(x) = n=0 1 3 (x 2) n is differentiable and n f (x) = n=0 n 1 3 n (x 2)n Uniqueness of Power Series Expansion Example What is the Taylor s series for sin(x) about x = 0? Solution. sin(0) = 0, sin (0) = cos 0 = 1, sin (2) (0) = sin 0 = 0, cos (3) (0) = cos 0 = 1, By induction, sin (2n) (0) = 0, sin (2n+1) = ( 1) n. Hence the Taylor series of sin(x) at 0 is ( 1) k 1 (2k + 1)! x2k+1. k=0 This series converges everywhere and equals sin(x). The following proposition shows that we would have needed no computation to compute the Taylor series of sin x at 0: Proposition (Uniqueness of Power Series expansion) Let x 0 (a, b). Suppose that f : (a, b) (x 0 R, x 0 + R) R is such that f(x) = a n (x x 0 ) n, n=0 where R is the radius of convergence. Then Proof a 0 = f(x 0 ), a 1 = f (x 0 ),..., a k = f (k) (x 0 ),.... k! Take x = x 0 in f(x) = a n (x x 0 ) n = a 0 + a 1 (x x 0 ) + a 2 (x x 0 ) , n=0 135

137 it is clear that f(x 0 ) = a 0. By term by term differentiation, f (x) = a 1 + 2a 2 (x x 0 ) + 3a 3 (x x 0 ) Take x = x 0 to see that f (x 0 ) = a 1. Iterating this argument, we get f (k) (k + 2)! (x) = a k k!+a k+1 (k + 1)!(x x 0 )+a k+2 (x x 0 ) 2 (k + 3)! +a k+3 (x x 0 ) ! 3! Take x = x 0 to see that a k = f (k) (x 0 ). k! Moral of the proposition: if the function is already in the form of a power series centred at x 0, this is the Taylor series of f about x 0. Example What is the Taylor series of f(x) = x 2 + x about x 0 = 2? Solution 1. f(2) = 6, f (2) = 5, f (2) = 2. Since f (x) = 0 for all x, the Taylor series about x 0 = 2 is (x 2) + 2 2! (x 2) = 6 + 5(x 2) + (x 2)2. Solution 2. On the other hand, we may re-write f(x) = x 2 + x = (x 2 + 2) 2 + (x 2 + 2) = (x 2) 2 + 5(x 2) + 6. The Taylor series for f at x 0 = 2 is (x 2) 2 + 5(x 2) + 6. The Taylor series is identical to f and so Taylor s formula holds. Example Find the Taylor series for f : (0, ) R, f(x) = ln(x), about x 0 = 1. We compute the derivatives: f (x) = 1 x, f (x) = 1 x 2, f (3) (x) = 2 1 x 3. Inducting on k to see that f (k) (x) = ( 1) k+1 (k 1)! 1 x k, f (k) (1) = ( 1) (k+1) (k 1)!. Hence the Taylor series is: f (k) (1) (x 1) k = k! k k=0 k=1 ( 1) (k+1) (x 1) k = (x 1) 1 2 (x 1) (x 1)3.... It has radius of convergence R = 1. Question: Does Taylor s formula hold? i.e. is it true that ( 1) (k+1) ln(x) = (x 1) k? k k=1 136

138 11.3 Taylor s Theorem with remainder in Lagrange form Given knowledge on f at x 0, Taylor s original idea is to approximate the value of f(x) by the value at x of the degree n Taylor polynomial about x 0. Let P n,x0 (x) = n k=0 R n (x) = f(x) f (k) (x 0 ) (x x 0 ) k k! n k=0 f (k) (x 0 ) (x x 0 ) k k! be the error term. We assume that f is C. If for some x, lim n R n (x) = 0 then lim n n k=0 f (k) (x 0 ) (x x 0 ) k k! exists and equals f(x). This means, precisely, that the infinite series is convergent and k=0 k=0 f (k) (x 0 ) (x x 0 ) k = lim k! n f (k) (x 0 ) (x x 0 ) k k! n k=0 On the other hand if Taylor s formula holds then f(x) = R n (x) = k=0 n=k+1 f (k) (x 0 ) (x x 0 ) k = f(x). k! f (k) (x 0 ) (x x 0 ) k k! f (k) (x 0 ) (x x 0 ) k. k! Since x is in the domain of convergence of the Taylor series, lim n R n (x) = 0. The following is a theorem of Lagrange (1797). 137

139 Theorem [Taylor s theorem with Lagrange Remainder Form] 1. Let x > x 0. Suppose that f is n times continuously differentiable on [x 0, x] and n + 1 times differentiable on (x 0, x). Then where some ξ (x 0, x). f(x) = n k=0 f (k) (x 0 ) (x x 0 ) k + R n (x) k! R n (x) = (x x 0) n+1 f (n+1) (ξ) (n + 1)! 2. The conclusion holds also for x < x 0, if f is (n+1) times continuously differentiable on [x, x 0 ] and n + 1 times differentiable on (x, x 0 ). Proof Let us vary the starting point of the interval and consider [y, x] for any y [x 0, x]. We will regard x as fixed (it does not move!). We define a function h : [x 0, x] R by h(y) = f(x) n f (k) (y)(x y) k R n (x) k! (x x 0 ) n+1 (x y)n+1. k=0 The middle term if the degree n Taylor polynomial about y. We note the following: h is continuous on [x 0, x]. This statement follows from the continuity of f (k), k n and that power functions. ) h is differentiable on (x 0, x). (This statement follows from the differentiability of f (k), k n, and the power functions. ) h(x 0 ) = h(x) = 0. By Rolle s Theorem, applied to h on [x 0, x], there exists ξ (x 0, x) such that h (ξ) = 0. The derivative of h can be calculated (exercise): h (y) = f (n+1) (y) (x y) n R n (x) + n! (x x 0 ) n+1 (n + 1)(x y)n. 138

140 Replace ξ in the above and use h (ξ) = 0, h (ξ) = f (n+1) (ξ) (x ξ) n R n (x) + n! (x x 0 ) n+1 (n + 1)(x ξ)n = 0. Since x 0 we may divide the expression by (x ξ) n and the formula for R n (x) is obtained. Remark Take n = 0. Taylor s theorem gives the Mean Value Theorem: if f is continuous on [x 0, x] and differentiable on (x 0, x), then f(x) = f(x 0 ) + f (ξ)(x x 0 ) for some ξ (x x 0 ). Take n = 1, Taylor s theorem gives the error, R 1 (x) = f (ξ) (x x 0) 2 2, for the tangent line approximation: f(x) = f(x 0 ) + f (x 0 )(x x 0 ) + R 1 (x). Corollary Let x 0 (a, b). If f is C on [a, b] and the derivatives of f are uniformly bounded on (a, b), i.e. there exists some K such that for all k and for all x [a, b], f (n) (x) K then Taylor s formula holds, i.e. for each x, x 0 [a, b] f(x) = f (k) (x 0 )(x x 0 ) k, x [a, b]. k! k=0 In other words f has a Taylor series expansion at every point of (a, b). In yet other words, f is real analytic in [a, b]. Proof For each x [a, b], R n (x) = (x x 0 ) n+1 f n+1 (ξ) (n + 1)! K x x 0 n+1 (n + 1)! and this tends to 0 as n. This means n f (k) (x 0 )(x x 0 ) k lim = f(x) n k! k=0 139

141 as required. Now we can answer the question we posed at the end of Section 10. Consider the sine and cosine functions defined using elementary geometry. We have seen in Example that cosine is the derivative of sine and the derivative of cosine is minus sine. Since both sine and cosine are bounded by 1 on all of R, both sine and cosine satisfy the conditions of Corollary , with K = 1. It follows that both sine and cosine are analytic on all of R. Power series 2 and 3 of Definition?? are the Taylor series of sine and cosine about x 0 = 0. Corollary tells us that they converge to sine and cosine. So the definition by means of power series gives the same function as the definition in elementary geometry. Just in case you think that elementary geometry is for children and that grown-ups prefer power series, try proving, directly from the power series definition, that sine and cosine are periodic with period 2π. Example Compute sin(1) to the precision Solution: For some ξ (0, 1), sin (n+1) (ξ) R n (1) = 1 n+1 1 (n + 1)! (n + 1)!. We wish to have 1 (n + 1)! = Take n such that n! Take n = 7, 7! = 5040 is sufficient. Then sin(1) [x x3 3! + x5 5! x7 7! ] x=1 = 1 1 3! + 1 5! 1 7!. Now use your calculator if needed. cos x 1 Example Show that lim x 0 x = 0. By Taylor s Theorem cos x 1 = x2 2 + R 3(x). For x 0 we may assume that x [ 1, 1]. Then for some c ( 1, 1), It follows that as x 0. R 3 (x) = cos(3) (c) 3! x 3 = sin(c) x 3 x 3. 3! cos x 1 x x2 /2 + x 3 = x/2 + x 2 0, x 140

142 How do we prove that lim n r n n! = 0 for any r > 0? Note that n=0 convergenes by the Ratio Test, and so its individual terms rn n! must converge to 0. Example We can determine the Taylor series for e x about x 0 = 0 d using only the facts that dx ex = e x and e 0 = 1. For r n n! d 2 dx 2 ex = d ( ) d dx dx ex d n = d dx ex = e x, and so, inductively, dx e x = e x for all n N. Hence, evaluating these n derivatives at x = 0, all take the value 1, and the Taylor series is k=0 x k k!. The Taylor series has radius of convergence. For any x R, there is ξ ( x, x) s.t. R n (x) = x n+1 (n + 1)! eξ max(ex, e x x n+1 ) (n + 1)! 0 as n. So for all x. e x = k=0 xk k! = exp(x) Example The Taylor series for ln(1 + x) about 0 is k=1 with R = 1. Also for x ( 1, 1), ln(1 + x) = ( 1) (k+1) x k k k=1 ( 1) (k+1) k x k? 141

143 and Solution. Let f(x) = ln(1 + x), then f (n) n+1 (n 1)! (x) = ( 1) (1 + x) n. R n (x) = = = If 0 < x < 1, then ξ (0, x) and f (n+1) (ξ) (n + 1)! xn+1 ( 1) n+2 n! (n + 1)!(1 + ξ) 1 x (n + 1) 1 + ξ x 1+ξ < 1. n+1 xn+1 n+1 lim R 1 n(x) lim n n (n + 1) = 0. *For the case of 1 < x < 0, we use the integral form of the remainder term R n (x). Since and x < 0, Since t x 1+t d dt f (n+1) (x) = ( 1) n 1 n! (1 + x) n+1, x (x t) n R n (x) = f (n+1) (t)dt 0 n! x = ( 1) n (x t) n 1 dt (1 + t) n+1 = 0 x 0 (t x) n 1 dt. (1 + t) n+1 ( ) t x (1 + t) (t x) = 1 + t (1 + t) 2 = is an increasing function in t and t x max x t t = 0 x = x > (1 + x) (1 + t) 2 > 0,

144 Note that x > 0. t x min x t t = x x 1 + x = 0. R n (x) = 0 x 0 (x t) n 1 dt (1 + t) n+1 ( x) n x (1 + t) dt = ( x) n [ ln(1 + x)]. Since 0 < x + 1 < 1, ln(1 + x) < 0 and 0 < x < 1. It follows that lim R n(x) lim n n ( x)n [ ln(1 + x)] = 0. Thus Taylor s formula hold for x ( 1, 1) Taylor s Theorem in Integral Form* This section is not included in the lectures nor in the exam for this module. The remainder term R n(x) in the integral form is the best formulation. But we need to know something of the Theory of Integration (Analysis III). With what we learnt in school and a little bit flexibility we should be able to digest the information. Please read on. Theorem (Taylor s Theorem with remainder in integral form*) If f is C n+1 on [a, x], then Thus Proof f(x) = n k=0 f (k) (a) x (x a) k + k a R n(x) = x a (x t) n f (n+1) (t)dt. n! (x t) n f (n+1) (t)dt. n! By the fundamental Theorem of calculus from Analysis III: Integration by parts gives x a f(x) = f(a) + x x a f (t)dt. (11.4.1) f (t)dt = f (t) d (x t)dt a dt = [ f (t)(x t) ] x x + (x t)f (t)dt a = f (a)(x a) + x a a (x t)f (t)dt. 143

145 So by (11.4.1), f(x) = f(a) + f (a)(x a) + x a (x t)f (t)dt. This proves the theorem when n = 1. Induction on n proves the case for all n: suppose that x (x t) n f(x) = P n(x) + f (n+1) (t)dt. a n! Then, using the same trick as in the case where n = 1, we have ] x f(x) = P n(x) + [ f (n+1) (x x t)n+1 (t) + f (n+2) (x t)n+1 (t) dt (n + 1)! (n + 1)! = P n+1(x) + x a f (n+2) (t) = P n+1(x) + R n+1(x). a (x t)n+1 dt (n + 1)! a Lemma (Intermediate Value Theorem in Integral Form*) If g is continuous on [a, b] and f is positive everywhere and integrable, then for some c (a, b), b a f(x)g(x)dx = g(c) b a f(x)dx. Proof By the extreme value theorem, there are numbers x 1, x 2 [a, b] such that m = g(x 1), M = g(x 2) and m g(x) M. Since f(x) 0, multiplication by f(x) preserves the order: mf(x) f(x)g(x) Mf(x). Integrate each term in the above inequality from a to b: m b a f(x)dx b a f(x)g(x)dx M b a f(x)dx If b f(x)dx = 0, the theorem is proved. Otherwise b f(x)dx >= 0, and a a 1 b m b f(x)dx f(x)g(x)dx M. a By the Intermediate Value Theorem, applied to g, there exists c [x 1, x 2] such that a 1 b g(c) = b f(x)dx f(x)g(x)dx. a a Remark It is necessary for the lemma that f does not change sign. For example if f(x) = sin(x), x [0, 2π], 2π sin(x)dx = 0, but 2π (sin x)(sin x)dx = 1 2π ( cos(2x))dx

146 By this lemma, the integral remainder term in Theorem can take the folllowing form: x (x t) n R n(x) = f (n+1) (t)dt a n! x = f (n+1) (x t) n (c) dt n! = f (n+1) (c) (n + 1)! (x a)(n+1). This is the Lagrange form of the remainder term R n(x) in Taylor s Theorem, as shown in below. We have just deduced Theorem from Theorem In the next section which we will prove it again by the Mean Value Theorem, without borrowing theorems from the theory of integration. The Cauchy form for the remainder R n(x) is R n = f (n+1) (c) This can be obtained in the same way: x Other integral techniques: a (x c)n (x a). n! (x t) n x R n(x) = f (n+1) (t)dt = f (n+1) (c)(x c) n 1 a n! a n! dt = f (n+1) (c)(x c) n x a. n! Example For x < 1, by term by term integration (proved in second year modules) x dy x x arctan x = 1 + y = ( y 2 ) n dy = ( y 2 ) n dy 2 = 0 0 ( 1) n y 2n+1 2n + 1 n=0 n=0 y=x y=0 = n=0 0 ( 1) n x 2n+1 2n + 1. n= Cauchy s Mean Value Theorem Lemma (Cauchy s Mean Value Theorem) If f and g are continuous on [a, b] and differentiable on (a, b) then there exists c (a, b) with If g (c) 0 g(b) g(a), then (f(b) f(a))g (c) = f (c)(g(b) g(a)). (11.4.2) f(b) f(a) g(b) g(a) = f (c) g (c). 145

147 Proof Set h(x) = g(x)(f(b) f(a)) f(x)(g(b) g(a)). Then h(a) = h(b), h is continuous on [a, b] and differentiable on (a, b). By Rolle s Theorem, there is a point c (a, b) such that 0 = h (c) = g (c)(f(b) f(a)) f (c)(g(b) g(a)), hence the conclusion. Remark Analytic intuition: given functions f and g, find a function h of the form h(x) = αf(x) + βg(x) such that h(a) = h(b). Then β f(a) f(b) = α g(b) f(a). Geometric Interpretation: The function γ : [a, b] R 2 parameterises a curve C in the plane. Proposition Let L be the line through the points γ(a) = (f(a), g(a)) and γ(b) = (f(b), g(b)). Then there is a point c (a, b) such that the tangent to the curve C at γ(c) is parallel to L. L γ( b) γ( a) γ( c) Proof The tangent to C at γ(c) has direction γ (c), which is equal to (f (c), g (c)). The line L has direction (f(b) f(a), g(b) g(a)). The two are parallel if and only if det f (c) f(b) f(a) g (c) g(b) g(a) =

148 This equation is equivalent to (11.4.2). Using Cauchy s Mean Value Theorem we now give a third proof of Taylor s Theorem (Theorem ) Proof We compare g n (x) := (x x 0) n+1 (n + 1)! with the remainder term R n (x) = f(x) n k=0 f (k) (x 0 ) (x x 0 ) k. k! The first n derivatives of g n (x) obviously vanish at x 0, and g n (n+1) (x 0 ) = 1. Let n f (k) (x 0 ) P n (x) = (x x 0 ) k, k! then This gives k=0 P (k) n (x 0 ) = f (k) (x 0 ) for 0 k n, P (n+1) n (x) = 0 for all x. R (k) n (x 0 ) = 0 for 0 k n, R (n+1) n (x) = f (n+1) (x) for all x. Now apply Cauchy s Mean Value Theorem to R n, g n on [x 0, x] to obtain ξ 1 (x 0, x) such that R n (x) g n (x) = R n(ξ 1 ) g n(ξ 1 ) (we are using the fact that R n (x 0 ) = g n (x 0 ) = 0). Apply the Cauchy MVT again to R n, g n on [x 0, ξ 1 ] to obtain ξ 2 (x 0, ξ 1 ) such that R n(ξ 1 ) g n(ξ 1 ) = R n(ξ 2 ) g n(ξ 2 ), apply it to R n and g n on [x 0, ξ 2 ] to obtain ξ 3 (x 0, ξ 2 ) such that and so on, arriving eventually at R n(ξ 2 ) g n(ξ 2 ) = R (3) n (ξ 3 ) g n (3) (ξ 3 ) R n (n) (ξ n ) (ξ n ) = R g (n) n (n+1) n (ξ n+1 ) g (n+1) 147 n (ξ n+1 )

149 (each time, at the k th step we have used the fact that R n (k) (x 0 ) = g n (k) (x 0 ) = 0). Thus we have R n (x) g n (x) = R n(ξ 1 ) g n(ξ 1 ) = R(2) n (ξ 2 ) g (2) (ξ 2 ) = = R n = f (n+1) (ξ n+1 ). where ξ k (x 0, ξ k 1 ), k = 1,..., n + 1. Taking ξ = ξ n+1, (n+1) n (ξ n+1 ) g (n+1) n (ξ n+1 ) R n (x) = g n (x)f (n+1) (ξ) = (x x 0) n+1 f (n+1) (ξ). (n + 1)! 11.5 A Table Table of standard Taylor expansions: 1 1 x = x n = 1 + x + x 2 + x , x < 1 e x = log(1 + x) = sin x = cos x = arctan x = n=0 x n n! = 1 + x + x2 2! + x3 3! +..., x R ( 1) n xn+1 n + 1 = x x2 2 + x3 3 x , 1 < x 1 ( 1) n x2n+1 (2n + 1)! = x x3 3! + x5 5!..., x R ( 1) n x2n (2n)! = 1 x2 2! + x4 4! x6 6! +..., x R ( 1) n x2n+1 (2n + 1) = x x3 3 + x5..., 5 x R, x 1 n=0 n=0 n=0 n=0 n=0 Stirling s Formula: n! = ( n ) n a 1 2πn (1 + e n + a 2 n ). 148

150 Chapter 12 Techniques for Evaluating Limits Let c be an extended real number (i.e. c R or c = ± ). Suppose that lim x c f(x) and lim x c g(x) are both equal to 0, or both equal to, or both equal to. What can we say about lim x c f(x)/g(x)? Which function tend to 0 (respectively ± ) faster? We cover two techniques for answering this kind of question: Taylor s Theorem and L Hôpital s Theorem Use of Taylor s Theorem How do we compute limits using Taylor expansions? Let r < s be two real numbers and a (r, s), and suppose that f, g C n ([r, s]). Suppose f and g are n + 1 times differentiable on (r, s). Assume that f(a) = f (1) (a) =... f (n 1) (a) = 0 g(a) = g (1) (a) =... g (n 1) (a) = 0. Suppose that f (n) (a) = a n, g (n) (a) = b n 0. By Taylor s Theorem, Here f(x) = a n n! (x a)n + R f n(x) g(x) = b n n! (x a)n + R g n(x). R f n(x) = f (n+1) (ξ f ) (x a)n+1, R g (n + 1)! n(x) = g (n+1) (x a)n+1 (ξ g ) (n + 1)! 149

151 for some ξ f and ξ g between a and x. Theorem Suppose that f, g C n+1 ([r, s]), a (r, s). Suppose that Suppose that g (n) (a) 0. Then f(a) = f (1) (a) =... f (n 1) (a) = 0, g(a) = g (1) (a) =... g (n 1) (a) = 0. f(x) lim x a g(x) = f (n) (a) g (n) (a). Proof Here Write g (n) (a) = b n and f (n) (a) = a n. By Taylor s Theorem, f(x) = a n n! (x a)n + R f n(x) g(x) = b n n! (x a)n + R g n(x). R f n(x) = f (n+1) (ξ f ) (x a)n+1, R g (n + 1)! n(x) = g (n+1) (x a)n+1 (ξ g ), (n + 1)! where ξ f, ξ g [r, s]. If f (n+1) and g (n+1) are continuous, they are bounded on [r, s] (by the Extreme Value Theorem). Then lim f (n+1) (x a) (ξ) x a (n + 1) (x a) lim x a g(n+1) (ξ) (n + 1) = 0, = 0. So f(x) lim x a g(x) = lim x a a n n! (x a) n + R f n(x) b n n! (x a) n + R g n(x) (x a) (n+1) a n + f (n+1) (ξ) = lim x a b n + g (n+1) (ξ) (x a) = a n b n. (n+1) 150

152 Example Since cos x 1 = x L Hôpital s Rule cos x 1 lim = 0 x 0 x + = 0x x cos x 1 lim = 0 x 0 x 2 = 0. f Question: Suppose that lim (x) x c g (x) = l where l R {± }. Suppose that lim x c f(x) and lim x c g(x) are both equal to 0, or to ±. Can we f(x) say something about lim x c g(x)? We call limits where both f and g tend to 0, 0 0 -type limits, and limits where both f and g tend to ± -type limits. We will deal with both and - type limits and would like to conclude in both cases that f(x) lim x c g(x) = lim f (x) x c g (x) = l. Theorem (A simple L Hôpital s rule) Let x 0 (a, b). Suppose that f, g are differentiable on (a, b). Suppose lim x x0 f(x) = lim x x0 g(x) = 0 and g (x 0 ) 0. Then f(x) lim x x 0 g(x) = f (x 0 ) g (x 0 ) Proof Since g (x 0 ) 0, there is an interval (x 0 r, x 0 + r) {x 0 } (a, b), some positive number r, on which g(x) g(x 0 ) 0. See Proposition for a proof for the case of f (x 0 ) > 0. The case of f (x 0 ) < 0 can be proved similarly. By the algebra of limits, for x (x 0 r, x 0 + r), f(x) lim x x 0 g(x) = lim x x 0 f(x) f(x 0 ) x x 0 = g(x) g(x 0 ) x x 0 f(x) f(x lim 0 ) x x0 x x 0 = f (x 0 ) g(x) g(x lim 0 ) x x0 g (x 0 ). x x 0 sin x Example Evaluate lim x 0 x. This is a 0 0 type limit. Moreover, taking g(x) = x, we have g (x 0 ) 0. So L Hôpital s rule applies, and sin x lim x 0 x = cos x 1 x=0 = 1 1 =

153 x Example Evaluate lim 2 x 0 x 2 +sin x. Since x 2 x=0 = 0 and (x 2 + sin x) x=0 = 0 we identify this as 0 0 L Hôpital s rule, type. By lim x 0 x 2 x 2 + sin x = 2x 2x + cos x x=0 = = 0. In the following version of L Hôpital s rule, we do not assume that f or g are defined at x 0. Theorem Let x 0 (a, b). Consider f, g : (a, b) {x 0 } R and assume that they are differentiable at every point of (a, b) {x 0 }. Suppose that g (x) 0 for all x (a, b) {x 0 }. 1. Suppose that If lim x x0 f (x) g (x) 2. Suppose that lim x x 0 f(x) = 0 = lim x x 0 g(x). = l R {± }, then f(x) lim x x 0 g(x) = lim f (x) x x 0 g = l. (12.2.1) (x) lim f(x) = ±, x x 0 lim g(x) = ±. x x 0 If lim x x0 f (x) g (x) = l R {± }, then (12.2.1) holds. Proof The assumption that g (x) 0 in (a, b) {x 0 } implies that g(x) g(y) if x y and x, y (a, x 0 ) (or x, y (x 0, b)). Because if g(x) = g(y) there would be some point ξ between x and y with g (ξ) = 0, by the Mean Value Theorem, whereas by hypothesis g is never zero on (a, b) {x 0 }. 1. For part 1, we prove only the case where l R. The case where l = ± is left as an exercise. (a) We may assume that f, g are defined and continuous at x 0 and that f(x 0 ) = g(x 0 ) = 0. For otherwise we simply define f(x 0 ) = 0 = lim x x 0 f(x), g(x 0 ) = 0 = lim x x 0 g(x). 152

154 (b) Let x > x 0. Our functions are continuous on [x 0, x] and differentiable on (x 0, x). Apply Cauchy s Mean Value Theorem to see there exists ξ (x 0, x) with Hence f(x) lim x x 0 + g(x) = f(x) f(x 0 ) g(x) g(x 0 ) = f (ξ) g (ξ). lim f(x) f(x 0 ) x x 0 + g(x) g(x 0 ) = This is because as x x 0, also ξ (x 0, x) x 0. lim f (ξ) x x 0 + g (ξ) = l. (c) Let x < x 0. The same Cauchy s Mean Value Theorem argument shows that f(x) lim x x 0 g(x) = lim f(x) f(x 0 ) x x 0 g(x) g(x 0 ) = In conclusion lim x x0 f(x) g(x) = l. 2. For part 2. First the case l = lim x x0 f (x) g (x) R. (a) There exists δ > 0 such that if 0 < x x 0 < δ, l ε < f (x) g (x) lim f (ξ) x x 0 g (ξ) = l. < l + ε. (12.2.2) Since g (x) 0 on (a, b) {x 0 }, the above expression makes sense. (b) Take x, y (x 0 δ, x 0 ) or x, y (x 0, x 0 + δ) such that x y. Then g(x) g(y) 0. Thus ( ) ( ) f(x) f(x) f(y) g(x) g(y) g(x) = + f(y) (12.2.3) g(x) g(y) g(x) g(x) (c) Fix any such y and let x approach x 0. Since lim x x0 f(x) = lim x x0 g(x) = +, we have f(y) lim x x 0 f(x) = 0, lim g(x) g(y) = 1. x x 0 g(x) (exercise) 153

155 (d) By Cauchy s Mean Value Theorem,(12.2.3) can be written ( f(x) f ) ( ) g(x) = (ξ) g(x) g(y) g + f(y) (12.2.4) (ξ) g(x) g(x) for some ξ between x and y. By choosing δ small enough, and x and y as in (b), we can make f (ξ) g (ξ) as close as we wish to l. By taking x close enough to x 0 while keeping y fixed, we can make g(x) g(y) g(x) as close as we like to 1, and f(y) g(x) as close as we wish to 0. It follows from (12.2.4) that f(x) lim x x 0 g(x) = l. Part 2), case l = +. For any A > 0 there exists δ > 0 such that f (x) g (x) > A whenever 0 < x x 0 < δ. Consider the case x > x 0. Fix y > x 0 with 0 < y x 0 < δ. By Cauchy s mean value theorem, for some c between x and y. Clearly f(y) lim x x 0 f(x) = 0, f(x) f(y) g(x) g(y) = f (c) g (c) lim g(x) g(y) = 1. x x 0 g(x) It follows (taking ε = 1/2 in the definition of limit) that there exists δ 1 such that if 0 < x x 0 < δ 1, 1 g(x) g(y) = 1 ε < < 1 + ε = 3 2 g(x) 2, 1 f(y) = ε < 2 f(x) < ε = 1 2. Thus if 0 < x x 0 < δ, f(x) g(x) = ( f(x) f(y) g(x) g(y) ) g(x) g(y) g(x) + f(y) g(x) A

156 Since A can be made arbitrarily large we proved that f(x) lim x x 0 + g(x) = +. The proof for the case lim x x0 f(x) g(x) = + is similar. cos x 1 Example Evaluate lim x 0 x. This is of 0 0 type. cos x 1 sin x lim = lim = 0 x 0 x x = 1. arctan(e 2. Evaluate lim x 1) x 0 x. arctan(e The limit lim x 1) x 0 x is of 0 0 type, since ex 1 0 and arctan(e x 1) arctan(0) = 0 by the continuity of arctan. arctan(e x 1) lim x 0 x = lim x 0 = 1 1+(e x 1) 2 e x 1 e (e 0 1) 2 = 1. e Example Evaluate lim 2(x 1) 1 x 1 ln x. All functions concerned are continuous, e2(x 1) 1 ln x = e2(1 1) 1 ln 1 = 0 0. e 2(x 1) 1 lim = lim x 1 ln x 2e 2(x 1) x 1 1 x = 2. Example Don t get carried away with l Hôpital s rule. Is the following correct and why? sin x cos x lim x 2 x 2 = lim x 2 2x = lim sin x x 2 2 = sin 2? 2 cos x 1 Example Evaluate lim x 0. This is of 0 x 2 0 type. cos x 1 sin x lim x 0 x 2 = lim x 0 2x cos x = lim x 0 2 =

157 sin x The limit lim x 0 2x is again of 0 0 rule. type and again we applied L Hôpital s Example Take f(x) = sin(x 1) and { x 1, x 1, g(x) = 0, x = 1 Hence lim f(x) = 0, lim g(x) = 0. x 1 x 1 f (x) lim x 1 g (x) = 1. f(x) lim x 1 g(x) = 1. Theorem Suppose that f and g are differentiable on (a, x 0 ) and g(x) 0, g (x) 0 for all x (a, x 0 ). Suppose is either 0 or infinity. Then provided that lim x x0 f (x) g (x) exists. lim f(x) = lim g(x) x x 0 x x 0 f(x) lim x x 0 g(x) = lim f (x) x x 0 g (x). Example Evaluate lim x 0+ x ln x. Since lim x 0+ ln x = we identify this as a 0 - type limit. We have not studied such limits, but can transform it into a -type limit, which we know how to deal with: ln x lim x ln x = lim x 0+ x 0+ 1 = lim x 0+ x Exercise Evaluate lim y ln y y. 1 x 1 x 2 = lim ( x) = 0. x 0+ Theorem Suppose f and g are differentiable on (a, ) and g(x) 0, g (x) 0 for all x sufficiently large. 156

158 1. Suppose Then lim f(x) = lim g(x) = 0 x + x + f(x) lim x + g(x) = lim f (x) x g (x). provided that lim x + f (x) g (x) exists. 2. Suppose Then lim f(x) = lim g(x) = x + x + f(x) lim x + g(x) = lim f (x) x g (x). provided that lim x + f (x) g (x) exists. Proof Idea of proof: Let t = 1 x, F (t) = f(1 t ) and G(t) = g( 1 t ) Also lim f(x) = lim f(1 x + t 0 t ) = lim F (t). t 0 lim g(x) = lim g(1 x + t 0 t ) = lim G(t). t 0 F (t) G (t) = f ( 1 t )( 1 ) t 2 g ( 1 t )( 1 ) = f (x) g t (x). 2 Note that the limit as x is into a limit as t ) +, which we do know how to deal with. Example Evaluate lim x x 2 e x. We identify this as type. x 2 lim x e x = lim 2x x e x = lim 2 x e x = 0. We have to apply L Hôpital s rule twice since the result after applying L Hôpital s 2x rule the first time is lim x e, which is again x type. Proposition Suppose that f, g : (a, b) R are differentiable, and f f(c) = g(c) = 0 for some c (a, b). Suppose that lim (x) x c g (x) = +. Then f(x) lim x c g(x) = + 157

159 Proof For any M > 0 there is δ > 0 such that if 0 < x c < δ, f (x) g (x) > M. First take x (c, c + δ). By Cauchy s mean value theorem applied to the interval [c, x], there exists ξ with ξ c < δ such that f(x) g(x) f(x) f(c) = f(x) g(c) = f (ξ) g (ξ) > M. hence lim x x0 + f(x) g(x) = +. A similar argument shows that lim x x 0 f(x) g(x) = + Remark We summarise the cases: Let x 0 R {± } and I be an open interval which either contains x 0 or has x 0 as an end point. Suppose f and g are differentiable on I and g(x) 0, g (x) 0 for all x I. Suppose Then lim f(x) = lim g(x) = 0 x x 0 x x 0 f(x) lim x x 0 g(x) = lim f (x 0 ) x x 0 g (x 0 ). provided that lim x x0 f (x) g (x) exists. Suitable one sided limits are used if x 0 is an end-point of the interval I. Recall that if α is a number, for x > 0 we define x α = e α log x. So d dx xα = e α log x α x = xα α x = αxα 1. Example For α > 0, log x lim x x α = lim x 1 x = lim αxα 1 x 1 αx α = 0. Conclusion: As x +, log x goes to infinity slower than x α any α >

160 Example Evaluate lim x 0 ( 1 x 1 sin x ). This is apparently of -type. We have lim ( 1 x 0 x 1 sin x ) = lim sin x x x 0 x sin x cos x 1 = lim x 0 = lim x 0 sin x + x cos x sin x 2 cos x x sin x = = 0. Example Evaluate lim x 0 ( 1 x 1 sin x ). We try Taylor s method. For some ξ, sin x = x x3 3! cos ξ: lim ( 1 x 0 x 1 sin x ) = lim sin x x x 0 x sin x = lim x 0 = lim x 0 x3 3! cos ξ x(x x3 3! x 3! cos ξ cos ξ) (1 x = 0. 3! cos ξ) 159

161 Chapter 13 Unfinished Business Theorem (The Inverse Function Theorem) Suppose that f : (a, b) R is of type C k, for some k N { }, and x 0 (a, b). 1. If f (x 0 ) > 0, then there exists r > 0 such that (x 0 r, x 0 + r) (a, b) and such that f : (x 0 r, x 0 + r) (f(x 0 r), f(x 0 + r)) is a bijection with inverse of type C k. 2. If f (x 0 ) < 0, then there exists r > 0 such that (x 0 r, x 0 + r) (a, b) and such that f : (x 0 r, x 0 + r) (f(x 0 + r), f(x 0 r)) is a bijection with inverse of type C k. Proof We only consider the case f (x 0 ) > 0. Since f is continuous there exists r > 0 such that f (x) > 0 for all x (x 0 r, x 0 + r). The function f is increasing on [x 0 r, x 0 + r], by Corollary Hence, by the Intermediate Value Theorem, f is a continuous bijection from (x 0 r, x 0 + r) to (f(x 0 r), f(x 0 + r)). [Comment: It has a continuous inverse by the continuous version of the Inverse Function Theorem (Theorem 5.4.1).] By the differential version of the Inverse Function Theorem (Thm ), f 1 is differentiable at any y (f(x 0 r), f(x 0 + r)) and (f 1 ) (y) = 1 f (f 1 (y)). Since f is continuous and f 1 is continuous, the composition (f 1 ) of the two continuous functions is continuous. This proves the case k = 1 of the theorem. If k 2 this we use induction. Suppose that we have proved that if f is of type C (k 1) then f 1 is of type C (k 1). We now show that this holds for 160

162 k. If f is C (k), then since (f 1 ) 1 (y) = f (f 1 (y)), we see that (f 1 ) is the composition of two C (k 1) functions and is therefore C (k 1). Thus (f 1 ) is C (k 1) which means that f 1 is C (k). 161