1 Derivatives of Piecewise Defined Functions


 Wesley Washington
 1 years ago
 Views:
Transcription
1 MATH 1010E University Matematics Lecture Notes (week 4) Martin Li 1 Derivatives of Piecewise Define Functions For piecewise efine functions, we often ave to be very careful in computing te erivatives. Te i erentiation rules (prouct, quotient, cain rules) can only be applie if te function is efine by ONE formula in a neigboroo of te point were we evaluate te erivative. If we want to calculate te erivative at a point were two i erent formulas meet, ten we must use te efinition of erivative as it of i erence quotient to correctly evaluate te erivative. Let us illustrate tis by te following example. Example 1.1 Fin te erivative f 0 (x) at every x 2 R for te piecewise efine function 5 2x wen x<0, f(x) = x 2 2x +5 wen x 0. Solution: We separate into 3 cases: x<0, x>0 an x = 0. For te first two cases, te function f(x) is efine by a single formula, so we coul just apply i erentiation rules to i erentiate te function. f 0 (x) =(5 2x) 0 = 2 for x<0, f 0 (x) =(x 2 2x + 5) 0 =2x 2 for x>0. At x = 0, we ave to use te efinition of erivative as it of i erence quotient. First of all, f(0) = 0 2 2(0) + 5 = 5. Ten we calculate te leftan an rigtan its: f() f(0) (5 2) 5 2= 2, f() f(0) ( ) ) = 2. +( 1
2 Since bot of tem exists an are equal, we ave f 0 f() f(0) (0) = 2. Terefore, putting all of tese togeter, we see tat f is i erentiable for every x 2 R an f 0 2 wen x apple 0, (x) = 2x 2 wen x>0. Remark 1.2 From te example above, we see tat te erivative f 0 (x) is still a continuous function (ceck tis!). Tis is not always true for any function! (Have you seen a counterexample? See Homework 2) Example 1.3 Consier te function efine by ax + b wen x apple 1, f(x) = ax 3 + x +2b wen x> 1, for wat value(s) of a, b 2 R is te function f i erentiable at every x 2 R? Solution: First, it is easy to see tat for ANY a, b 2 R, te function f is i erentiable at every x 6= 1sincef is efine by a polynomial on ( 1, +1) an ( 1, 1). Te only catc is at te point x = 1. If f is i erentiable at x = 1, it must also be continuous at x = 1. Terefore, we nee f(x) =f( x! 1 1). Now, f( 1) = a + b an te leftan an rigtan its are f(x) =a( x! 1 1)3 +( 1) + 2b = a +2b 1, f(x) =a( 1) + b = a + b. x! 1 + If f is continuous, ten bot of tese its must be te same an equal to f( 1). Hence, we ave a + b = a +2b 1 ) b =1. Now, we take b = 1. To fin te value of a wic make f i erentiable at x = 1, we require te it f( 1+) f( 1) 2
3 to exists, wic is equivalent to te statement tat te leftan an rigtan its exist an are equal. Te left an it is f( 1+) f( 1) Te rigt an it is f( 1+) f( 1) + [a( 1+) + 1] ( a + 1) a = a. [a( 1+) 3 +( 1+) + 2] ( a + 1) + a[( 1+) 3 + 1] + + a[( 1+) 2 ( 1+) + 1] + + = 3a +1. Terefore, if we set tem equal to eac oter, we obtain te conition a =3a +1 ) a = 1 2. In summary, we ave a = x 2 R. 1/2 an b =1iff is i erentiable at every 2 Di erentiation Rules II: Prouct an Quotient Rules Teorem 2.1 If f an g are i erentiable functions, ten bot teir prouct fg an quotient f/g are i erentiable an we ave (1) Prouct Rule: (2) Quotient Rule: provie tat g(x) 6= 0. [f(x)g(x)] 0 = f 0 (x)g(x)+f(x)g 0 (x), f(x) 0 = g(x)f 0 (x) f(x)g 0 (x) g(x) (g(x)) 2, Remark 2.2 Observe tat te i erentiation rule [kf(x)] 0 = kf 0 (x) were k is a constant is just a special case of prouct rule by taking g(x) k, wic as g 0 (x) =0. 3
4 Before we go into te proof of tese rules, let us first look at a few examples of ow to apply tese rules to elp us calculate erivatives. Example 2.3 x [(x + 2)(x2 + 1)] = (x + 2) 0 (x 2 + 1) + (x + 2)(x 2 + 1) 0 = (1)(x 2 + 1) + (x + 2)(2x) = 3x 2 +4x +1. x [sin x cos x] = (sinx)0 cos x +sinx(cos x) 0 = cos x cos x +sinx( sin x) = cos 2x. x x 2 +1 x +1 = (x + 1)(x2 + 1) 0 (x 2 + 1)(x + 1) 0 (x + 1) 2 = (x + 1)(2x) (x2 + 1)(1) (x + 1) 2 = x2 +2x 1 (x + 1) 2. x sin x x = x(sin x)0 sin x(x) 0 = x 2 x cos x sin x x 2. Note tat for te last two examples, te calculation is vali only for x 6= 1 an x 6= 0respectively. sin x Question: Doesteit x!0 x x exist? If we efine sin x f(x) = x wen x 6= 0, 1 wen x =0, ten oes f 0 (0) exists? If so, is it relate to te it at te beginning? Question: Calculation te erivatives of all te trigonometric an yperbolic functions! Now, we come back to te proof of te prouct rule an quotient rule. 4
5 Proof of Prouct Rule: Using te efinition of erivative, [f(x)g(x)] 0 f(x + )g(x + ) f(x)g(x) [f(x + )g(x + ) f(x)g(x + )] + [f(x)g(x + ) f(x)g(x)] f(x + ) g(x + ) f(x) g(x + ) + f(x) g(x) = g(x)f 0 (x)+f(x)g 0 (x). In te last equality, we ave also use tat g(x + ) =g(x) sinceg is continuous (even i erentiable) by assumption. Proof of Quotient Rule: Using te efinition of erivative, f(x) 0 f(x+) g(x+ g(x) f(x) g(x) f(x + )g(x) f(x)g(x + ) g(x)g(x + ) f(x+)g(x) f(x)g(x) + g(x)g(x + ) f(x) g(x)g(x + ) = g(x)f 0 (x) f(x)g 0 (x) (g(x)) 2. g(x) f(x+) f(x) f(x)g(x) f(x)g(x+) g(x+) g(x) Again we ave use te continuity of g in te last equality. 3 Composite Functions Apart from aition, subtraction, multiplication an ivision to get new functions, tere is anoter useful way to obtain new functions from ol calle composition. Definition 3.1 Given two functions f : D! E an g : E! F,wecan efine te composite function g f : D! F by g f(x) :=g(f(x)). 5
6 If we tink of functions as assignments, ten g f means assigning x to f(x) first an ten we furter assign f(x) tog(f(x)). Note tat in orer for te composition g f to be well efine, te omain of g must contain te image of f. However, we o not require eiter f or g to be injective or surjective. Question: Wic of te following statement is true? f an g are injective ) g f an g are injective ) g f is injective? f is injective? g is not injective ) g g is not surjective ) g f is not injective? f is not surjective? Let us look at one example. Consier two functions efine by f : R! R, f(x) := cos x, te composition g g : R! R, g(y) :=y 2, f : R! R is ence g f(x) =g(f(x)) = g(cos x) = cos 2 x. Note tat we can also o te composition in a i erent orer for tis example. We can form f g : R! R wic is efine by f g(y) =f(g(y)) = f(y 2 ) = cos y 2. Note tat even in tis case bot g f an f g are efine, tey are NOT equal to eac oter. Terefore, te orering is important wen we talk about composition: Remark 3.2 In general, we ave g f 6= f g even wen bot are wellefine. 6
7 Te notation g f may be confusing sometimes since we are writing g first an ten f but te composition means oing f first an ten g. One goo way to memorize tis is tat wen g f acts on x, wewriteg f(x). It is f wic its x first, an ten followe by g. Tis is te reason wy we use tis convention. Also, in te above example, we write g f as a function of x an f g as a function of y. Wen comparing tem as functions, te name of te variable (x an y) are irrelevant. It is te rule of assignment tat etermines te function. For example, we consier f(x) =x an g(y) =y as te same function. It will become clear later tat it is useful to keep using x for elements in te omain of f an y as elements in te coomain of f. Composition is a rater nice operation wic preserves many of te analytic properties of f an g. Teorem 3.3 If (i) f is continuous at x 0,an (ii) g is continuous at y 0 := f(x 0 ), ten g f is continuous at x 0. In oter wors, composition of continuous functions is continuous. Te proof is rater intuitive. Wen x is close to x 0,teny := f(x) is also close to y 0 := f(x 0 ) by te continuity of f. On te oter an, since y is close to y 0,wemustaveg(y) close to g(y 0 ). Tis is te same as saying tat g(f(x)) is close to g(f(x 0 )). A rigorous proof can given using te efinition of continuity. Intereste stuents can try to write own a complete proof as an exercise. We ave alreay use tis teorem implicitly many times wen we evaluate its. For example, wen we compute x!0 cos2 x = (cos 0) 2 =1, we ave use te continuity of cos an square function wic justifies te irect substitution wit x = 0 to compute te it. 4 Di erentiation Rules III: Cain Rule Composition of i erentiable functions is also i erentiable. Teorem 4.1 (Cain Rule) If 7
8 (i) f is i erentiable at x 0,an (ii) g is i erentiable at y 0 := f(x 0 ), ten g f is i erentiable at x 0 an (g f) 0 (x 0 )=g 0 (f(x 0 )) f 0 (x 0 ). Remark 4.2 Te cain rule simply says tat te erivative of composition is just te prouct of erivatives. Yet tis is one of te most confusing rules among all te i erentiation rules we ave seen so far. Te reason is tat we ave to be careful at wic points we are evaluating te erivatives! We are NOT evaluating g 0 at x 0, but at f(x 0 ) instea. If you recall te stepbystep assignment picture of composite functions, you see tat it is inee te only way for te formula to make sense since g 0 (x 0 ) is not even efine as x 0 is not in te omain of g! Let us turn to some examples. Example 4.3 Calculate te erivatives of te following functions: (i) cos x 2, (ii) (x 2 + 1) 7, (iii) e sin x, (iv) e sin x2. Solution: (i) Let f(x) =x 2 an g(y) = cos y, teng te function we want to i erentiate. Note tat f(x) = cos x 2 is f 0 (x) =2x an g 0 (y) = sin y. Terefore, applying cain rule we get (g f) 0 (x) =g 0 (f(x))f 0 (x) =( sin(x 2 )) (2x) = 2x sin x 2. (ii) Let f(x) =x 2 +1 an g(y) =y 7.Weavef 0 (x) =2x an g 0 (y) =7y 6, Terefore, x (x2 + 1) 7 = 7(x 2 + 1) 6 (2x) = 14x(x 2 + 1) 6. 8
9 Terefore, te cain rule basically says tat we can i erentiate layerbylayer, starting from te outermost layer. In tis example, we first i erentiation te function of raising to power 7 an ten i erentiate into te bracket. (iii) Using te concept of i erentiating layerbylayer, we o not nee to efine f an g every time. Since we know te erivative of e y is just e y an te erivative of sin x is cos x, we get x esin x = e sin x cos x. (iv) For functions involving more tan two layers, we just i erentiate tem one by one. Here, te outer layer is exponential function, te mile layer is sin an te inner later is x 2,so x esin x2 = e sin x2 (cos x 2 ) (2x). Note tat wen i erentiating te outer layers, you keep te inner layers uncange. Tis makes sure tat you are evaluating at te correct point as iscusse in Remark 4.2. Question: Use layer by layer i erentiation to evaluate te following erivatives: q x + p x x an x x p 1+x 2 Proof of Cain Rule: By efinition of erivative, (g f) 0 g(f(x)) g(f(x 0 )) (x 0 ) x!x 0 x x apple 0 g(f(x)) g(f(x0 )) x!x 0 f(x) f(x 0 ). f(x) f(x 0 ) x x 0 g(f(x)) g(f(x 0 )) f(x) f(x 0 ). x!x 0 f(x) f(x 0 ) x!x 0 x x 0 Let y = f(x) an y = f(x 0 ), since f is continuous at x 0,weavey! y 0 as x! x 0, terefore, we can rewrite te first it in te last line above in terms of y: g(y) g(y 0 ) f(x) f(x 0 ) = g 0 (y 0 )f 0 (x 0 )=g 0 (f(x 0 ))f 0 (x 0 ). y!y 0 y y 0 x!x 0 x x 0 Tis proves te cain rule. Question: Tere is a loopole in te above proof. Can you fin it an fix it? 9
10 5 Inverse Di erentiation We can use te cain rule to fin te erivative of te inverse f 1 of a function, if it exists. Teorem 5.1 Let f :(a, b)! (c, ) be a bijective i erentiable function wose erivative f 0 is continuous. Ten, te inverse function g :(c, )! (a, b) is i erentiable at every y suc tat f 0 (g(y)) 6= 0an g 0 (y) = 1 f 0 (g(y)). Remark 5.2 Note tat te inverse may not be i erentiable at were f 0 = 0. For example, consier f : R! R efine by f(x) =x 3, ten its inverse g(y) =y 1/3 exists but is not i erentiable at y =0. Proof of Teorem 5.1 Te proof tat g is i erentiable is muc more involve so we skip it ere. By te efinition of inverse, f(g(y)) = y for all y. Since te above equation ols for ALL y, we can i erentiate te wole equation on bot sie wit respect to y, applying cain rule on te left an sie, we obtain f 0 (g(y))g 0 (y) =1. If f 0 (g(y)) 6= 0, we can ivie it to te oter sie to obtain te formula of g 0 (y). Example 5.3 Sow tat x (ln y) = 1 y. Solution: Recall tat te exponential function exp(x) : R! (0, 1) is 11 an onto wose inverse is ln y :(0, 1)! R. We ave alreay seen tat xexp(x) =exp(x). Terefore, using inverse i erentiation, keeping in min tat y =exp(x), we ave y ln(y) = 1 exp(x) = 1 y. Example 5.4 Sow tat x sin 1 y = p 1. 1 y 2 10
11 Solution: Recall tat sin(x) :R! R is neiter 11 nor onto so te inverse oes not exists. However, if we restrict its omain an coomain to sin(x) : ( 2, 2 )! ( 1, 1), ten it becomes 11 an onto an ence tere is an inverse sin 1 y : ( 1, 1)! ( 2, 2 ). Using inverse i erentiation, since y =sinx, weave y sin 1 1 y = x sin x = 1 cos x = 1 p 1 sin 2 x = 1 p. 1 y 2 Note tat we coul ave restrict sin(x) to a i erent omain e.g. ( 2, 3 2 ). Te inverse function sin 1 y woul be i erent, but te erivative is te same! Te coice of sin 1 is a penomenon calle brancing. Question: Wat appens to y sin 1 y wen y!±1? Wat oes it correspon to in terms of te slope of te graps? Question: Discuss te omains an coomains of te inverses of oter trigonometric an yperbolic functions. Wat are teir erivatives? 6 More Examples We give a few more examples illustrating te use of all te i erentiation rules we iscusse. x (ln(x + p 1+x 2 )) = = = 1 x + p 1+x 2 x (x + p 1+x 2 ) 1 x + p 1+x [1 + x(1 + 2 x2 ) 1 2 ] 1 p. 1+x 2 We can use exp an ln to efine an arbitrary exponential function a b for ANY a>0 an b 2 R: a b := exp(b ln a). Hence, we can calculate te erivatives of tese general exponential functions: x 3x = x (exp(x ln 3)) = exp(x ln 3) ln 3 = 3x ln 3. x xx = x (exp(x ln x)) = exp(x ln x) [(1)(ln x)+(x)( 1 x )] = (1+ ln x)xx. Question: Calculate x (xxx ). 11
Proof of the Power Rule for Positive Integer Powers
Te Power Rule A function of te form f (x) = x r, were r is any real number, is a power function. From our previous work we know tat x x 2 x x x x 3 3 x x In te first two cases, te power r is a positive
More informationSection 3.3. Differentiation of Polynomials and Rational Functions. Difference Equations to Differential Equations
Difference Equations to Differential Equations Section 3.3 Differentiation of Polynomials an Rational Functions In tis section we begin te task of iscovering rules for ifferentiating various classes of
More informationMAT1A01: Differentiation of Polynomials & Exponential Functions + the Product & Quotient Rules
MAT1A01: Differentiation of Polynomials & Exponential Functions + te Prouct & Quotient Rules Dr Craig 17 April 2013 Reminer Mats Learning Centre: CRing 512 My office: CRing 533A (Stats Dept corrior)
More informationDerivatives Math 120 Calculus I D Joyce, Fall 2013
Derivatives Mat 20 Calculus I D Joyce, Fall 203 Since we ave a good understanding of its, we can develop derivatives very quickly. Recall tat we defined te derivative f x of a function f at x to be te
More informationCHAPTER 8: DIFFERENTIAL CALCULUS
CHAPTER 8: DIFFERENTIAL CALCULUS 1. Rules of Differentiation As we ave seen, calculating erivatives from first principles can be laborious an ifficult even for some relatively simple functions. It is clearly
More informationMath 229 Lecture Notes: Product and Quotient Rules Professor Richard Blecksmith richard@math.niu.edu
Mat 229 Lecture Notes: Prouct an Quotient Rules Professor Ricar Blecksmit ricar@mat.niu.eu 1. Time Out for Notation Upate It is awkwar to say te erivative of x n is nx n 1 Using te prime notation for erivatives,
More informationUnderstanding the Derivative Backward and Forward by Dave Slomer
Understanding te Derivative Backward and Forward by Dave Slomer Slopes of lines are important, giving average rates of cange. Slopes of curves are even more important, giving instantaneous rates of cange.
More informationf(x + h) f(x) h as representing the slope of a secant line. As h goes to 0, the slope of the secant line approaches the slope of the tangent line.
Derivative of f(z) Dr. E. Jacobs Te erivative of a function is efine as a limit: f (x) 0 f(x + ) f(x) We can visualize te expression f(x+) f(x) as representing te slope of a secant line. As goes to 0,
More informationLecture 10: What is a Function, definition, piecewise defined functions, difference quotient, domain of a function
Lecture 10: Wat is a Function, definition, piecewise defined functions, difference quotient, domain of a function A function arises wen one quantity depends on anoter. Many everyday relationsips between
More information6. Differentiating the exponential and logarithm functions
1 6. Differentiating te exponential and logaritm functions We wis to find and use derivatives for functions of te form f(x) = a x, were a is a constant. By far te most convenient suc function for tis purpose
More informationCHAPTER 7. Di erentiation
CHAPTER 7 Di erentiation 1. Te Derivative at a Point Definition 7.1. Let f be a function defined on a neigborood of x 0. f is di erentiable at x 0, if te following it exists: f 0 fx 0 + ) fx 0 ) x 0 )=.
More informationDifferential Calculus: Differentiation (First Principles, Rules) and Sketching Graphs (Grade 12)
OpenStaxCNX moule: m39313 1 Differential Calculus: Differentiation (First Principles, Rules) an Sketcing Graps (Grae 12) Free Hig Scool Science Texts Project Tis work is prouce by OpenStaxCNX an license
More informationDifferentiable Functions
Capter 8 Differentiable Functions A differentiable function is a function tat can be approximated locally by a linear function. 8.. Te derivative Definition 8.. Suppose tat f : (a, b) R and a < c < b.
More informationSAT Subject Math Level 1 Facts & Formulas
Numbers, Sequences, Factors Integers:..., 3, 2, 1, 0, 1, 2, 3,... Reals: integers plus fractions, decimals, and irrationals ( 2, 3, π, etc.) Order Of Operations: Aritmetic Sequences: PEMDAS (Parenteses
More information7.6 Complex Fractions
Section 7.6 Comple Fractions 695 7.6 Comple Fractions In tis section we learn ow to simplify wat are called comple fractions, an eample of wic follows. 2 + 3 Note tat bot te numerator and denominator are
More informationSections 3.1/3.2: Introducing the Derivative/Rules of Differentiation
Sections 3.1/3.2: Introucing te Derivative/Rules of Differentiation 1 Tangent Line Before looking at te erivative, refer back to Section 2.1, looking at average velocity an instantaneous velocity. Here
More information4.4 The Derivative. 51. Disprove the claim: If lim f (x) = L, then either lim f (x) = L or. 52. If lim x a. f (x) = and lim x a. g(x) =, then lim x a
Capter 4 Real Analysis 281 51. Disprove te claim: If lim f () = L, ten eiter lim f () = L or a a lim f () = L. a 52. If lim a f () = an lim a g() =, ten lim a f + g =. 53. If lim f () = an lim g() = L
More informationInstantaneous Rate of Change:
Instantaneous Rate of Cange: Last section we discovered tat te average rate of cange in F(x) can also be interpreted as te slope of a scant line. Te average rate of cange involves te cange in F(x) over
More information20. Product rule, Quotient rule
20. Prouct rule, 20.1. Prouct rule Prouct rule, Prouct rule We have seen that the erivative of a sum is the sum of the erivatives: [f(x) + g(x)] = x x [f(x)] + x [(g(x)]. One might expect from this that
More informationMath 113 HW #5 Solutions
Mat 3 HW #5 Solutions. Exercise.5.6. Suppose f is continuous on [, 5] and te only solutions of te equation f(x) = 6 are x = and x =. If f() = 8, explain wy f(3) > 6. Answer: Suppose we ad tat f(3) 6. Ten
More informationTrapezoid Rule. y 2. y L
Trapezoid Rule and Simpson s Rule c 2002, 2008, 200 Donald Kreider and Dwigt Lar Trapezoid Rule Many applications of calculus involve definite integrals. If we can find an antiderivative for te integrand,
More informationACT Math Facts & Formulas
Numbers, Sequences, Factors Integers:..., 3, 2, 1, 0, 1, 2, 3,... Rationals: fractions, tat is, anyting expressable as a ratio of integers Reals: integers plus rationals plus special numbers suc as
More informationf(a + h) f(a) f (a) = lim
Lecture 7 : Derivative AS a Function In te previous section we defined te derivative of a function f at a number a (wen te function f is defined in an open interval containing a) to be f (a) 0 f(a + )
More informationThis supplement is meant to be read after Venema s Section 9.2. Throughout this section, we assume all nine axioms of Euclidean geometry.
Mat 444/445 Geometry for Teacers Summer 2008 Supplement : Similar Triangles Tis supplement is meant to be read after Venema s Section 9.2. Trougout tis section, we assume all nine axioms of uclidean geometry.
More informationFinite Difference Approximations
Capter Finite Difference Approximations Our goal is to approximate solutions to differential equations, i.e., to find a function (or some discrete approximation to tis function) tat satisfies a given relationsip
More informationLecture 17: Implicit differentiation
Lecture 7: Implicit ifferentiation Nathan Pflueger 8 October 203 Introuction Toay we iscuss a technique calle implicit ifferentiation, which provies a quicker an easier way to compute many erivatives we
More informationLecture 10. Limits (cont d) Onesided limits. (Relevant section from Stewart, Seventh Edition: Section 2.4, pp. 113.)
Lecture 10 Limits (cont d) Onesided its (Relevant section from Stewart, Sevent Edition: Section 2.4, pp. 113.) As you may recall from your earlier course in Calculus, we may define onesided its, were
More informationMath WarmUp for Exam 1 Name: Solutions
Disclaimer: Tese review problems do not represent te exact questions tat will appear te exam. Tis is just a warmup to elp you begin studying. It is your responsibility to review te omework problems, webwork
More informationarcsine (inverse sine) function
Inverse Trigonometric Functions c 00 Donal Kreier an Dwight Lahr We will introuce inverse functions for the sine, cosine, an tangent. In efining them, we will point out the issues that must be consiere
More informationLecture 13: Differentiation Derivatives of Trigonometric Functions
Lecture 13: Differentiation Derivatives of Trigonometric Functions Derivatives of the Basic Trigonometric Functions Derivative of sin Derivative of cos Using the Chain Rule Derivative of tan Using the
More informationnparameter families of curves
1 nparameter families of curves For purposes of this iscussion, a curve will mean any equation involving x, y, an no other variables. Some examples of curves are x 2 + (y 3) 2 = 9 circle with raius 3,
More informationMATH 125: LAST LECTURE
MATH 5: LAST LECTURE FALL 9. Differential Equations A ifferential equation is an equation involving an unknown function an it s erivatives. To solve a ifferential equation means to fin a function that
More informationIn other words the graph of the polynomial should pass through the points
Capter 3 Interpolation Interpolation is te problem of fitting a smoot curve troug a given set of points, generally as te grap of a function. It is useful at least in data analysis (interpolation is a form
More informationRules for Finding Derivatives
3 Rules for Fining Derivatives It is teious to compute a limit every time we nee to know the erivative of a function. Fortunately, we can evelop a small collection of examples an rules that allow us to
More informationf(x) = a x, h(5) = ( 1) 5 1 = 2 2 1
Exponential Functions an their Derivatives Exponential functions are functions of the form f(x) = a x, where a is a positive constant referre to as the base. The functions f(x) = x, g(x) = e x, an h(x)
More information1.6. Analyse Optimum Volume and Surface Area. Maximum Volume for a Given Surface Area. Example 1. Solution
1.6 Analyse Optimum Volume and Surface Area Estimation and oter informal metods of optimizing measures suc as surface area and volume often lead to reasonable solutions suc as te design of te tent in tis
More informationDiscovering Area Formulas of Quadrilaterals by Using Composite Figures
Activity: Format: Ojectives: Related 009 SOL(s): Materials: Time Required: Directions: Discovering Area Formulas of Quadrilaterals y Using Composite Figures Small group or Large Group Participants will
More informationOptimal Pricing Strategy for Second Degree Price Discrimination
Optimal Pricing Strategy for Second Degree Price Discrimination Alex O Brien May 5, 2005 Abstract Second Degree price discrimination is a coupon strategy tat allows all consumers access to te coupon. Purcases
More informationTangent Lines and Rates of Change
Tangent Lines and Rates of Cange 922005 Given a function y = f(x), ow do you find te slope of te tangent line to te grap at te point P(a, f(a))? (I m tinking of te tangent line as a line tat just skims
More information2 Limits and Derivatives
2 Limits and Derivatives 2.7 Tangent Lines, Velocity, and Derivatives A tangent line to a circle is a line tat intersects te circle at exactly one point. We would like to take tis idea of tangent line
More informationNew Vocabulary volume
. Plan Objectives To find te volume of a prism To find te volume of a cylinder Examples Finding Volume of a Rectangular Prism Finding Volume of a Triangular Prism 3 Finding Volume of a Cylinder Finding
More informationAn Interest Rate Model
An Interest Rate Model Concepts and Buzzwords Building Price Tree from Rate Tree Lognormal Interest Rate Model Nonnegativity Volatility and te Level Effect Readings Tuckman, capters 11 and 12. Lognormal
More informationExam 2 Review. . You need to be able to interpret what you get to answer various questions.
Exam Review Exam covers 1.6,.1.3, 1.5, 4.14., and 5.15.3. You sould know ow to do all te omework problems from tese sections and you sould practice your understanding on several old exams in te exam
More informationGeometric Stratification of Accounting Data
Stratification of Accounting Data Patricia Gunning * Jane Mary Horgan ** William Yancey *** Abstract: We suggest a new procedure for defining te boundaries of te strata in igly skewed populations, usual
More information1 Density functions, cummulative density functions, measures of central tendency, and measures of dispersion
Density functions, cummulative density functions, measures of central tendency, and measures of dispersion densityfunctionsintro.tex October, 9 Note tat tis section of notes is limitied to te consideration
More information2 HYPERBOLIC FUNCTIONS
HYPERBOLIC FUNCTIONS Chapter Hyperbolic Functions Objectives After stuying this chapter you shoul unerstan what is meant by a hyperbolic function; be able to fin erivatives an integrals of hyperbolic functions;
More informationACTIVITY: Deriving the Area Formula of a Trapezoid
4.3 Areas of Trapezoids a trapezoid? How can you derive a formula for te area of ACTIVITY: Deriving te Area Formula of a Trapezoid Work wit a partner. Use a piece of centimeter grid paper. a. Draw any
More informationSolutions to modified 2 nd Midterm
Math 125 Solutions to moifie 2 n Miterm 1. For each of the functions f(x) given below, fin f (x)). (a) 4 points f(x) = x 5 + 5x 4 + 4x 2 + 9 Solution: f (x) = 5x 4 + 20x 3 + 8x (b) 4 points f(x) = x 8
More informationSolution Derivations for Capa #7
Solution Derivations for Capa #7 1) Consider te beavior of te circuit, wen various values increase or decrease. (Select Iincreases, Ddecreases, If te first is I and te rest D, enter IDDDD). A) If R1
More informationM(0) = 1 M(1) = 2 M(h) = M(h 1) + M(h 2) + 1 (h > 1)
Insertion and Deletion in VL Trees Submitted in Partial Fulfillment of te Requirements for Dr. Eric Kaltofen s 66621: nalysis of lgoritms by Robert McCloskey December 14, 1984 1 ackground ccording to Knut
More information2.1: The Derivative and the Tangent Line Problem
.1.1.1: Te Derivative and te Tangent Line Problem Wat is te deinition o a tangent line to a curve? To answer te diiculty in writing a clear deinition o a tangent line, we can deine it as te iting position
More informationThe Derivative as a Function
Section 2.2 Te Derivative as a Function 200 Kiryl Tsiscanka Te Derivative as a Function DEFINITION: Te derivative of a function f at a number a, denoted by f (a), is if tis limit exists. f (a) f(a+) f(a)
More information4.1 Rightangled Triangles 2. 4.2 Trigonometric Functions 19. 4.3 Trigonometric Identities 36. 4.4 Applications of Trigonometry to Triangles 53
ontents 4 Trigonometry 4.1 Rigtangled Triangles 4. Trigonometric Functions 19 4.3 Trigonometric Identities 36 4.4 pplications of Trigonometry to Triangles 53 4.5 pplications of Trigonometry to Waves 65
More informationME422 Mechanical Control Systems Modeling Fluid Systems
Cal Poly San Luis Obispo Mecanical Engineering ME422 Mecanical Control Systems Modeling Fluid Systems Owen/Ridgely, last update Mar 2003 Te dynamic euations for fluid flow are very similar to te dynamic
More informationDetermine the perimeter of a triangle using algebra Find the area of a triangle using the formula
Student Name: Date: Contact Person Name: Pone Number: Lesson 0 Perimeter, Area, and Similarity of Triangles Objectives Determine te perimeter of a triangle using algebra Find te area of a triangle using
More informationChapter 7 Numerical Differentiation and Integration
45 We ave a abit in writing articles publised in scientiþc journals to make te work as Þnised as possible, to cover up all te tracks, to not worry about te blind alleys or describe ow you ad te wrong idea
More information19.2. First Order Differential Equations. Introduction. Prerequisites. Learning Outcomes
First Orer Differential Equations 19.2 Introuction Separation of variables is a technique commonly use to solve first orer orinary ifferential equations. It is socalle because we rearrange the equation
More informationTo differentiate logarithmic functions with bases other than e, use
To ifferentiate logarithmic functions with bases other than e, use 1 1 To ifferentiate logarithmic functions with bases other than e, use log b m = ln m ln b 1 To ifferentiate logarithmic functions with
More informationFinite Volume Discretization of the Heat Equation
Lecture Notes 3 Finite Volume Discretization of te Heat Equation We consider finite volume discretizations of te onedimensional variable coefficient eat equation, wit Neumann boundary conditions u t x
More informationSurface Areas of Prisms and Cylinders
12.2 TEXAS ESSENTIAL KNOWLEDGE AND SKILLS G.10.B G.11.C Surface Areas of Prisms and Cylinders Essential Question How can you find te surface area of a prism or a cylinder? Recall tat te surface area of
More informationIntroduction to Integration Part 1: AntiDifferentiation
Mathematics Learning Centre Introuction to Integration Part : AntiDifferentiation Mary Barnes c 999 University of Syney Contents For Reference. Table of erivatives......2 New notation.... 2 Introuction
More informationVerifying Numerical Convergence Rates
1 Order of accuracy Verifying Numerical Convergence Rates We consider a numerical approximation of an exact value u. Te approximation depends on a small parameter, suc as te grid size or time step, and
More informationTrigonometry CheatSheet
Trigonometry CheatSheet 1 How to use this ocument This ocument is not meant to be a list of formulas to be learne by heart. The first few formulas are very basic (they escen from the efinition an/or Pythagoras
More informationM147 Practice Problems for Exam 2
M47 Practice Problems for Exam Exam will cover sections 4., 4.4, 4.5, 4.6, 4.7, 4.8, 5., an 5.. Calculators will not be allowe on the exam. The first ten problems on the exam will be multiple choice. Work
More informationSAT Math MustKnow Facts & Formulas
SAT Mat MustKnow Facts & Formuas Numbers, Sequences, Factors Integers:..., 3, 2, 1, 0, 1, 2, 3,... Rationas: fractions, tat is, anyting expressabe as a ratio of integers Reas: integers pus rationas
More informationComputer Science and Engineering, UCSD October 7, 1999 GoldreicLevin Teorem Autor: Bellare Te GoldreicLevin Teorem 1 Te problem We æx a an integer n for te lengt of te strings involved. If a is an nbit
More informationThe EOQ Inventory Formula
Te EOQ Inventory Formula James M. Cargal Matematics Department Troy University Montgomery Campus A basic problem for businesses and manufacturers is, wen ordering supplies, to determine wat quantity of
More informationInverse Trig Functions
Inverse Trig Functions c A Math Support Center Capsule February, 009 Introuction Just as trig functions arise in many applications, so o the inverse trig functions. What may be most surprising is that
More informationWriting Mathematics Papers
Writing Matematics Papers Tis essay is intended to elp your senior conference paper. It is a somewat astily produced amalgam of advice I ave given to students in my PDCs (Mat 4 and Mat 9), so it s not
More informationArea of a Parallelogram
Area of a Parallelogram Focus on After tis lesson, you will be able to... φ develop te φ formula for te area of a parallelogram calculate te area of a parallelogram One of te sapes a marcing band can make
More informationPyramidization of Polygonal Prisms and Frustums
European International Journal of cience and ecnology IN: 0499 www.eijst.org.uk Pyramidization of Polygonal Prisms and Frustums Javad Hamadani Zade Department of Matematics Dalton tate College Dalton,
More informationDistances in random graphs with infinite mean degrees
Distances in random graps wit infinite mean degrees Henri van den Esker, Remco van der Hofstad, Gerard Hoogiemstra and Dmitri Znamenski April 26, 2005 Abstract We study random graps wit an i.i.d. degree
More informationInverse Trig Functions
Inverse Trig Functions Trig functions are not onetoone, so we can not formally get an inverse. To efine the notion of inverse trig functions we restrict the omains to obtain onetoone functions: [ Restrict
More informationCollege Planning Using Cash Value Life Insurance
College Planning Using Cas Value Life Insurance CAUTION: Te advisor is urged to be extremely cautious of anoter college funding veicle wic provides a guaranteed return of premium immediately if funded
More informationSept 20, 2011 MATH 140: Calculus I Tutorial 2. ln(x 2 1) = 3 x 2 1 = e 3 x = e 3 + 1
Sept, MATH 4: Calculus I Tutorial Solving Quadratics, Dividing Polynomials Problem Solve for x: ln(x ) =. ln(x ) = x = e x = e + Problem Solve for x: e x e x + =. Let y = e x. Then we have a quadratic
More informationarxiv:math/ v1 [math.pr] 20 Feb 2003
JUGGLING PROBABILITIES GREGORY S. WARRINGTON arxiv:mat/0302257v [mat.pr] 20 Feb 2003. Introduction Imagine yourself effortlessly juggling five balls in a ig, lazy pattern. Your rigt and catces a ball and
More informationMath Test Sections. The College Board: Expanding College Opportunity
Taking te SAT I: Reasoning Test Mat Test Sections Te materials in tese files are intended for individual use by students getting ready to take an SAT Program test; permission for any oter use must be sougt
More informationModule 2. The Science of Surface and Ground Water. Version 2 CE IIT, Kharagpur
Module Te Science of Surface and Ground Water Version CE IIT, Karagpur Lesson 6 Principles of Ground Water Flow Version CE IIT, Karagpur Instructional Objectives On completion of te lesson, te student
More informationSection 3.1 Worksheet NAME. f(x + h) f(x)
MATH 1170 Section 3.1 Worksheet NAME Recall that we have efine the erivative of f to be f (x) = lim h 0 f(x + h) f(x) h Recall also that the erivative of a function, f (x), is the slope f s tangent line
More informationIs Gravity an Entropic Force?
Is Gravity an Entropic Force? San Gao Unit for HPS & Centre for Time, SOPHI, University of Sydney, Sydney, NSW 006, Australia; EMail: sgao7319@uni.sydney.edu.au Abstract: Te remarkable connections between
More informationMath 230.01, Fall 2012: HW 1 Solutions
Math 3., Fall : HW Solutions Problem (p.9 #). Suppose a wor is picke at ranom from this sentence. Fin: a) the chance the wor has at least letters; SOLUTION: All wors are equally likely to be chosen. The
More informationCan a LumpSum Transfer Make Everyone Enjoy the Gains. from Free Trade?
Can a LumpSum Transfer Make Everyone Enjoy te Gains from Free Trade? Yasukazu Icino Department of Economics, Konan University June 30, 2010 Abstract I examine lumpsum transfer rules to redistribute te
More informationTheoretical calculation of the heat capacity
eoretical calculation of te eat capacity Principle of equipartition of energy Heat capacity of ideal and real gases Heat capacity of solids: DulongPetit, Einstein, Debye models Heat capacity of metals
More informationThe modelling of business rules for dashboard reporting using mutual information
8 t World IMACS / MODSIM Congress, Cairns, Australia 37 July 2009 ttp://mssanz.org.au/modsim09 Te modelling of business rules for dasboard reporting using mutual information Gregory Calbert Command, Control,
More information 1  Handout #22 May 23, 2012 Huffman Encoding and Data Compression. CS106B Spring 2012. Handout by Julie Zelenski with minor edits by Keith Schwarz
CS106B Spring 01 Handout # May 3, 01 Huffman Encoding and Data Compression Handout by Julie Zelenski wit minor edits by Keit Scwarz In te early 1980s, personal computers ad ard disks tat were no larger
More informationAdvice for Undergraduates on Special Aspects of Writing Mathematics
Advice for Undergraduates on Special Aspects of Writing Matematics Abstract Tere are several guides to good matematical writing for professionals, but few for undergraduates. Yet undergraduates wo write
More informationFINITE DIFFERENCE METHODS
FINITE DIFFERENCE METHODS LONG CHEN Te best known metods, finite difference, consists of replacing eac derivative by a difference quotient in te classic formulation. It is simple to code and economic to
More informationCHAPTER 2. Inequalities
CHAPTER 2 Inequalities In this section we add the axioms describe the behavior of inequalities (the order axioms) to the list of axioms begun in Chapter 1. A thorough mastery of this section is essential
More informationThe Derivative. Not for Sale
3 Te Te Derivative 3. Limits 3. Continuity 3.3 Rates of Cange 3. Definition of te Derivative 3.5 Grapical Differentiation Capter 3 Review Etended Application: A Model for Drugs Administered Intravenously
More informationPolynomials can be added or subtracted simply by adding or subtracting the corresponding terms, e.g., if
1. Polynomials 1.1. Definitions A polynomial in x is an expression obtained by taking powers of x, multiplying them by constants, and adding them. It can be written in the form c 0 x n + c 1 x n 1 + c
More informationOption Pricing Using the Binomial Model
Finance 400 A. Penati  G. Pennacci Option Pricing Using te Binomial Moel Te CoxRossRubinstein (CRR) tecnique is useful for valuing relatively complicate options, suc as tose aving American (early exercise)
More informationSAT Math Facts & Formulas
Numbers, Sequences, Factors SAT Mat Facts & Formuas Integers:..., 3, 2, 1, 0, 1, 2, 3,... Reas: integers pus fractions, decimas, and irrationas ( 2, 3, π, etc.) Order Of Operations: Aritmetic Sequences:
More informationf(x) f(a) x a Our intuition tells us that the slope of the tangent line to the curve at the point P is m P Q =
Lecture 6 : Derivatives and Rates of Cange In tis section we return to te problem of finding te equation of a tangent line to a curve, y f(x) If P (a, f(a)) is a point on te curve y f(x) and Q(x, f(x))
More informationSolutions by: KARATUĞ OZAN BiRCAN. PROBLEM 1 (20 points): Let D be a region, i.e., an open connected set in
KOÇ UNIVERSITY, SPRING 2014 MATH 401, MIDTERM1, MARCH 3 Instructor: BURAK OZBAGCI TIME: 75 Minutes Solutions by: KARATUĞ OZAN BiRCAN PROBLEM 1 (20 points): Let D be a region, i.e., an open connected set
More informationHyperbolic functions (CheatSheet)
Hyperbolic functions (CheatSheet) 1 Intro For historical reasons hyperbolic functions have little or no room at all in the syllabus of a calculus course, but as a matter of fact they have the same ignity
More informationVolumes of Pyramids and Cones. Use the Pythagorean Theorem to find the value of the variable. h 2 m. 1.5 m 12 in. 8 in. 2.5 m
5 Wat You ll Learn To find te volume of a pramid To find te volume of a cone... And W To find te volume of a structure in te sape of a pramid, as in Eample Volumes of Pramids and Cones Ceck Skills You
More informationTOPIC 3: CONTINUITY OF FUNCTIONS
TOPIC 3: CONTINUITY OF FUNCTIONS. Absolute value We work in the field of real numbers, R. For the study of the properties of functions we need the concept of absolute value of a number. Definition.. Let
More informationLecture 5 : Continuous Functions Definition 1 We say the function f is continuous at a number a if
Lecture 5 : Continuous Functions Definition We say the function f is continuous at a number a if f(x) = f(a). (i.e. we can make the value of f(x) as close as we like to f(a) by taking x sufficiently close
More informationAnswers to the Practice Problems for Test 2
Answers to the Practice Problems for Test 2 Davi Murphy. Fin f (x) if it is known that x [f(2x)] = x2. By the chain rule, x [f(2x)] = f (2x) 2, so 2f (2x) = x 2. Hence f (2x) = x 2 /2, but the lefthan
More information