5.3 Improper Integrals Involving Rational and Exponential Functions

Size: px
Start display at page:

Download "5.3 Improper Integrals Involving Rational and Exponential Functions"

Transcription

1 Section 5.3 Improper Integrals Involving Rational and Exponential Functions dθ +a cos θ =, < a <. a sin θ ( dθ = a a b a + b cos θ b ), dθ a + b cos θ =, <a. a a + b dθ a + b sin θ = a a + b, <a. < b <a dθ a cos θ + b sin θ + c = dθ a cos θ + b sin θ + c = c a b, a + b <c. (a + c)(b + c), < c < a, c < b. 5.3 Improper Integrals Involving Rational and Exponential Functions In this section we present a useful technique to evaluate improper integrals involving rational and exponential functions. Let a and b be arbitrary real numbers. Consider the integrals () b f(x) and a f(x), where in each case f is continuous in the interval of integration and at its finite endpoint. These are called improper integrals, because the interval of integration is infinite. The integral a f(x) is convergent if b lim b a f(x) exists as a finite number. Similarly, b f(x) is convergent if lim a a b f(x) exists as a finite number. Now let f(x) be continuous on the real line. The integral f(x) is also improper since the interval of integration is infinite, but here it is infinite in both the positive and negative direction. Such an integral is said to be convergent if both f(x) and f(x) are convergent. In this case, we set (Figure )

2 3 Chapter 5 Residue Theory Figure Splitting an improper integral over the line. () f(x) = lim a a f(x) + lim b b f(x). We define the Cauchy principal value of the integral (Figure ) a (3) P.V. f(x) = lim f(x), if the limit exists. a a f(x) to be The Cauchy principal value of an integral may exist even though the integral itself is not convergent. For example, a a x = for all a, which implies that P.V. x =, but the integral itself is clearly not convergent since x =. However, whenever f(x) is convergent, then P.V. f(x) exists, and the two integrals will be the same. This is because lim a a f(x) and lim a a f(x) both exist, and so Figure The Cauchy principal value of the integral. P.V. f(x) = lim a = lim a a ( a = lim = a a f(x) f(x) + a f(x). a f(x) + lim a ) f(x) a f(x) Because of this fact, we can compute a convergent integral over the real line by computing its principal value, which can often be obtained by use of complex methods and the residue theorem. The following test of convergence for improper integrals is similar to tests that we have proved for infinite series. We omit the proof.

3 Section 5.3 Improper Integrals Involving Rational and Exponential Functions 3 PROPOSITION INEQUALITIES FOR IMPROPER INTEGRALS Let B A f(x) represent an improper integral as in (), where A = or B =. (i) If B A f(x) is convergent, then B A f(x) is convergent and we have B A f(x) B A f(x). (ii) If f(x) g(x) for all A < x < B and B A g(x) is convergent, then B A f(x) is convergent and we have B A f(x) B A g(x). You are encouraged to use Proposition or any other test from calculus (such as the limit comparison test) to show that an improper integral is convergent. Once the convergence is determined, we may compute the integral via its principal value, as we will illustrate in the examples. EXAMPLE Evaluate Improper integrals and residues: the main ideas I = x x 4 +. Solution To highlight the main ideas, we present the solution in basic steps. Step : Show that the improper integral is convergent. Because the integrand is continuous on the real line, it is enough to show that the integral outside a x finite interval, say [, ], is convergent. For x, we have x 4 + x x = 4 x, and since x is convergent, it follows by Proposition that x x 4 + and x x 4 + are convergent. Thus x is convergent, and so x 4 + x R x (4) x 4 = lim + R x 4 +. Step : Set up the contour integral. We will replac by z and consider the function f(z) = z z 4 +. The general guideline is to integrate this function over a contour that consists partly of the interval [, R], so as to recapture the integral R x x 4 + as part of the contour integral on γ R. Choosing the appropriate contour is not x obvious in general. For a rational function, such as, where the denominator x 4 + does not vanish on th-axis and the degree of the denominator is two more than the degree of the numerator, a closed semi-circle γ R as in Figure 3 will work. Since γ R consists of the interval [, R] followed by the semi-circle σ R, using the additive property of path integrals (Proposition (iii), Section 3.), we write (5) I γr = f(z) dz = f(z) dz + f(z) dz = I R + I σr. γ R [, R] σ R Figure 3 The path and poles for the contour integral in Example. For z = x in [, R], we have f(z) =f(x) = R x x 4 + x x 4 + and dz =, and so I R =, which according to (4) converges to the desired integral as R. So, in order to compute the desired integral, we must get a handle on the other quantities, I γr and I σr, in (5). Our strategy is as follows. In Step 3, we compute I γr by the residue theorem; and in Step 4, we show that lim R I σr =. This

4 3 Chapter 5 Residue Theory will give us the necessary ingredients to complete the solution in Step 5. Step 3: Compute I γr by the residue theorem. For R>, the function f(z) = z z 4 + has two poles inside γ R. These are the roots of z 4 + = in the upper half-plane. To solve z 4 =, we write =e i ; then using the formula for the nth roots from Section.3, we find the four roots z = +i, z = +i, z 3 = i, z 4 = i (see Figure 3). Since these are simple roots, we have simple poles at z and z and the residues there are (Proposition (ii), Section 5.) Res (z )= and similarly z d = z dz (z4 + ) 4z 3 = z=z 4z = z=z z=z Res (z )= 4z So, by the residue theorem, for all R> (6) I γr = γr z z=z = i 4. 4( + i) = i 4, z 4 + dz =i( Res (z ) + Res (z ) ) =i i 4 =. Step 4: Show that lim R I σr =. For z on σ R, we have z = R, and so z z 4 + R R 4 = M R. Appealing to the integral inequality (Theorem, Section 3.), we have I σr = z 4 + dz l(σ R)M R = R σr z R R 4 = R /R 3, Step 5: Compute the improper integral. Using (5) and (6), we obtain = I R + I σr. as R. Let R, then I R x x 4 + and I σ R, and so x x 4 + =. Figure 4 An alternative path for the integral in Example. In Example, we could have used the contour γ R in the lower half-plane in Figure 4. In this case, it is easiest to take the orientation of γ R to be negative in order to coincide with the orientation of the interval [, R]. The five steps that we used in the solution of Example can be used to prove a the following geberal result.

5 Section 5.3 Improper Integrals Involving Rational and Exponential Functions 33 PROPOSITION INTEGRALS OF RATIONAL FUNCTIONS Let f(x) = p(x) q(x) be a rational function with degree q(x) + degree p(x), and let σ R denote an arc of the circle centered at with radius R>. Then (7) lim p(z) R σ R q(z) dz =. Moreover, if q(x) has no real roots and z, z,..., z N denote all the poles in the upper half-plane, then of p(z) q(z) (8) p(x) N q(x) =i j= Res ( p(z) q(z),z ) j. Here is a straightforward application. Figure 5 The path and poles for the contour integral in Example. EXAMPLE Evaluate Improper integral of a rational function (x + )(x + 4). Solution The integrand satisfies the two conditions of Proposition : degree q(x) = 4 + degree p(x) =, and the denominator q(x) = (x + )(x + 4) has no real roots. So we may apply (8). We have (z + )(z + 4) = (z + i)(z i)(z +i)(z i), and hence has simple poles at i and i in the upper half-plane (Figure 5). The residues there (z +)(z +4) are and Res (i) = lim(z i) z i (z i)(z + i)(z + 4) = (i)( + 4) = i 6 According to (8), Res (i) = lim (z i) z i (z + )(z i)(z +i) = i. (x + )(x + 4) =i( i 6 + i ) = 6. Integrals Involving Exponential Functions The success of the method of contour integration depends crucially on the choice of contours. In the previous examples, we used expanding semicircular contours. To evaluate integrals involving exponential functions, we will use rectangular contours. We will consider integrals of the form e ax e bx, where a < b, and c>. + c

6 34 Chapter 5 Residue Theory Since c> and e ax >, the denominator does not equal zero. Also the condition a < b guarantees that the integral is convergent (Exercise 8). EXAMPLE 3 Integral involving exponential functions Let be a real number >. Establish the identity (9) e x + = sin. Solution Step : As noted before, since >, the integral converges. The integrand leads us to consider the function f(z) = ez e z +, whose poles are the roots of e z + =. Since the exponential function is i-periodic, then e z = =e i z = i +ki z k = (k +) i, k =, ±, ±,.... Thus f(z) = (Figure 6). ez e z + has infinitely many poles at z k, all lying on the imaginary axis Step : Selecting the contour of integration. Our contour should expand in the x-direction in order to cover the entir-axis. To avoid including infinitely many poles on the y-axis, we will not expand the contour in the upper half-plane, as we did with the semi-circles in the previous examples. Instead, we will use a rectangular contour γ R consisting of the paths γ, γ, γ 3, and γ 4, as in Figure 6, and let I j denote the path integral of f(z) over γ j (j =,..., 4). As R, γ R will expand in the horizontal direction, but the length of the vertical sides remains fixed at. To understand the reason for our choice of the vertical length, let us consider I and I 3. On γ, we have z = x, dz =, Figure 6 The path and poles for the contour integral in Example 3. () I = R e x +, and so I converges to the desired integral I as R. On γ 3, we have z = x+i, dz =, and using the i-periodicity of the exponential function, we get () I 3 = R +i e (x+i ) + = e i R = e e x i I. + This last equality explains the choice of the vertical sides: They are chosen so that the integral on the returning horizontal side γ 3 is equal to a constant multiple of the integral on γ. From here the solution is straightforward. Step 3: Applying the residue theorem. From Step, we have only one pole of f(z) inside γ R at z = i. Since ez + has a simple root, this is a simple pole. Using Proposition (ii), Section 5., we find and so by the residue theorem () I + I + I 3 + I 4 = Res ( e z e z +, i) = e i e i = e i i = e ei, γr e z ( e z e z dz =i Res + e z +, i) = i e i.

7 Section 5.3 Improper Integrals Involving Rational and Exponential Functions 35 Step 4: Show that the integrals on the vertical sides tend to as R. For I, z = R + iy ( y ) on γ, hence e z = e R e iy = e R, e z + e z =e R e z + e R, and so for z on γ f(z) = e z e z + e R+iy e R = er e R = e ( )R e. Consequently, by the ML-inequality for path integrals, I = e γ z e z + dz l(γ ) e ( )R e =, as R. e ( )R e The proof that I 4 as R is done similarly; we omit the details. Step 5: Compute the desired integral (9). Using (), (), and (), we find that I ( e i) + I + I 4 = i e i. Letting R and using the result of Step 4, we get and after simplifying which implies (9). ( e i) e x + = e x + i = ie, i ( e i e i) = sin, There are interesting integrals of rational functions that are not computable using semi-circular contours as in Example, such as (3) x 3 +. This integral can be reduced to an integral invovling expoential functions, by using the substitution x = e t. We outline this useful technique in the following example. EXAMPLE 4 The substitution x = e t Establish the identity (4) x + = sin, any real number >. Solution Step : Show that the integral converges. The integrand is continuous, so it is enough to show that the integral converges on [, ). We have x +, x

8 36 Chapter 5 Residue Theory Figure 7 Splitting an improper integral. and the integral is convergent since x <. Step : Apply the substitution x = e t. Let x = e t, = e t dt, and note that as x varies from to, t varies from to, and so I = x + = e t e t + dt = e x +, where, for convenience, in the last integral we have used x as a variable of integration instead of t. Identity (4) follows now from Example 3. The tricky part in Example 3 is choosing the contour. Let us clarify this part with one more example. We will compute the integral ln x. The x 4 + integral is improper because the interval is infinite and because the integrand tends to as x. To define the convergence of such integrals, we follow the general procedure of taking all one-sided limits one at a time. Thus (Figure 7) ln x x 4 + = lim ɛ ɛ ln x b x 4 + lim + b ln x x 4 +. It is not difficult to show that both limits exist and hence that the integral converges. We will use the substitution x = e t to solve the problem. If you like, you could instead check the convergence after changing variables. EXAMPLE 5 Derive (5) Solution An integral involving ln x ln x = x Let x = e t, ln x = t, = e t dt. This transforms the integral into t e 4t + et dt = x e 4x +, Figure 8 The path and poles for the contour integral in Example 4. where we have renamed the variabl, just for convenience. In evaluating this integral, we will integrate f(z) = zez e 4z + over the rectangular contour γ R in Figure 8, and let I j denote the integral of f(z) on γ j. Here again, we chose the vertical sides of γ R so that on the returning path γ 3 the denominator equals to e 4x +. As we will see momentarily, this will enable us to relate I 3 to I. Let us now compute I γr = γ R f(z) dz. As you can check, f(z) has one (simple) pole at z = i 4 inside γ R. By Proposition (ii), Section 5., the residue there is So by the residue theorem ( ) i( + i) (6) I γr =i 6 Res ( ze z e 4z +,i ) i = 4 ei 4 + i) = i( 4 4ei 6. = ( + i) 8 = I + I + I 3 + I 4.

9 Section 5.3 Improper Integrals Involving Rational and Exponential Functions 37 Moving to each I j (j =,..., 4), we have I = γ ze z e 4z + dz = R x e 4x +. On γ 3, z = x + i, dz =, so using e i = i, we get I 3 = = i γ3 ze z e 4z + dz = R = ii + R x e 4x + + R (x + i )ex e i R e 4x +. e 4x + e 4x + To show that I and I 4 tend to as R, we proceed as in Step 5 of Example 3. For z = R + iy ( y ) on γ, we have z R + y R +, and so, as in Example 3, f(z) = ze z e 4z + = z e z e 4z + (R + ) e R e 4R = R + e 3R e. Consequently, by the ML-inequality for path integrals I = ze γ z e 4z + dz l(γ )(R + ) e 3R e = (R + ), as R. e 3R e The proof that I 4 as R is done similarly; we omit the details. Substituting our finding into (6) and taking the limit as R, we get ( + i) 8 ( = lim I + I ii + R R x = ( i) e 4x + + e 4x + + I 4 e 4x +. ) Taking imaginary parts of both sides, we obtain our answer x e 4x + = 8. If we take real parts of both sides we get the value of the integral in (4) that corresponds to = 4. The substitution x = e t is also useful even when we do not use complex methods to evaluate the integral in t. For example, ln x x + = te t e t dt = + since the integral is convergent and the integrand function of t. Exercises 5.3 tet e t + = t e t +e t is an odd

10 38 Chapter 5 Residue Theory In Exercises 8, evaluate the given improper integral by the method of Example x 4 + =. (x + )(x 4 + ) = (x + ) 3 = (4x + )(x i) = 3 i 8. x 4 + x + = 3 (x i)(x +3i) = (x 4 + ) = 3 4 (x + i)(x 3i) = In Exercises 9, evaluate the given improper integral by the method of Example e 3x + = 3 3 x = = + e x e ax e bx + = 3 b sin a b ( < a < b) In Exercises 3, evaluate the given improper integral by the method of Example 5. In some cases, the integral follows from more general formulas that we derived earlier. You may use these formulas to verify your answer x 3 + = 3 3 x x 5 + = x x 3 + = 3 x 3 = sin ( 5 4. ) ln(x) x +4 = ln. ( ) ln x 3 x = Use the contour γ R in Figure 9 to evaluate x 3 +. (n)! = ( + x ) n+ n (n!) x x + = sin ( ) ( > ) x = ( < < ) (x + ) sin x ln(x) x 3 + = 7 + ln 3 3 ln(ax) x + b = ln(ab) (a, b > ) b (ln x) 3 x 3 = Figure 9 The path and poles for the contour integral in Exercise. [Hint: I γr = I + I + I 3 ; I 3 = e i 3 I ; and I as R.] 6. Follow the steps in Example to prove Proposition. 7. A property of the gamma function. (a) Show that for < <, x +x = sin.

Review for Calculus Rational Functions, Logarithms & Exponentials

Review for Calculus Rational Functions, Logarithms & Exponentials Definition and Domain of Rational Functions A rational function is defined as the quotient of two polynomial functions. F(x) = P(x) / Q(x) The domain of F is the set of all real numbers except those for

More information

MATH 311: COMPLEX ANALYSIS CONTOUR INTEGRALS LECTURE

MATH 311: COMPLEX ANALYSIS CONTOUR INTEGRALS LECTURE MATH 3: COMPLEX ANALYSIS CONTOUR INTEGRALS LECTURE Recall the Residue Theorem: Let be a simple closed loop, traversed counterclockwise. Let f be a function that is analytic on and meromorphic inside. Then

More information

6. Define log(z) so that π < I log(z) π. Discuss the identities e log(z) = z and log(e w ) = w.

6. Define log(z) so that π < I log(z) π. Discuss the identities e log(z) = z and log(e w ) = w. hapter omplex integration. omplex number quiz. Simplify 3+4i. 2. Simplify 3+4i. 3. Find the cube roots of. 4. Here are some identities for complex conjugate. Which ones need correction? z + w = z + w,

More information

2 Complex Functions and the Cauchy-Riemann Equations

2 Complex Functions and the Cauchy-Riemann Equations 2 Complex Functions and the Cauchy-Riemann Equations 2.1 Complex functions In one-variable calculus, we study functions f(x) of a real variable x. Likewise, in complex analysis, we study functions f(z)

More information

MA107 Precalculus Algebra Exam 2 Review Solutions

MA107 Precalculus Algebra Exam 2 Review Solutions MA107 Precalculus Algebra Exam 2 Review Solutions February 24, 2008 1. The following demand equation models the number of units sold, x, of a product as a function of price, p. x = 4p + 200 a. Please write

More information

1 if 1 x 0 1 if 0 x 1

1 if 1 x 0 1 if 0 x 1 Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or

More information

Practice with Proofs

Practice with Proofs Practice with Proofs October 6, 2014 Recall the following Definition 0.1. A function f is increasing if for every x, y in the domain of f, x < y = f(x) < f(y) 1. Prove that h(x) = x 3 is increasing, using

More information

TOPIC 4: DERIVATIVES

TOPIC 4: DERIVATIVES TOPIC 4: DERIVATIVES 1. The derivative of a function. Differentiation rules 1.1. The slope of a curve. The slope of a curve at a point P is a measure of the steepness of the curve. If Q is a point on the

More information

MATH 52: MATLAB HOMEWORK 2

MATH 52: MATLAB HOMEWORK 2 MATH 52: MATLAB HOMEWORK 2. omplex Numbers The prevalence of the complex numbers throughout the scientific world today belies their long and rocky history. Much like the negative numbers, complex numbers

More information

TOPIC 3: CONTINUITY OF FUNCTIONS

TOPIC 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 information

2. Length and distance in hyperbolic geometry

2. Length and distance in hyperbolic geometry 2. Length and distance in hyperbolic geometry 2.1 The upper half-plane There are several different ways of constructing hyperbolic geometry. These different constructions are called models. In this lecture

More information

The Method of Partial Fractions Math 121 Calculus II Spring 2015

The Method of Partial Fractions Math 121 Calculus II Spring 2015 Rational functions. as The Method of Partial Fractions Math 11 Calculus II Spring 015 Recall that a rational function is a quotient of two polynomials such f(x) g(x) = 3x5 + x 3 + 16x x 60. The method

More information

Solutions to Homework 10

Solutions to Homework 10 Solutions to Homework 1 Section 7., exercise # 1 (b,d): (b) Compute the value of R f dv, where f(x, y) = y/x and R = [1, 3] [, 4]. Solution: Since f is continuous over R, f is integrable over R. Let x

More information

Lecture VI. Review of even and odd functions Definition 1 A function f(x) is called an even function if. f( x) = f(x)

Lecture VI. Review of even and odd functions Definition 1 A function f(x) is called an even function if. f( x) = f(x) ecture VI Abstract Before learning to solve partial differential equations, it is necessary to know how to approximate arbitrary functions by infinite series, using special families of functions This process

More information

x a x 2 (1 + x 2 ) n.

x a x 2 (1 + x 2 ) n. Limits and continuity Suppose that we have a function f : R R. Let a R. We say that f(x) tends to the limit l as x tends to a; lim f(x) = l ; x a if, given any real number ɛ > 0, there exists a real number

More information

Some Notes on Taylor Polynomials and Taylor Series

Some Notes on Taylor Polynomials and Taylor Series Some Notes on Taylor Polynomials and Taylor Series Mark MacLean October 3, 27 UBC s courses MATH /8 and MATH introduce students to the ideas of Taylor polynomials and Taylor series in a fairly limited

More information

Chapter 2 Limits Functions and Sequences sequence sequence Example

Chapter 2 Limits Functions and Sequences sequence sequence Example Chapter Limits In the net few chapters we shall investigate several concepts from calculus, all of which are based on the notion of a limit. In the normal sequence of mathematics courses that students

More information

Complex Numbers and the Complex Exponential

Complex Numbers and the Complex Exponential Complex Numbers and the Complex Exponential Frank R. Kschischang The Edward S. Rogers Sr. Department of Electrical and Computer Engineering University of Toronto September 5, 2005 Numbers and Equations

More information

7. Cauchy s integral theorem and Cauchy s integral formula

7. Cauchy s integral theorem and Cauchy s integral formula 7. Cauchy s integral theorem and Cauchy s integral formula 7.. Independence of the path of integration Theorem 6.3. can be rewritten in the following form: Theorem 7. : Let D be a domain in C and suppose

More information

Undergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics

Undergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics Undergraduate Notes in Mathematics Arkansas Tech University Department of Mathematics An Introductory Single Variable Real Analysis: A Learning Approach through Problem Solving Marcel B. Finan c All Rights

More information

Asymptotes. Definition. The vertical line x = a is called a vertical asymptote of the graph of y = f(x) if

Asymptotes. Definition. The vertical line x = a is called a vertical asymptote of the graph of y = f(x) if Section 2.1: Vertical and Horizontal Asymptotes Definition. The vertical line x = a is called a vertical asymptote of the graph of y = f(x) if, lim x a f(x) =, lim x a x a x a f(x) =, or. + + Definition.

More information

3 Contour integrals and Cauchy s Theorem

3 Contour integrals and Cauchy s Theorem 3 ontour integrals and auchy s Theorem 3. Line integrals of complex functions Our goal here will be to discuss integration of complex functions = u + iv, with particular regard to analytic functions. Of

More information

ANALYTICAL MATHEMATICS FOR APPLICATIONS 2016 LECTURE NOTES Series

ANALYTICAL MATHEMATICS FOR APPLICATIONS 2016 LECTURE NOTES Series ANALYTICAL MATHEMATICS FOR APPLICATIONS 206 LECTURE NOTES 8 ISSUED 24 APRIL 206 A series is a formal sum. Series a + a 2 + a 3 + + + where { } is a sequence of real numbers. Here formal means that we don

More information

Again, the limit must be the same whichever direction we approach from; but now there is an infinity of possible directions.

Again, the limit must be the same whichever direction we approach from; but now there is an infinity of possible directions. Chapter 4 Complex Analysis 4.1 Complex Differentiation Recall the definition of differentiation for a real function f(x): f f(x + δx) f(x) (x) = lim. δx 0 δx In this definition, it is important that the

More information

MATH : HONORS CALCULUS-3 HOMEWORK 6: SOLUTIONS

MATH : HONORS CALCULUS-3 HOMEWORK 6: SOLUTIONS MATH 16300-33: HONORS CALCULUS-3 HOMEWORK 6: SOLUTIONS 25-1 Find the absolute value and argument(s) of each of the following. (ii) (3 + 4i) 1 (iv) 7 3 + 4i (ii) Put z = 3 + 4i. From z 1 z = 1, we have

More information

Definition of Vertical Asymptote The line x = a is called a vertical asymptote of f (x) if at least one of the following is true: f (x) =

Definition of Vertical Asymptote The line x = a is called a vertical asymptote of f (x) if at least one of the following is true: f (x) = Vertical Asymptotes Definition of Vertical Asymptote The line x = a is called a vertical asymptote of f (x) if at least one of the following is true: lim f (x) = x a lim f (x) = lim x a lim f (x) = x a

More information

Differentiation and Integration

Differentiation and Integration This material is a supplement to Appendix G of Stewart. You should read the appendix, except the last section on complex exponentials, before this material. Differentiation and Integration Suppose we have

More information

HOMEWORK 5 SOLUTIONS. n!f n (1) lim. ln x n! + xn x. 1 = G n 1 (x). (2) k + 1 n. (n 1)!

HOMEWORK 5 SOLUTIONS. n!f n (1) lim. ln x n! + xn x. 1 = G n 1 (x). (2) k + 1 n. (n 1)! Math 7 Fall 205 HOMEWORK 5 SOLUTIONS Problem. 2008 B2 Let F 0 x = ln x. For n 0 and x > 0, let F n+ x = 0 F ntdt. Evaluate n!f n lim n ln n. By directly computing F n x for small n s, we obtain the following

More information

Using a table of derivatives

Using a table of derivatives Using a table of derivatives In this unit we construct a Table of Derivatives of commonly occurring functions. This is done using the knowledge gained in previous units on differentiation from first principles.

More information

MATH 2300 review problems for Exam 3 ANSWERS

MATH 2300 review problems for Exam 3 ANSWERS MATH 300 review problems for Exam 3 ANSWERS. Check whether the following series converge or diverge. In each case, justify your answer by either computing the sum or by by showing which convergence test

More information

Notes on metric spaces

Notes on metric spaces Notes on metric spaces 1 Introduction The purpose of these notes is to quickly review some of the basic concepts from Real Analysis, Metric Spaces and some related results that will be used in this course.

More information

Main page. Given f ( x, y) = c we differentiate with respect to x so that

Main page. Given f ( x, y) = c we differentiate with respect to x so that Further Calculus Implicit differentiation Parametric differentiation Related rates of change Small variations and linear approximations Stationary points Curve sketching - asymptotes Curve sketching the

More information

MA4001 Engineering Mathematics 1 Lecture 10 Limits and Continuity

MA4001 Engineering Mathematics 1 Lecture 10 Limits and Continuity MA4001 Engineering Mathematics 1 Lecture 10 Limits and Dr. Sarah Mitchell Autumn 2014 Infinite limits If f(x) grows arbitrarily large as x a we say that f(x) has an infinite limit. Example: f(x) = 1 x

More information

CHAPTER 2 FOURIER SERIES

CHAPTER 2 FOURIER SERIES CHAPTER 2 FOURIER SERIES PERIODIC FUNCTIONS A function is said to have a period T if for all x,, where T is a positive constant. The least value of T>0 is called the period of. EXAMPLES We know that =

More information

Math 21A Brian Osserman Practice Exam 1 Solutions

Math 21A Brian Osserman Practice Exam 1 Solutions Math 2A Brian Osserman Practice Exam Solutions These solutions are intended to indicate roughly how much you would be expected to write. Comments in [square brackets] are additional and would not be required.

More information

5 Indefinite integral

5 Indefinite integral 5 Indefinite integral The most of the mathematical operations have inverse operations: the inverse operation of addition is subtraction, the inverse operation of multiplication is division, the inverse

More information

Understanding Poles and Zeros

Understanding Poles and Zeros MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING 2.14 Analysis and Design of Feedback Control Systems Understanding Poles and Zeros 1 System Poles and Zeros The transfer function

More information

The Math Circle, Spring 2004

The Math Circle, Spring 2004 The Math Circle, Spring 2004 (Talks by Gordon Ritter) What is Non-Euclidean Geometry? Most geometries on the plane R 2 are non-euclidean. Let s denote arc length. Then Euclidean geometry arises from the

More information

Inner Product Spaces

Inner Product Spaces Math 571 Inner Product Spaces 1. Preliminaries An inner product space is a vector space V along with a function, called an inner product which associates each pair of vectors u, v with a scalar u, v, and

More information

106 Chapter 5 Curve Sketching. If f(x) has a local extremum at x = a and. THEOREM 5.1.1 Fermat s Theorem f is differentiable at a, then f (a) = 0.

106 Chapter 5 Curve Sketching. If f(x) has a local extremum at x = a and. THEOREM 5.1.1 Fermat s Theorem f is differentiable at a, then f (a) = 0. 5 Curve Sketching Whether we are interested in a function as a purely mathematical object or in connection with some application to the real world, it is often useful to know what the graph of the function

More information

Lectures 5-6: Taylor Series

Lectures 5-6: Taylor Series Math 1d Instructor: Padraic Bartlett Lectures 5-: Taylor Series Weeks 5- Caltech 213 1 Taylor Polynomials and Series As we saw in week 4, power series are remarkably nice objects to work with. In particular,

More information

COMPLEX NUMBERS AND SERIES. Contents

COMPLEX NUMBERS AND SERIES. Contents COMPLEX NUMBERS AND SERIES MIKE BOYLE Contents 1. Complex Numbers Definition 1.1. A complex number is a number z of the form z = x + iy, where x and y are real numbers, and i is another number such that

More information

Residues and Contour Integration Problems

Residues and Contour Integration Problems Residues and ontour Integration Problems lassify the singularity of f(z) at the indicated point.. f(z) = cot(z) at z =. Ans. Simple pole. Solution. The test for a simple pole at z = is that lim z zcot(z)

More information

Area Between Curves. The idea: the area between curves y = f(x) and y = g(x) (if the graph of f(x) is above that of g(x)) for a x b is given by

Area Between Curves. The idea: the area between curves y = f(x) and y = g(x) (if the graph of f(x) is above that of g(x)) for a x b is given by MATH 42, Fall 29 Examples from Section, Tue, 27 Oct 29 1 The First Hour Area Between Curves. The idea: the area between curves y = f(x) and y = g(x) (if the graph of f(x) is above that of g(x)) for a x

More information

Taylor Polynomials and Taylor Series Math 126

Taylor Polynomials and Taylor Series Math 126 Taylor Polynomials and Taylor Series Math 26 In many problems in science and engineering we have a function f(x) which is too complicated to answer the questions we d like to ask. In this chapter, we will

More information

88 CHAPTER 2. VECTOR FUNCTIONS. . First, we need to compute T (s). a By definition, r (s) T (s) = 1 a sin s a. sin s a, cos s a

88 CHAPTER 2. VECTOR FUNCTIONS. . First, we need to compute T (s). a By definition, r (s) T (s) = 1 a sin s a. sin s a, cos s a 88 CHAPTER. VECTOR FUNCTIONS.4 Curvature.4.1 Definitions and Examples The notion of curvature measures how sharply a curve bends. We would expect the curvature to be 0 for a straight line, to be very small

More information

1. Periodic Fourier series. The Fourier expansion of a 2π-periodic function f is:

1. Periodic Fourier series. The Fourier expansion of a 2π-periodic function f is: CONVERGENCE OF FOURIER SERIES 1. Periodic Fourier series. The Fourier expansion of a 2π-periodic function f is: with coefficients given by: a n = 1 π f(x) a 0 2 + a n cos(nx) + b n sin(nx), n 1 f(x) cos(nx)dx

More information

www.mathsbox.org.uk ab = c a If the coefficients a,b and c are real then either α and β are real or α and β are complex conjugates

www.mathsbox.org.uk ab = c a If the coefficients a,b and c are real then either α and β are real or α and β are complex conjugates Further Pure Summary Notes. Roots of Quadratic Equations For a quadratic equation ax + bx + c = 0 with roots α and β Sum of the roots Product of roots a + b = b a ab = c a If the coefficients a,b and c

More information

55x 3 + 23, f(x) = x2 3. x x 2x + 3 = lim (1 x 4 )/x x (2x + 3)/x = lim

55x 3 + 23, f(x) = x2 3. x x 2x + 3 = lim (1 x 4 )/x x (2x + 3)/x = lim Slant Asymptotes If lim x [f(x) (ax + b)] = 0 or lim x [f(x) (ax + b)] = 0, then the line y = ax + b is a slant asymptote to the graph y = f(x). If lim x f(x) (ax + b) = 0, this means that the graph of

More information

Chapter 9. Miscellaneous Applications

Chapter 9. Miscellaneous Applications hapter 9 53 Miscellaneous Applications In this chapter we consider selected methods where complex variable techniques can be employed to solve various types of problems In particular, we consider the summation

More information

Tangent and normal lines to conics

Tangent and normal lines to conics 4.B. Tangent and normal lines to conics Apollonius work on conics includes a study of tangent and normal lines to these curves. The purpose of this document is to relate his approaches to the modern viewpoints

More information

sin(x) < x sin(x) x < tan(x) sin(x) x cos(x) 1 < sin(x) sin(x) 1 < 1 cos(x) 1 cos(x) = 1 cos2 (x) 1 + cos(x) = sin2 (x) 1 < x 2

sin(x) < x sin(x) x < tan(x) sin(x) x cos(x) 1 < sin(x) sin(x) 1 < 1 cos(x) 1 cos(x) = 1 cos2 (x) 1 + cos(x) = sin2 (x) 1 < x 2 . Problem Show that using an ɛ δ proof. sin() lim = 0 Solution: One can see that the following inequalities are true for values close to zero, both positive and negative. This in turn implies that On the

More information

M344 - ADVANCED ENGINEERING MATHEMATICS Lecture 9: Orthogonal Functions and Trigonometric Fourier Series

M344 - ADVANCED ENGINEERING MATHEMATICS Lecture 9: Orthogonal Functions and Trigonometric Fourier Series M344 - ADVANCED ENGINEERING MATHEMATICS ecture 9: Orthogonal Functions and Trigonometric Fourier Series Before learning to solve partial differential equations, it is necessary to know how to approximate

More information

SOLVING TRIGONOMETRIC INEQUALITIES (CONCEPT, METHODS, AND STEPS) By Nghi H. Nguyen

SOLVING TRIGONOMETRIC INEQUALITIES (CONCEPT, METHODS, AND STEPS) By Nghi H. Nguyen SOLVING TRIGONOMETRIC INEQUALITIES (CONCEPT, METHODS, AND STEPS) By Nghi H. Nguyen DEFINITION. A trig inequality is an inequality in standard form: R(x) > 0 (or < 0) that contains one or a few trig functions

More information

4. Complex integration: Cauchy integral theorem and Cauchy integral formulas. Definite integral of a complex-valued function of a real variable

4. Complex integration: Cauchy integral theorem and Cauchy integral formulas. Definite integral of a complex-valued function of a real variable 4. Complex integration: Cauchy integral theorem and Cauchy integral formulas Definite integral of a complex-valued function of a real variable Consider a complex valued function f(t) of a real variable

More information

EE2 Mathematics : Complex Variables

EE2 Mathematics : Complex Variables EE2 Mathematics : omplex Variables http://www2.imperial.ac.uk/ nsjones These notes are not identical word-for-word with my lectures which will be given on the blackboard. Some of these notes may contain

More information

Metric Spaces. Chapter 7. 7.1. Metrics

Metric Spaces. Chapter 7. 7.1. Metrics Chapter 7 Metric Spaces A metric space is a set X that has a notion of the distance d(x, y) between every pair of points x, y X. The purpose of this chapter is to introduce metric spaces and give some

More information

MATH 132: CALCULUS II SYLLABUS

MATH 132: CALCULUS II SYLLABUS MATH 32: CALCULUS II SYLLABUS Prerequisites: Successful completion of Math 3 (or its equivalent elsewhere). Math 27 is normally not a sufficient prerequisite for Math 32. Required Text: Calculus: Early

More information

MATH 381 HOMEWORK 2 SOLUTIONS

MATH 381 HOMEWORK 2 SOLUTIONS MATH 38 HOMEWORK SOLUTIONS Question (p.86 #8). If g(x)[e y e y ] is harmonic, g() =,g () =, find g(x). Let f(x, y) = g(x)[e y e y ].Then Since f(x, y) is harmonic, f + f = and we require x y f x = g (x)[e

More information

SECOND-ORDER LINEAR HOMOGENEOUS DIFFERENTIAL EQUATIONS

SECOND-ORDER LINEAR HOMOGENEOUS DIFFERENTIAL EQUATIONS L SECOND-ORDER LINEAR HOOGENEOUS DIFFERENTIAL EQUATIONS SECOND-ORDER LINEAR HOOGENEOUS DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS A second-order linear differential equation is one of the form d

More information

Homework # 3 Solutions

Homework # 3 Solutions Homework # 3 Solutions February, 200 Solution (2.3.5). Noting that and ( + 3 x) x 8 = + 3 x) by Equation (2.3.) x 8 x 8 = + 3 8 by Equations (2.3.7) and (2.3.0) =3 x 8 6x2 + x 3 ) = 2 + 6x 2 + x 3 x 8

More information

Lecture 5 : Continuous Functions Definition 1 We say the function f is continuous at a number a if

Lecture 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 information

Lecture 7 : Inequalities 2.5

Lecture 7 : Inequalities 2.5 3 Lecture 7 : Inequalities.5 Sometimes a problem may require us to find all numbers which satisfy an inequality. An inequality is written like an equation, except the equals sign is replaced by one of

More information

APPLICATIONS OF DIFFERENTIATION

APPLICATIONS OF DIFFERENTIATION 4 APPLICATIONS OF DIFFERENTIATION APPLICATIONS OF DIFFERENTIATION So far, we have been concerned with some particular aspects of curve sketching: Domain, range, and symmetry (Chapter 1) Limits, continuity,

More information

Math 209 Solutions to Assignment 7. x + 2y. 1 x + 2y i + 2. f x = cos(y/z)), f y = x z sin(y/z), f z = xy z 2 sin(y/z).

Math 209 Solutions to Assignment 7. x + 2y. 1 x + 2y i + 2. f x = cos(y/z)), f y = x z sin(y/z), f z = xy z 2 sin(y/z). Math 29 Solutions to Assignment 7. Find the gradient vector field of the following functions: a fx, y lnx + 2y; b fx, y, z x cosy/z. Solution. a f x x + 2y, f 2 y x + 2y. Thus, the gradient vector field

More information

The Limit of a Sequence of Numbers: Infinite Series

The Limit of a Sequence of Numbers: Infinite Series Connexions module: m36135 1 The Limit of a Sequence of Numbers: Infinite Series Lawrence Baggett This work is produced by The Connexions Project and licensed under the Creative Commons Attribution License

More information

Math 120 Final Exam Practice Problems, Form: A

Math 120 Final Exam Practice Problems, Form: A Math 120 Final Exam Practice Problems, Form: A Name: While every attempt was made to be complete in the types of problems given below, we make no guarantees about the completeness of the problems. Specifically,

More information

CHAPTER 13. Definite Integrals. Since integration can be used in a practical sense in many applications it is often

CHAPTER 13. Definite Integrals. Since integration can be used in a practical sense in many applications it is often 7 CHAPTER Definite Integrals Since integration can be used in a practical sense in many applications it is often useful to have integrals evaluated for different values of the variable of integration.

More information

PRACTICE FINAL. Problem 1. Find the dimensions of the isosceles triangle with largest area that can be inscribed in a circle of radius 10cm.

PRACTICE FINAL. Problem 1. Find the dimensions of the isosceles triangle with largest area that can be inscribed in a circle of radius 10cm. PRACTICE FINAL Problem 1. Find the dimensions of the isosceles triangle with largest area that can be inscribed in a circle of radius 1cm. Solution. Let x be the distance between the center of the circle

More information

Learning Objectives for Math 165

Learning Objectives for Math 165 Learning Objectives for Math 165 Chapter 2 Limits Section 2.1: Average Rate of Change. State the definition of average rate of change Describe what the rate of change does and does not tell us in a given

More information

k=1 k2, and therefore f(m + 1) = f(m) + (m + 1) 2 =

k=1 k2, and therefore f(m + 1) = f(m) + (m + 1) 2 = Math 104: Introduction to Analysis SOLUTIONS Alexander Givental HOMEWORK 1 1.1. Prove that 1 2 +2 2 + +n 2 = 1 n(n+1)(2n+1) for all n N. 6 Put f(n) = n(n + 1)(2n + 1)/6. Then f(1) = 1, i.e the theorem

More information

since by using a computer we are limited to the use of elementary arithmetic operations

since by using a computer we are limited to the use of elementary arithmetic operations > 4. Interpolation and Approximation Most functions cannot be evaluated exactly: x, e x, ln x, trigonometric functions since by using a computer we are limited to the use of elementary arithmetic operations

More information

Math 4310 Handout - Quotient Vector Spaces

Math 4310 Handout - Quotient Vector Spaces Math 4310 Handout - Quotient Vector Spaces Dan Collins The textbook defines a subspace of a vector space in Chapter 4, but it avoids ever discussing the notion of a quotient space. This is understandable

More information

Math 41: Calculus Final Exam December 7, 2009

Math 41: Calculus Final Exam December 7, 2009 Math 41: Calculus Final Exam December 7, 2009 Name: SUID#: Select your section: Atoshi Chowdhury Yuncheng Lin Ian Petrow Ha Pham Yu-jong Tzeng 02 (11-11:50am) 08 (10-10:50am) 04 (1:15-2:05pm) 03 (11-11:50am)

More information

Math 113 HW #9 Solutions

Math 113 HW #9 Solutions Math 3 HW #9 Solutions 4. 50. Find the absolute maximum and absolute minimum values of on the interval [, 4]. f(x) = x 3 6x 2 + 9x + 2 Answer: First, we find the critical points of f. To do so, take the

More information

6.8 Taylor and Maclaurin s Series

6.8 Taylor and Maclaurin s Series 6.8. TAYLOR AND MACLAURIN S SERIES 357 6.8 Taylor and Maclaurin s Series 6.8.1 Introduction The previous section showed us how to find the series representation of some functions by using the series representation

More information

Real Roots of Univariate Polynomials with Real Coefficients

Real Roots of Univariate Polynomials with Real Coefficients Real Roots of Univariate Polynomials with Real Coefficients mostly written by Christina Hewitt March 22, 2012 1 Introduction Polynomial equations are used throughout mathematics. When solving polynomials

More information

Series Convergence Tests Math 122 Calculus III D Joyce, Fall 2012

Series Convergence Tests Math 122 Calculus III D Joyce, Fall 2012 Some series converge, some diverge. Series Convergence Tests Math 22 Calculus III D Joyce, Fall 202 Geometric series. We ve already looked at these. We know when a geometric series converges and what it

More information

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES Contents 1. Random variables and measurable functions 2. Cumulative distribution functions 3. Discrete

More information

Math 317 HW #5 Solutions

Math 317 HW #5 Solutions Math 317 HW #5 Solutions 1. Exercise 2.4.2. (a) Prove that the sequence defined by x 1 = 3 and converges. x n+1 = 1 4 x n Proof. I intend to use the Monotone Convergence Theorem, so my goal is to show

More information

Roots and Coefficients of a Quadratic Equation Summary

Roots and Coefficients of a Quadratic Equation Summary Roots and Coefficients of a Quadratic Equation Summary For a quadratic equation with roots α and β: Sum of roots = α + β = and Product of roots = αβ = Symmetrical functions of α and β include: x = and

More information

Zeros of a Polynomial Function

Zeros of a Polynomial Function Zeros of a Polynomial Function An important consequence of the Factor Theorem is that finding the zeros of a polynomial is really the same thing as factoring it into linear factors. In this section we

More information

Chapter 5 Polar Coordinates; Vectors 5.1 Polar coordinates 1. Pole and polar axis

Chapter 5 Polar Coordinates; Vectors 5.1 Polar coordinates 1. Pole and polar axis Chapter 5 Polar Coordinates; Vectors 5.1 Polar coordinates 1. Pole and polar axis 2. Polar coordinates A point P in a polar coordinate system is represented by an ordered pair of numbers (r, θ). If r >

More information

PROBLEM SET. Practice Problems for Exam #1. Math 1352, Fall 2004. Oct. 1, 2004 ANSWERS

PROBLEM SET. Practice Problems for Exam #1. Math 1352, Fall 2004. Oct. 1, 2004 ANSWERS PROBLEM SET Practice Problems for Exam # Math 352, Fall 24 Oct., 24 ANSWERS i Problem. vlet R be the region bounded by the curves x = y 2 and y = x. A. Find the volume of the solid generated by revolving

More information

5. Möbius Transformations

5. Möbius Transformations 5. Möbius Transformations 5.1. The linear transformation and the inversion. In this section we investigate the Möbius transformation which provides very convenient methods of finding a one-to-one mapping

More information

INTRODUCTION TO THE CONVERGENCE OF SEQUENCES

INTRODUCTION TO THE CONVERGENCE OF SEQUENCES INTRODUCTION TO THE CONVERGENCE OF SEQUENCES BECKY LYTLE Abstract. In this paper, we discuss the basic ideas involved in sequences and convergence. We start by defining sequences and follow by explaining

More information

6. Metric spaces. In this section we review the basic facts about metric spaces. d : X X [0, )

6. Metric spaces. In this section we review the basic facts about metric spaces. d : X X [0, ) 6. Metric spaces In this section we review the basic facts about metric spaces. Definitions. A metric on a non-empty set X is a map with the following properties: d : X X [0, ) (i) If x, y X are points

More information

Fourier Series. Chapter Some Properties of Functions Goal Preliminary Remarks

Fourier Series. Chapter Some Properties of Functions Goal Preliminary Remarks Chapter 3 Fourier Series 3.1 Some Properties of Functions 3.1.1 Goal We review some results about functions which play an important role in the development of the theory of Fourier series. These results

More information

Sept 20, 2011 MATH 140: Calculus I Tutorial 2. ln(x 2 1) = 3 x 2 1 = e 3 x = e 3 + 1

Sept 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 information

COMPLEX NUMBERS. a bi c di a c b d i. a bi c di a c b d i For instance, 1 i 4 7i 1 4 1 7 i 5 6i

COMPLEX NUMBERS. a bi c di a c b d i. a bi c di a c b d i For instance, 1 i 4 7i 1 4 1 7 i 5 6i COMPLEX NUMBERS _4+i _-i FIGURE Complex numbers as points in the Arg plane i _i +i -i A complex number can be represented by an expression of the form a bi, where a b are real numbers i is a symbol with

More information

Applications of Integration Day 1

Applications of Integration Day 1 Applications of Integration Day 1 Area Under Curves and Between Curves Example 1 Find the area under the curve y = x2 from x = 1 to x = 5. (What does it mean to take a slice?) Example 2 Find the area under

More information

Free Response Questions Compiled by Kaye Autrey for face-to-face student instruction in the AP Calculus classroom

Free Response Questions Compiled by Kaye Autrey for face-to-face student instruction in the AP Calculus classroom Free Response Questions 1969-005 Compiled by Kaye Autrey for face-to-face student instruction in the AP Calculus classroom 1 AP Calculus Free-Response Questions 1969 AB 1 Consider the following functions

More information

Course 221: Analysis Academic year , First Semester

Course 221: Analysis Academic year , First Semester Course 221: Analysis Academic year 2007-08, First Semester David R. Wilkins Copyright c David R. Wilkins 1989 2007 Contents 1 Basic Theorems of Real Analysis 1 1.1 The Least Upper Bound Principle................

More information

11.7 Polar Form of Complex Numbers

11.7 Polar Form of Complex Numbers 11.7 Polar Form of Complex Numbers 989 11.7 Polar Form of Complex Numbers In this section, we return to our study of complex numbers which were first introduced in Section.. Recall that a complex number

More information

Chapter 12. The Straight Line

Chapter 12. The Straight Line 302 Chapter 12 (Plane Analytic Geometry) 12.1 Introduction: Analytic- geometry was introduced by Rene Descartes (1596 1650) in his La Geometric published in 1637. Accordingly, after the name of its founder,

More information

Section 1. Statements and Truth Tables. Definition 1.1: A mathematical statement is a declarative sentence that is true or false, but not both.

Section 1. Statements and Truth Tables. Definition 1.1: A mathematical statement is a declarative sentence that is true or false, but not both. M3210 Supplemental Notes: Basic Logic Concepts In this course we will examine statements about mathematical concepts and relationships between these concepts (definitions, theorems). We will also consider

More information

Calculus 1: Sample Questions, Final Exam, Solutions

Calculus 1: Sample Questions, Final Exam, Solutions Calculus : Sample Questions, Final Exam, Solutions. Short answer. Put your answer in the blank. NO PARTIAL CREDIT! (a) (b) (c) (d) (e) e 3 e Evaluate dx. Your answer should be in the x form of an integer.

More information

FIRST YEAR CALCULUS. Chapter 7 CONTINUITY. It is a parabola, and we can draw this parabola without lifting our pencil from the paper.

FIRST YEAR CALCULUS. Chapter 7 CONTINUITY. It is a parabola, and we can draw this parabola without lifting our pencil from the paper. FIRST YEAR CALCULUS WWLCHENW L c WWWL W L Chen, 1982, 2008. 2006. This chapter originates from material used by the author at Imperial College, University of London, between 1981 and 1990. It It is is

More information

Polynomial and Rational Functions

Polynomial and Rational Functions Polynomial and Rational Functions Quadratic Functions Overview of Objectives, students should be able to: 1. Recognize the characteristics of parabolas. 2. Find the intercepts a. x intercepts by solving

More information

Section 2-5 Quadratic Equations and Inequalities

Section 2-5 Quadratic Equations and Inequalities -5 Quadratic Equations and Inequalities 5 a bi 6. (a bi)(c di) 6. c di 63. Show that i k, k a natural number. 6. Show that i k i, k a natural number. 65. Show that i and i are square roots of 3 i. 66.

More information