Evaluate the iterated integral. 1 0 s4 cos(s5) dt ds 0

Exercises and Problems in Calculus

John M. Erdman

Portland State University

Version August 1, 2013

c©2010 John M. Erdman

E-mail address: erdman@pdx.edu

Contents

Preface ix

Part 1. PRELIMINARY MATERIAL 1

Chapter 1. INEQUALITIES AND ABSOLUTE VALUES 3 1.1. Background 3 1.2. Exercises 4 1.3. Problems 5 1.4. Answers to Odd-Numbered Exercises 6

Chapter 2. LINES IN THE PLANE 7 2.1. Background 7 2.2. Exercises 8 2.3. Problems 9 2.4. Answers to Odd-Numbered Exercises 10

Chapter 3. FUNCTIONS 11 3.1. Background 11 3.2. Exercises 12 3.3. Problems 15 3.4. Answers to Odd-Numbered Exercises 17

Part 2. LIMITS AND CONTINUITY 19

Chapter 4. LIMITS 21 4.1. Background 21 4.2. Exercises 22 4.3. Problems 24 4.4. Answers to Odd-Numbered Exercises 25

Chapter 5. CONTINUITY 27 5.1. Background 27 5.2. Exercises 28 5.3. Problems 29 5.4. Answers to Odd-Numbered Exercises 30

Part 3. DIFFERENTIATION OF FUNCTIONS OF A SINGLE VARIABLE 31

Chapter 6. DEFINITION OF THE DERIVATIVE 33 6.1. Background 33 6.2. Exercises 34 6.3. Problems 36 6.4. Answers to Odd-Numbered Exercises 37

Chapter 7. TECHNIQUES OF DIFFERENTIATION 39

iii

iv CONTENTS

7.1. Background 39 7.2. Exercises 40 7.3. Problems 45 7.4. Answers to Odd-Numbered Exercises 47

Chapter 8. THE MEAN VALUE THEOREM 49 8.1. Background 49 8.2. Exercises 50 8.3. Problems 51 8.4. Answers to Odd-Numbered Exercises 52

Chapter 9. L’HÔPITAL’S RULE 53 9.1. Background 53 9.2. Exercises 54 9.3. Problems 56 9.4. Answers to Odd-Numbered Exercises 57

Chapter 10. MONOTONICITY AND CONCAVITY 59 10.1. Background 59 10.2. Exercises 60 10.3. Problems 65 10.4. Answers to Odd-Numbered Exercises 66

Chapter 11. INVERSE FUNCTIONS 69 11.1. Background 69 11.2. Exercises 70 11.3. Problems 72 11.4. Answers to Odd-Numbered Exercises 74

Chapter 12. APPLICATIONS OF THE DERIVATIVE 75 12.1. Background 75 12.2. Exercises 76 12.3. Problems 82 12.4. Answers to Odd-Numbered Exercises 84

Part 4. INTEGRATION OF FUNCTIONS OF A SINGLE VARIABLE 87

Chapter 13. THE RIEMANN INTEGRAL 89 13.1. Background 89 13.2. Exercises 90 13.3. Problems 93 13.4. Answers to Odd-Numbered Exercises 95

Chapter 14. THE FUNDAMENTAL THEOREM OF CALCULUS 97 14.1. Background 97 14.2. Exercises 98 14.3. Problems 102 14.4. Answers to Odd-Numbered Exercises 105

Chapter 15. TECHNIQUES OF INTEGRATION 107 15.1. Background 107 15.2. Exercises 108 15.3. Problems 115 15.4. Answers to Odd-Numbered Exercises 118

CONTENTS v

Chapter 16. APPLICATIONS OF THE INTEGRAL 121 16.1. Background 121 16.2. Exercises 122 16.3. Problems 127 16.4. Answers to Odd-Numbered Exercises 130

Part 5. SEQUENCES AND SERIES 131

Chapter 17. APPROXIMATION BY POLYNOMIALS 133 17.1. Background 133 17.2. Exercises 134 17.3. Problems 136 17.4. Answers to Odd-Numbered Exercises 137

Chapter 18. SEQUENCES OF REAL NUMBERS 139 18.1. Background 139 18.2. Exercises 140 18.3. Problems 143 18.4. Answers to Odd-Numbered Exercises 144

Chapter 19. INFINITE SERIES 145 19.1. Background 145 19.2. Exercises 146 19.3. Problems 148 19.4. Answers to Odd-Numbered Exercises 149

Chapter 20. CONVERGENCE TESTS FOR SERIES 151 20.1. Background 151 20.2. Exercises 152 20.3. Problems 155 20.4. Answers to Odd-Numbered Exercises 156

Chapter 21. POWER SERIES 157 21.1. Background 157 21.2. Exercises 158 21.3. Problems 164 21.4. Answers to Odd-Numbered Exercises 166

Part 6. SCALAR FIELDS AND VECTOR FIELDS 169

Chapter 22. VECTOR AND METRIC PROPERTIES of Rn 171 22.1. Background 171 22.2. Exercises 174 22.3. Problems 177 22.4. Answers to Odd-Numbered Exercises 179

Chapter 23. LIMITS OF SCALAR FIELDS 181 23.1. Background 181 23.2. Exercises 182 23.3. Problems 184 23.4. Answers to Odd-Numbered Exercises 185

Part 7. DIFFERENTIATION OF FUNCTIONS OF SEVERAL VARIABLES 187

vi CONTENTS

Chapter 24. PARTIAL DERIVATIVES 189 24.1. Background 189 24.2. Exercises 190 24.3. Problems 192 24.4. Answers to Odd-Numbered Exercises 193

Chapter 25. GRADIENTS OF SCALAR FIELDS AND TANGENT PLANES 195 25.1. Background 195 25.2. Exercises 196 25.3. Problems 199 25.4. Answers to Odd-Numbered Exercises 201

Chapter 26. MATRICES AND DETERMINANTS 203 26.1. Background 203 26.2. Exercises 207 26.3. Problems 210 26.4. Answers to Odd-Numbered Exercises 213

Chapter 27. LINEAR MAPS 215 27.1. Background 215 27.2. Exercises 217 27.3. Problems 219 27.4. Answers to Odd-Numbered Exercises 221

Chapter 28. DEFINITION OF DERIVATIVE 223 28.1. Background 223 28.2. Exercises 224 28.3. Problems 226 28.4. Answers to Odd-Numbered Exercises 227

Chapter 29. DIFFERENTIATION OF FUNCTIONS OF SEVERAL VARIABLLES 229 29.1. Background 229 29.2. Exercises 232 29.3. Problems 234 29.4. Answers to Odd-Numbered Exercises 237

Chapter 30. MORE APPLICATIONS OF THE DERIVATIVE 239 30.1. Background 239 30.2. Exercises 241 30.3. Problems 243 30.4. Answers to Odd-Numbered Exercises 244

Part 8. PARAMETRIZED CURVES 245

Chapter 31. PARAMETRIZED CURVES 247 31.1. Background 247 31.2. Exercises 248 31.3. Problems 255 31.4. Answers to Odd-Numbered Exercises 256

Chapter 32. ACCELERATION AND CURVATURE 259 32.1. Background 259 32.2. Exercises 260 32.3. Problems 263

CONTENTS vii

32.4. Answers to Odd-Numbered Exercises 265

Part 9. MULTIPLE INTEGRALS 267

Chapter 33. DOUBLE INTEGRALS 269 33.1. Background 269 33.2. Exercises 270 33.3. Problems 274 33.4. Answers to Odd-Numbered Exercises 275

Chapter 34. SURFACES 277 34.1. Background 277 34.2. Exercises 278 34.3. Problems 280 34.4. Answers to Odd-Numbered Exercises 281

Chapter 35. SURFACE AREA 283 35.1. Background 283 35.2. Exercises 284 35.3. Problems 286 35.4. Answers to Odd-Numbered Exercises 287

Chapter 36. TRIPLE INTEGRALS 289 36.1. Background 289 36.2. Exercises 290 36.3. Answers to Odd-Numbered Exercises 293

Chapter 37. CHANGE OF VARIABLES IN AN INTEGRAL 295 37.1. Background 295 37.2. Exercises 296 37.3. Problems 298 37.4. Answers to Odd-Numbered Exercises 299

Chapter 38. VECTOR FIELDS 301 38.1. Background 301 38.2. Exercises 302 38.3. Answers to Odd-Numbered Exercises 304

Part 10. THE CALCULUS OF DIFFERENTIAL FORMS 305

Chapter 39. DIFFERENTIAL FORMS 307 39.1. Background 307 39.2. Exercises 309 39.3. Problems 310 39.4. Answers to Odd-Numbered Exercises 311

Chapter 40. THE EXTERIOR DIFFERENTIAL OPERATOR 313 40.1. Background 313 40.2. Exercises 315 40.3. Problems 316 40.4. Answers to Odd-Numbered Exercises 317

Chapter 41. THE HODGE STAR OPERATOR 319 41.1. Background 319 41.2. Exercises 320

viii CONTENTS

41.3. Problems 321 41.4. Answers to Odd-Numbered Exercises 322

Chapter 42. CLOSED AND EXACT DIFFERENTIAL FORMS 323 42.1. Background 323 42.2. Exercises 324 42.3. Problems 325 42.4. Answers to Odd-Numbered Exercises 326

Part 11. THE FUNDAMENTAL THEOREM OF CALCULUS 327

Chapter 43. MANIFOLDS AND ORIENTATION 329 43.1. Background—The Language of Manifolds 329 Oriented points 330 Oriented curves 330 Oriented surfaces 330 Oriented solids 331 43.2. Exercises 332 43.3. Problems 334 43.4. Answers to Odd-Numbered Exercises 335

Chapter 44. LINE INTEGRALS 337 44.1. Background 337 44.2. Exercises 338 44.3. Problems 342 44.4. Answers to Odd-Numbered Exercises 343

Chapter 45. SURFACE INTEGRALS 345 45.1. Background 345 45.2. Exercises 346 45.3. Problems 348 45.4. Answers to Odd-Numbered Exercises 349

Chapter 46. STOKES’ THEOREM 351 46.1. Background 351 46.2. Exercises 352 46.3. Problems 356 46.4. Answers to Odd-Numbered Exercises 358

Bibliography 359

Index 361

Preface

This is a set of exercises and problems for a (more or less) standard beginning calculus sequence. While a fair number of the exercises involve only routine computations, many of the exercises and most of the problems are meant to illuminate points that in my experience students have found confusing.

Virtually all of the exercises have fill-in-the-blank type answers. Often an exercise will end

with something like, “ . . . so the answer is a √

3 + π

b where a = and b = .” One

advantage of this type of answer is that it makes it possible to provide students with feedback on a substantial number of homework exercises without a huge investment of time. More importantly, it gives students a way of checking their work without giving them the answers. When a student

works through the exercise and comes up with an answer that doesn’t look anything like a √

3 + π

b ,

he/she has been given an obvious invitation to check his/her work. The major drawback of this type of answer is that it does nothing to promote good communi-

cation skills, a matter which in my opinion is of great importance even in beginning courses. That is what the problems are for. They require logically thought through, clearly organized, and clearly written up reports. In my own classes I usually assign problems for group work outside of class. This serves the dual purposes of reducing the burden of grading and getting students involved in the material through discussion and collaborative work.

This collection is divided into parts and chapters roughly by topic. Many chapters begin with a “background” section. This is most emphatically not intended to serve as an exposition of the relevant material. It is designed only to fix notation, definitions, and conventions (which vary widely from text to text) and to clarify what topics one should have studied before tackling the exercises and problems that follow.

The flood of elementary calculus texts published in the past half century shows, if nothing else, that the topics discussed in a beginning calculus course can be covered in virtually any order. The divisions into chapters in these notes, the order of the chapters, and the order of items within a chapter is in no way intended to reflect opinions I have about the way in which (or even if) calculus should be taught. For the convenience of those who might wish to make use of these notes I have simply chosen what seems to me one fairly common ordering of topics. Neither the exercises nor the problems are ordered by difficulty. Utterly trivial problems sit alongside ones requiring substantial thought.

Each chapter ends with a list of the solutions to all the odd-numbered exercises. The great majority of the “applications” that appear here, as in most calculus texts, are best

regarded as jests whose purpose is to demonstrate in the very simplest ways some connections between physical quantities (area of a field, volume of a silo, speed of a train, etc.) and the mathematics one is learning. It does not make these “real world” problems. No one seriously imagines that some Farmer Jones is really interested in maximizing the area of his necessarily rectangular stream-side pasture with a fixed amount of fencing, or that your friend Sally just happens to notice that the train passing her is moving at 54.6 mph. To my mind genuinely interesting “real world” problems require, in general, way too much background to fit comfortably into an already overstuffed calculus course. You will find in this collection just a very few serious applications, problem 15 in Chapter 29, for example, where the background is either minimal or largely irrelevant to the solution of the problem.

ix

x PREFACE

I make no claims of originality. While I have dreamed up many of the items included here, there are many others which are standard calculus exercises that can be traced back, in one form or another, through generations of calculus texts, making any serious attempt at proper attribution quite futile. If anyone feels slighted, please contact me.

There will surely be errors. I will be delighted to receive corrections, suggestions, or criticism at

erdman@pdx.edu

I have placed the the LATEX source files on my web page so that anyone who wishes can download the material, edit it, add to it, and use it for any noncommercial purpose.

Part 1

PRELIMINARY MATERIAL

CHAPTER 1

INEQUALITIES AND ABSOLUTE VALUES

1.1. Background

Topics: inequalities, absolute values.

1.1.1. Definition. If x and a are two real numbers the distance between x and a is |x− a|. For most purposes in calculus it is better to think of an inequality like |x− 5| < 2 geometrically rather then algebraically. That is, think “The number x is within 2 units of 5,” rather than “The absolute value of x minus 5 is strictly less than 2.” The first formulation makes it clear that x is in the open interval (3, 7).

1.1.2. Definition. Let a be a real number. A neighborhood of a is an open interval (c, d) in R which contains a. An open interval (a − δ, a + δ) which is centered at a is a symmetric neighborhood (or a δ-neighborhood) of a.

1.1.3. Definition. A deleted (or punctured) neighborhood of a point a ∈ R is an open interval around a from which a has been deleted. Thus, for example, the deleted δ-neighborhood about 3 would be (3− δ, 3 + δ) \ {3} or, using different notation, (3− δ, 3) ∪ (3, 3 + δ).

1.1.4. Definition. A point a is an accumulation point of a set B ⊆ R if every deleted neigh- borhood of a contains at least one point of B.

1.1.5. Notation (For Set Operations). Let A and B be subsets of a set S. Then

(1) x ∈ A ∪B if x ∈ A or x ∈ B (union); (2) x ∈ A ∩B if x ∈ A and x ∈ B (intersection); (3) x ∈ A \B if x ∈ A and x /∈ B (set difference); and (4) x ∈ Ac if x ∈ S \A (complement).

If the set S is not specified, it is usually understood to be the set R of real numbers or, starting in Part 6, the set Rn, Euclidean n-dimensional space.

3

4 1. INEQUALITIES AND ABSOLUTE VALUES

1.2. Exercises

(1) The inequality |x − 2| < 6 can be expressed in the form a < x < b where a = and b = .

(2) The inequality −15 ≤ x ≤ 7 can be expressed in the form |x− a| ≤ b where a = and b = .

(3) Solve the equation |4x+ 23| = |4x− 9|. Answer: x = . (4) Find all numbers x which satisfy |x2 + 2| = |x2 − 11|.

Answer: x = and x = .

(5) Solve the inequality 3x

x2 + 2 ≥ 1 x− 1

. Express your answer in interval notation.

Answer: [ , ) ∪ [ 2 , ) . (6) Solve the equation |x− 2|2 + 3|x− 2| − 4 = 0.

Answer: x = and x = .

(7) The inequality −4 ≤ x ≤ 10 can be expressed in the form |x− a| ≤ b where a = and b = .

(8) Sketch the graph of the equation x− 2 = |y − 3|. (9) The inequality |x + 4| < 7 can be expressed in the form a < x < b where a = and

b = .

(10) Solve the inequality |3x+ 7| < 5. Express your answer in interval notation. Answer: ( , ).

(11) Find all numbers x which satisfy |x2 − 9| = |x2 − 5|. Answer: x = and x = .

(12) Solve the inequality

∣∣∣∣2x2 − 314 ∣∣∣∣ ≤ 12. Express your answer in interval notation.

Answer: [ , ].

(13) Solve the inequality |x− 3| ≥ 6. Express your answer in interval notation. Answer: ( , ] ∪ [ , ) .

(14) Solve the inequality x

x+ 2 ≥ x+ 3 x− 4

. Express your answer in interval notation.

Answer: ( , ) ∪ [ , ).

(15) In interval notation the solution set for the inequality x+ 1

x− 2 ≤ x+ 2 x+ 3

is (−∞, ) ∪ [ , 2 ).

(16) Solve the inequality 4x2 − x+ 19

x3 + x2 + 4x+ 4 ≥ 1. Express your answer in interval notation.

Answer: ( , ].

(17) Solve the equation 2|x+ 3|2 − 15|x+ 3|+ 7 = 0. Answer: x = , x = , x = , and x = .

(18) Solve the inequality x ≥ 1 + 2 x

. Express your answer in interval notation.

Answer: [ , 0 ) ∪ [ , ).

1.3. PROBLEMS 5

1.3. Problems

(1) Let a, b ∈ R. Show that | |a| − |b| | ≤ |a− b|. (2) Let a, b ∈ R. Show that |ab| ≤ 12(a

2 + b2).

6 1. INEQUALITIES AND ABSOLUTE VALUES

1.4. Answers to Odd-Numbered Exercises

(1) −4, 8

(3) −7 4

(5) [−12 , 1) ∪ [2,∞) (7) 3, 7

(9) −11, 3 (11) −

√ 7, √

7

(13) (−∞,−3] ∪ [9,∞) (15) (−∞,−3) ∪ [−74 , 2)

(17) −10, −7 2

, −5 2

, 4

CHAPTER 2

LINES IN THE PLANE

2.1. Background

Topics: equations of lines in the plane, slope, x- and y-intercepts, parallel and perpendicular lines.

2.1.1. Definition. Let (x1, y1) and (x2, y2) be points in the plane such that x1 6= x2. The slope of the (nonvertical straight) line L which passes through these points is

m L

:= y2 − y1 x2 − x1

.

The equation for L is y − y0 = mL(x− x0)

where (x0, y0) is any point lying on L. (If the line L is vertical (that is, parallel to the y-axis) it is common to say that it has infinite slope and write m

L = ∞. The equation for a vertical line is

x = x0 where (x0, y0) is any point lying on L.) Two nonvertical lines L and L ′ are parallel if their respective slopes m

L and m

L ′ are equal.

(Any two vertical lines are parallel.) They are perpendicular if their respective slopes are negative

reciprocals; that is, if m L ′

= 1

m L

. (Vertical lines are always perpendicular to horizontal lines.)

7

8 2. LINES IN THE PLANE

2.2. Exercises

(1) The equation of the line passing through the points (−7,−3) and (8, 2) is ay = x+b where a = and b =

(2) The equation of the perpendicular bisector of the line segment joining the points (2,−5) and (4, 3) is ax+ by + 1 = 0 where a = and b = .

(3) Let L be the line passing through the point (4, 9) with slope 34 . The x-intercept of L is and its y-intercept is .

(4) The equation of the line which passes through the point (4, 2) and is perpendicular to the line x+ 2y = 1 is ax+ by + 1 = 0 where a = and b = .

(5) The equation of the line which is parallel to the line x + 32y = 5 2 and passes through the

point (−1,−3) is 2x+ ay + b = 0 where a = and b = .

2.3. PROBLEMS 9

2.3. Problems

(1) The town of Plainfield is 4 miles east and 6 miles north of Burlington. Allentown is 8 miles west and 1 mile north of Plainfield. A straight road passes through Plainfield and Burlington. A second straight road passes through Allentown and intersects the first road at a point somewhere south and west of Burlington. The angle at which the roads intersect is π/4 radians. Explain how to find the location of the point of intersection and carry out the computation you describe.

(2) Prove that the line segment joining the midpoints of two sides of a triangle is half the length of the third side and is parallel to it. Hint. Try not to make things any more complicated than they need to be. A thoughtful choice of a coordinate system may be helpful. One possibility: orient the triangle so that one side runs along the x-axis and one vertex is at the origin.

10 2. LINES IN THE PLANE

2.4. Answers to Odd-Numbered Exercises

(1) 3, −2 (3) −8, 6 (1) 3, 11

CHAPTER 3

FUNCTIONS

3.1. Background

Topics: functions, domain, codomain, range, bounded above, bounded below, composition of functions.

3.1.1. Definition. If S and T are sets we say that f is a function from S to T if for every x in S there corresponds one and only one element f(x) in T . The set S is called the domain of f and is denoted by dom f . The set T is called the codomain of f . The range of f is the set of all f(x) such that x belongs to S. It is denoted by ran f . The words function, map, mapping, and transformation are synonymous.

A function f : A→ B is said to be real valued if B ⊆ R and is called a function of a real variable if A ⊆ R.

The notation f : S → T : x 7→ f(x) indicates that f is a function whose domain is S, whose codomain is T , and whose value at x is f(x). Thus, for example, f : R → R : x 7→ x2 defines the real valued function whose value at each real number x is given by f(x) = x2. We use dom f to denote the domain of f and ran f to denote its range.

3.1.2. Definition. A function f : S → R is bounded above by a number M is f(x) ≤ M for every x ∈ S, It is bounded below by a number K if K ≤ f(x) for every x ∈ S. And it is bounded if it is bounded both above and below; that is, if there exists N > 0 such that |f(x)| ≤ N for every x ∈ S.

3.1.3. Definition. Let f and g be real valued functions of a real variable. Define the composite of g and f , denoted by g ◦ f , by

(g ◦ f)(x) := g(f(x)) for all x ∈ dom f such that f(x) ∈ dom g. The operation ◦ is called composition.

For problem 2, the following fact may be useful.

3.1.4. Theorem. Every nonempty open interval in R contains both rational and irrational numbers.

11

12 3. FUNCTIONS

3.2. Exercises

(1) Let f(x) = 1

1 + 1

1 + 1

x

. Then:

(a) f(12) = .

(b) The domain of f is the set of all real numbers except , , and .

(2) Let f(x) = 7− √ x2 − 9√

25− x2 . Then dom f = ( , ] ∪ [ , ).

(3) Find the domain and range of the function f(x) = 2 √

4− x2 − 3. Answer: dom f = [ , ] and ran f = [ , ].

(4) Let f(x) = x3 − 4x2 − 11x− 190. The set of all numbers x such that |f(x)− 40| < 260 is ( , ) ∪ ( , ).

(5) Let f(x) = x+ 5, g(x) = √ x, and h(x) = x2. Then (g ◦ (h− (g ◦ f)))(4) = .

(6) Let f(x) = 1

1− 2

1 + 1

1− x

.

(a) Find f(1/2). Answer. .

(b) Find the domain of f . Answer. The domain of f is the set of all real numbers except , , and .

(7) Let f(x) =

√ x2 − 4

5− √

36− x2 . Then, in interval notation, that part of the domain of f which

is to the right of the origin is [2, a) ∪ (a, b] where a = and b = .

(8) Let f(x) = (−x2 − 7x− 10)−1/2. (a) Then f(−3) = . (b) The domain of f is ( , ) .

(9) Let f(x) = x3 − 4 for all real numbers x. Then for all x 6= 0 define a new function g by g(x) = (2x)−1(f(1 + x) − f(1 − x)). Then g(x) can be written in the form ax2 + bx + c where a = , b = , and c = .

(10) The cost of making a widget is 75 cents. If they are sold for $1.95 each, 3000 widgets can be sold. For every cent the price is lowered, 60 more widgets can be sold.

(a) If x is the price of a widget in cents, then the net profit is p(x) = ax2 + bx+ c where a = , b = , and c = .

(b) The “best” price (that is, the price that maximizes profit) is x = $ . .

(c) At this best price the profit is $ .

(11) Let f(x) = 3 √

25− x2 + 2. Then dom f = [ , ] and ran f = [ , ]. (12) Find a formula exhibiting the area A of an equilateral triangle as a function of the length

s of one of its sides.

Answer: A(s) = .

(13) Let f(x) = 4x3−18x2−4x+33. Find the largest set S on which the function f is bounded above by 15 and below by −15.

Answer: S = [ , ] ∪ [ , ] ∪ [ , ] .

3.2. EXERCISES 13

(14) Let f(x) = √ x, g(x) =

4

5− x , and h(x) = x2. Find (h ◦ ((h ◦ g ◦ f)− f))(4).

Answer: .

(15) Let f(x) = x+ 7, g(x) = √ x+ 2, and h(x) = x2. Find (h ◦ ((f ◦ g)− (g ◦ f)))(7).

Answer: .

(16) Let f(x) = √

5− x, g(x) = √ x+ 11, h(x) = 2(x− 1)−1, and j(x) = 4x− 1.

Then (f ◦ (g + (h ◦ g)(h ◦ j)))(5) = .

(17) Let f(x) = x2, g(x) = √

9 + x, and h(x) = 1

x− 2 . Then (h ◦ (f ◦ g− g ◦ f))(4) = .

(18) Let f(x) = x2, g(x) = √

9 + x, and h(x) = (x− 1)1/3. Then (h ◦ ((f ◦ g)(g ◦ f)))(4) = .

(19) Let f(x) = 5

x , g(x) =

√ x, and h(x) = x+ 1. Then (g(f ◦ g) + (g ◦ f ◦ h))(4) = .

(20) Let g(x) = 5− x2, h(x) = √ x+ 13, and j(x) =

1

x . Then (j ◦ h ◦ g)(3) = .

(21) Let h(x) = 1√ x+ 6

, j(x) = 1

x , and g(x) = 5− x2. Then (g ◦ j ◦ h)(3) = .

(22) Let f(x) = x2 + 2

x , g(x) =

2

2x+ 3 , and h(x) =

√ 2x. Then (h ◦ g ◦ f)(4) = .

(23) Let f(x) = 3(x+ 1)3, g(x) = x5 + x4

x+ 1 , and h(x) =

√ x.

Then (h ◦ (g + (h ◦ f)))(2) = . (24) Let f(x) = x2, g(x) =

√ x+ 11, h(x) = 2(x− 1)−1, and j(x) = 4x− 1.

Then (f ◦ ((h ◦ g) + (h ◦ j)))(5) = . (25) Let f(x) = x3−5x2+x−7. Find a function g such that (f ◦g)(x) = 27x3+90x2+78x−2.

Answer: g(x) = .

(26) Let f(x) = cosx and g(x) = x2 for all x. Write each of the following functions in terms of f and g. Example. If h(x) = cos2 x2, then h = g ◦ f ◦ g. (a) If h(x) = cosx2, then h = .

(b) If h(x) = cosx4, then h= .

(c) If h(x) = cos4 x2, then h = .

(d) If h(x) = cos(cos2 x), then h = .

(e) If h(x) = cos2(x4 + x2), then h = .

(27) Let f(x) = x3, g(x) = x − 2, and h(x) = sinx for all x. Write each of the following functions in terms of f , g, and h. Example. If k(x) = sin3(x− 2)3, then k = f ◦ h ◦ f ◦ g. (a) If k(x) = sin3 x, then k = .

(b) If k(x) = sinx3, then k = .

(c) If k(x) = sin(x3 − 2), then k = . (d) If k(x) = sin(sinx− 2), then k = . (e) If k(x) = sin3(sin3(x− 2), then k = . (f) If k(x) = sin9(x− 2), then k = . (g) If k(x) = sin(x3 − 8), then k = . (h) If k(x) = sin(x3 − 6x2 + 12x− 8), then k = .

14 3. FUNCTIONS

(28) Let g(x) = 3x− 2. Find a function f such that (f ◦ g)(x) = 18x2 − 36x+ 19. Answer: f(x) = .

(29) Let h(x) = arctanx for x ≥ 0, g(x) = cosx, and f(x) = (1−x2)−1. Find a number p such that (f ◦ g ◦ h)(x) = 1 + xp. Answer: p = .

(30) Let f(x) = 3x2 + 5x+ 1. Find a function g such that (f ◦g)(x) = 3x4 + 6x3−4x2−7x+ 3. Answer: g(x) = .

(31) Let g(x) = 2x− 1. Find a function f such that (f ◦ g)(x) = 8x3 − 28x2 + 28x− 14. Answer: f(x) = .

(32) Find two solutions to the equation

8 cos3(π(x2 + 83x+ 2)) + 16 cos 2(π(x2 + 83x+ 2)) + 16 cos(π(x

2 + 83x+ 2)) = 13.

Answer: and .

(33) Let f(x) = (x+ 4)−1/2, g(x) = x2 + 1, h(x) = (x− 3)1/2, and j(x) = x−1. Then (j ◦ ((g ◦ h)− (g ◦ f)))(5) = .

(34) Let g(x) = 3x− 2. Find a function f such that (f ◦ g)(x) = 18x2 − 36x+ 19. Answer: f(x) = .

(35) Let f(x) = x3−5x2+x−7. Find a function g such that (f ◦g)(x) = 27x3+90x2+78x−2. Answer: g(x) = .

(36) Let f(x) = x2 + 1. Find a function g such that (f ◦ g)(x) = 2 + 2 x

+ 1

x2 .

Answer: g(x) = .

(37) Let f(x) = x2 + 3x+ 4. Find two functions g such that (f ◦ g)(x) = 4x2 − 6x+ 4. Answer: g(x) = and g(x) = .

(38) Let h(x) = x−1 and g(x) = √ x + 1. Find a function f such that (f ◦ g ◦ h)(x) =

x−3/2 + 4x−1 + 2x−1/2 − 6. Answer: f(x) = .

(39) Let g(x) = x2 + x− 1. Find a function f such that (f ◦ g)(x) = x4 + 2x3 − 3x2 − 4x+ 6. Answer: f(x) = .

(40) Let S(x) = x2 and P (x) = 2x.

Then (S ◦ S ◦ S ◦ S ◦ P ◦ P )(−1) = .

3.3. PROBLEMS 15

3.3. Problems

(1) Do there exist functions f and g defined on R such that f(x) + g(y) = xy

for all real numbers x and y? Explain.

(2) Your friend Susan has become interested in functions f : R → R which preserve both the operation of addition and the operation of multiplication; that is, functions f which satisfy

f(x+ y) = f(x) + f(y) (3.1)

and

f(xy) = f(x)f(y) (3.2)

for all x, y ∈ R. Naturally she started her investigation by looking at some examples. The trouble is that she was able to find only two very simple examples: f(x) = 0 for all x and f(x) = x for all x. After expending considerable effort she was unable to find additional examples. She now conjectures that there are no other functions satisfying (3.1) and (3.2). Write Susan a letter explaining why she is correct.

Hint. You may choose to pursue the following line of argument. Assume that f is a function (not identically zero) which satisfies (3.1) and (3.2) above. (a) Show that f(0) = 0. [In (3.1) let y = 0.] (b) Show that if a 6= 0 and a = ab, then b = 1. (c) Show that f(1) = 1. [How do we know that there exists a number c such that

f(c) 6= 0? Let x = c and y = 1 in (3.2).] (d) Show that f(n) = n for every natural number n. (e) Show that f(−n) = −n for every natural number n. [Let x = n and y = −n in (3.1).

Use (d).] (f) Show that f(1/n) = 1/n for every natural number n. [Let x = n and y = 1/n

in (3.2).] (g) Show that f(r) = r for every rational number r. [If r ≥ 0 write r = m/n where

m and n are natural numbers; then use (3.2), (d), and (e). Next consider the case r < 0.]

(h) Show that if x ≥ 0, then f(x) ≥ 0. [Write x as √ x √ x and use (3.2).]

(i) Show that if x ≤ y, then f(x) ≤ f(y). [Show that f(−x) = −f(x) holds for all real numbers x. Use (h).]

(j) Now prove that f must be the identity function on R. [Argue by contradiction: Assume f(x) 6= x for some number x. Then there are two possibilities: either f(x) > x or f(x) < x. Show that both of these lead to a contradiction. Apply theorem 3.1.4 to the two cases f(x) > x and f(x) < x to obtain the contradiction f(x) < f(x).]

(3) Let f(x) = 1−x and g(x) = 1/x. Taking composites of these two functions in all possible ways (f ◦ f , g ◦ f , f ◦ g ◦ f ◦ f ◦ f , g ◦ g ◦ f ◦ g ◦ f ◦ f , etc.), how many distinct functions can be produced? Write each of the resulting functions in terms of f and g. How do you know there are no more? Show that each function on your list has an inverse which is also on your list. What is the common domain for these functions? That is, what is the largest set of real numbers for which all these functions are defined?

(4) Prove or disprove: composition of functions is commutative; that is g ◦ f = f ◦ g when both sides are defined.

(5) Let f , g, h : R→ R. Prove or disprove: f ◦ (g + h) = f ◦ g + f ◦ h. (6) Let f , g, h : R→ R. Prove or disprove: (f + g) ◦ h = (f ◦ h) + (g ◦ h).

16 3. FUNCTIONS

(7) Let a ∈ R be a constant and let f(x) = a− x for all x ∈ R. Show that f ◦ f = I (where I is the identity function on R: I(x) = x for all x).

3.4. ANSWERS TO ODD-NUMBERED EXERCISES 17

3.4. Answers to Odd-Numbered Exercises

(1) (a) 3

4

(b) −1, −1 2

, 0

(3) [−2, 2], [−3, 1] (5) √

13

(7) √

11, 6

(9) 1, 0, 3

(11) [−5, 5], [2, 17] (13) [−32 ,−1] ∪ [1, 2] ∪ [4,

9 2 ]

(15) 36

(17) 1

6

(19) 6

(21) −4 (23) 5

(25) 3x+ 5

(27) (a) f ◦ h (b) h ◦ f (c) h ◦ g ◦ f (d) h ◦ g ◦ h (e) f ◦ h ◦ f ◦ h ◦ g (f) f ◦ f ◦ h ◦ g (g) h ◦ g ◦ g ◦ g ◦ g ◦ f (h) h ◦ f ◦ g

(29) −2 (31) x3 − 4x2 + 3x− 6

(33) 9

17

(35) 3x+ 5

(37) −2x, 2x− 3 (39) x2 − 2x+ 3

Part 2

LIMITS AND CONTINUITY

CHAPTER 4

LIMITS

4.1. Background

Topics: limit of f(x) as x approaches a, limit of f(x) as x approaches infinity, left- and right-hand limits.

4.1.1. Definition. Suppose that f is a real valued function of a real variable, a is an accumulation point of the domain of f , and ` ∈ R. We say that ` is the limit of f(x) as x approaches a if for every neighborhood V of ` there exists a corresponding deleted neighborhood U of a which satisfies the following condition:

for every point x in the domain of f which lies in U the point f(x) lies in V .

Once we have convinced ourselves that in this definition it doesn’t matter if we work only with symmetric neighborhoods of points, we can rephrase the definition in a more conventional algebraic fashion: ` is the limit of f(x) as x approaches a provided that for every � > 0 there exists δ > 0 such that if 0 < |x− a| < δ and x ∈ dom f , then |f(x)− `| < �.

4.1.2. Notation. To indicate that a number ` is the limit of f(x) as x approaches a, we may write either

lim x→a

f(x) = l or f(x)→ ` as x→ a.

(See problem 2.)

21

22 4. LIMITS

4.2. Exercises

(1) lim x→3

x3 − 13x2 + 51x− 63 x3 − 4x2 − 3x+ 18

= a

5 where a = .

(2) lim x→0

√ x2 + 9x+ 9− 3

x = a

2 where a = .

(3) lim x→1

x3 − x2 + 2x− 2 x3 + 3x2 − 4x

= 3

a where a = .

(4) lim t→0

t√ 4− t− 2

= .

(5) lim x→0

√ x+ 9− 3

x =

1

a where a = .

(6) lim x→2

x3 − 3x2 + x+ 2 x3 − x− 6

= 1

a where a = .

(7) lim x→2

x3 − x2 − 8x+ 12 x3 − 10x2 + 28x− 24

= −a 4

where a = .

(8) lim x→0

√ x2 − x+ 4− 2 x2 + 3x

= −1 a

where a = .

(9) lim x→1

x3 + x2 − 5x+ 3 x3 − 4x2 + 5x− 2

= .

(10) lim x→3

x3 − 4x2 − 3x+ 18 x3 − 8x2 + 21x− 18

= .

(11) lim x→−1

x3 − x2 − 5x− 3 x3 + 6x2 + 9x+ 4

= −4 a

where a = .

(12) lim x→0

2x sinx

1− cosx = .

(13) lim x→0

1− cosx 3x sinx

= 1

a where a = .

(14) lim x→0

tan 3x− sin 3x x3

= a

2 where a = .

(15) lim h→0

sin 2h

5h2 + 7h = .

(16) lim h→0

cot 7h

cot 5h = .

(17) lim x→0

secx− cosx 3x2

= 1

a where a = .

(18) lim x→∞

(9x8 − 6x5 + 4)1/2

(64x12 + 14x7 − 7)1/3 = a

4 where a = .

(19) lim x→∞

√ x( √ x+ 3−

√ x− 2) = a

2 where a = .

(20) lim x→∞

7− x+ 2x2 − 3x3 − 5x4

4 + 3x− x2 + x3 + 2x4 = a

2 where a = .

(21) lim x→∞

(2x4 − 137)5

(x2 + 429)10 = .

4.2. EXERCISES 23

(22) lim x→∞

(5x10 + 32)3

(1− 2x6)5 = − a

32 where a = .

(23) lim x→∞

(√ x2 + x− x

) =

1

a where a = .

(24) lim x→∞

x(256x4 + 81x2 + 49)−1/4 = 1

a where a = .

(25) lim x→∞

x

(√ 3x2 + 22−

√ 3x2 + 4

) = a √ a where a = .

(26) lim x→∞

x 2 3 ( (x+ 1)

1 3 − x

1 3 )

= 1

a where a = .

(27) lim x→∞

(√ x+ √ x−

√ x− √ x

) = .

(28) Let f(x) =

 2x− 1, if x < 2;x2 + 1, if x > 2. Then limx→2− f(x) = and limx→2+ f(x) = . (29) Let f(x) =

|x− 1| x− 1

. Then lim x→1−

f(x) = and lim x→1+

f(x) = .

(30) Let f(x) =

 5x− 3, if x < 1;x2, if x ≥ 1. Then limx→1− f(x) = and limx→1+ f(x) = . (31) Let f(x) =

 3x+ 2, if x < −2;x2 + 3x− 1, if x ≥ −2. Then limx→−2− f(x) = and limx→−2+ f(x) = . (32) Suppose y = f(x) is the equation of a curve which always lies between the parabola

x2 = y − 1 and the hyperbola yx+ y − 1 = 0. Then lim x→0

f(x) = .

24 4. LIMITS

4.3. Problems

(1) Find lim x→0+

( e−1/x sin(1/x)− (x+ 2)3

) (if it exists) and give a careful argument showing

that your answer is correct.

(2) The notation limx→a f(x) = ` that we use for limits is somewhat optimistic. It assumes the uniqueness of limits. Prove that limits, if they exist, are indeed unique. That is, suppose that f is a real valued function of a real variable, a is an accumulation point of the domain of f , and `, m ∈ R. Prove that if f(x) → ` as x → a and f(x) → m as x → a, then l = m. (Explain carefully why it was important that we require a to be an accumulation point of the domain of f .)

(3) Let f(x) = sinπx

x+ 1 for all x 6= −1. The following information is known about a function g

defined for all real numbers x 6= 1:

(i) g = p

q where p(x) = ax2 + bx+ c and q(x) = dx+ e for some constants a, b, c, d, e;

(ii) the only x-intercept of the curve y = g(x) occurs at the origin; (iii) g(x) ≥ 0 on the interval [0, 1) and is negative elsewhere on its domain; (iv) g has a vertical asymptote at x = 1; and (v) g(1/2) = 3.

Either find lim x→1

g(x)f(x) or else show that this limit does not exist.

Hints. Write an explicit formula for g by determining the constants a . . . e. Use (ii) to find c; use (ii) and (iii) to find a; use (iv) to find a relationship between d and e; then use (v) to obtain an explicit form for g. Finally look at f(x)g(x); replace sinπx by sin(π(x− 1) + π) and use the formula for the sine of the sum of two numbers.

(4) Evaluate lim x→0

√ |x| cos (π1/x2) 2 + √ x2 + 3

(if it exists). Give a careful proof that your conclusion is

correct.

4.4. ANSWERS TO ODD-NUMBERED EXERCISES 25

4.4. Answers to Odd-Numbered Exercises

(1) −4 (3) 5

(5) 6

(7) 5

(9) −4 (11) 3

(13) 6

(15) 2

7

(17) 3

(19) 5

(21) 32

(23) 2

(25) 3

(27) 1

(29) −1, 1 (31) −4, −3

CHAPTER 5

CONTINUITY

5.1. Background

Topics: continuous functions, intermediate value theorem. extreme value theorem.

There are many ways of stating the intermediate value theorem. The simplest says that con- tinuous functions take intervals to intervals.

5.1.1. Definition. A subset J of the real line R is an interval if z ∈ J whenever a, b ∈ J and a < z < b.