John M. Erdman Portland State University Version August 1 .
Exercises and Problems in CalculusJohn M. ErdmanPortland State UniversityVersion August 1, 2013c 2010 John M. ErdmanE-mail address: erdman@pdx.edu
ContentsPrefacePart 1.ixPRELIMINARY MATERIAL1Chapter 1. INEQUALITIES AND ABSOLUTE VALUES1.1. Background1.2. Exercises1.3. Problems1.4. Answers to Odd-Numbered Exercises33456Chapter 2. LINES IN THE PLANE2.1. Background2.2. Exercises2.3. Problems2.4. Answers to Odd-Numbered Exercises778910Chapter 3. FUNCTIONS3.1. Background3.2. Exercises3.3. Problems3.4. Answers to Odd-Numbered Exercises1111121517Part 2.LIMITS AND CONTINUITY19Chapter 4. LIMITS4.1. Background4.2. Exercises4.3. Problems4.4. Answers to Odd-Numbered Exercises2121222425Chapter 5. CONTINUITY5.1. Background5.2. Exercises5.3. Problems5.4. Answers to Odd-Numbered Exercises2727282930Part 3.31DIFFERENTIATION OF FUNCTIONS OF A SINGLE VARIABLEChapter 6. DEFINITION OF THE DERIVATIVE6.1. Background6.2. Exercises6.3. Problems6.4. Answers to Odd-Numbered Exercises3333343637Chapter 7.39TECHNIQUES OF DIFFERENTIATIONiii
ivCONTENTS7.1. Background7.2. Exercises7.3. Problems7.4. Answers to Odd-Numbered Exercises39404547Chapter 8. THE MEAN VALUE THEOREM8.1. Background8.2. Exercises8.3. Problems8.4. Answers to Odd-Numbered Exercises4949505152Chapter 9. L’HÔPITAL’S RULE9.1. Background9.2. Exercises9.3. Problems9.4. Answers to Odd-Numbered Exercises5353545657Chapter10.1.10.2.10.3.10.4.10. MONOTONICITY AND CONCAVITYBackgroundExercisesProblemsAnswers to Odd-Numbered Exercises5959606566Chapter11.1.11.2.11.3.11.4.11. INVERSE FUNCTIONSBackgroundExercisesProblemsAnswers to Odd-Numbered Exercises6969707274Chapter12.1.12.2.12.3.12.4.12. APPLICATIONS OF THE DERIVATIVEBackgroundExercisesProblemsAnswers to Odd-Numbered Exercises7575768284Part 4.INTEGRATION OF FUNCTIONS OF A SINGLE VARIABLE87Chapter13.1.13.2.13.3.13.4.13. THE RIEMANN INTEGRALBackgroundExercisesProblemsAnswers to Odd-Numbered Exercises8989909395Chapter14.1.14.2.14.3.14.4.14. THE FUNDAMENTAL THEOREM OF CALCULUSBackgroundExercisesProblemsAnswers to Odd-Numbered . TECHNIQUES OF INTEGRATIONBackgroundExercisesProblemsAnswers to Odd-Numbered Exercises107107108115118
Chapter16.1.16.2.16.3.16.4.Part 5.CONTENTSv16. APPLICATIONS OF THE INTEGRALBackgroundExercisesProblemsAnswers to Odd-Numbered Exercises121121122127130SEQUENCES AND SERIES131Chapter17.1.17.2.17.3.17.4.17. APPROXIMATION BY POLYNOMIALSBackgroundExercisesProblemsAnswers to Odd-Numbered .18. SEQUENCES OF REAL NUMBERSBackgroundExercisesProblemsAnswers to Odd-Numbered .19. INFINITE SERIESBackgroundExercisesProblemsAnswers to Odd-Numbered .20. CONVERGENCE TESTS FOR SERIESBackgroundExercisesProblemsAnswers to Odd-Numbered .21. POWER SERIESBackgroundExercisesProblemsAnswers to Odd-Numbered Exercises157157158164166Part 6.SCALAR FIELDS AND VECTOR FIELDS169Chapter22.1.22.2.22.3.22.4.22. VECTOR AND METRIC PROPERTIES of RnBackgroundExercisesProblemsAnswers to Odd-Numbered .23. LIMITS OF SCALAR FIELDSBackgroundExercisesProblemsAnswers to Odd-Numbered Exercises181181182184185Part 7.DIFFERENTIATION OF FUNCTIONS OF SEVERAL VARIABLES187
viCONTENTSChapter24.1.24.2.24.3.24.4.24. PARTIAL DERIVATIVESBackgroundExercisesProblemsAnswers to Odd-Numbered .25. GRADIENTS OF SCALAR FIELDS AND TANGENT PLANESBackgroundExercisesProblemsAnswers to Odd-Numbered .26. MATRICES AND DETERMINANTSBackgroundExercisesProblemsAnswers to Odd-Numbered .27. LINEAR MAPSBackgroundExercisesProblemsAnswers to Odd-Numbered .28. DEFINITION OF DERIVATIVEBackgroundExercisesProblemsAnswers to Odd-Numbered .29. DIFFERENTIATION OF FUNCTIONS OF SEVERAL VARIABLLESBackgroundExercisesProblemsAnswers to Odd-Numbered .30. MORE APPLICATIONS OF THE DERIVATIVEBackgroundExercisesProblemsAnswers to Odd-Numbered Exercises239239241243244Part 8.PARAMETRIZED CURVES245Chapter31.1.31.2.31.3.31.4.31. PARAMETRIZED CURVESBackgroundExercisesProblemsAnswers to Odd-Numbered Exercises247247248255256Chapter32.1.32.2.32.3.32. ACCELERATION AND CURVATUREBackgroundExercisesProblems259259260263
CONTENTS32.4.Answers to Odd-Numbered ExercisesPart 9.MULTIPLE INTEGRALSvii265267Chapter33.1.33.2.33.3.33.4.33. DOUBLE INTEGRALSBackgroundExercisesProblemsAnswers to Odd-Numbered .34. SURFACESBackgroundExercisesProblemsAnswers to Odd-Numbered .35. SURFACE AREABackgroundExercisesProblemsAnswers to Odd-Numbered Exercises283283284286287Chapter36.1.36.2.36.3.36. TRIPLE INTEGRALSBackgroundExercisesAnswers to Odd-Numbered . CHANGE OF VARIABLES IN AN INTEGRALBackgroundExercisesProblemsAnswers to Odd-Numbered Exercises295295296298299Chapter38.1.38.2.38.3.38. VECTOR FIELDSBackgroundExercisesAnswers to Odd-Numbered Exercises301301302304Part 10.THE CALCULUS OF DIFFERENTIAL FORMS305Chapter39.1.39.2.39.3.39.4.39. DIFFERENTIAL FORMSBackgroundExercisesProblemsAnswers to Odd-Numbered .40. THE EXTERIOR DIFFERENTIAL OPERATORBackgroundExercisesProblemsAnswers to Odd-Numbered Exercises313313315316317Chapter 41. THE HODGE STAR OPERATOR41.1. Background41.2. Exercises319319320
viiiCONTENTS41.3. Problems41.4. Answers to Odd-Numbered ExercisesChapter42.1.42.2.42.3.42.4.42. CLOSED AND EXACT DIFFERENTIAL FORMSBackgroundExercisesProblemsAnswers to Odd-Numbered ExercisesPart 11.THE FUNDAMENTAL THEOREM OF CALCULUS321322323323324325326327Chapter 43. MANIFOLDS AND ORIENTATION43.1. Background—The Language of ManifoldsOriented pointsOriented curvesOriented surfacesOriented solids43.2. Exercises43.3. Problems43.4. Answers to Odd-Numbered .2.44.3.44.4.44. LINE INTEGRALSBackgroundExercisesProblemsAnswers to Odd-Numbered .45. SURFACE INTEGRALSBackgroundExercisesProblemsAnswers to Odd-Numbered .46. STOKES’ THEOREMBackgroundExercisesProblemsAnswers to Odd-Numbered Exercises351351352356358Bibliography359Index361
PrefaceThis 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 andmost of the problems are meant to illuminate points that in my experience students have foundconfusing.Virtually all of the exercises have fill-in-the-blank type answers. Often an exercise will end πwith something like, “ . . . so the answer is a 3 where a and b .” Onebadvantage of this type of answer is that it makes it possible to provide students with feedback on asubstantial 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 πworks through the exercise and comes up with an answer that doesn’t look anything like a 3 ,bhe/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 communication skills, a matter which in my opinion is of great importance even in beginning courses. Thatis what the problems are for. They require logically thought through, clearly organized, and clearlywritten 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 inthe material through discussion and collaborative work.This collection is divided into parts and chapters roughly by topic. Many chapters begin witha “background” section. This is most emphatically not intended to serve as an exposition of therelevant material. It is designed only to fix notation, definitions, and conventions (which varywidely from text to text) and to clarify what topics one should have studied before tackling theexercises 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. Thedivisions into chapters in these notes, the order of the chapters, and the order of items within achapter is in no way intended to reflect opinions I have about the way in which (or even if) calculusshould be taught. For the convenience of those who might wish to make use of these notes I havesimply chosen what seems to me one fairly common ordering of topics. Neither the exercises nor theproblems are ordered by difficulty. Utterly trivial problems sit alongside ones requiring substantialthought.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 bestregarded as jests whose purpose is to demonstrate in the very simplest ways some connectionsbetween physical quantities (area of a field, volume of a silo, speed of a train, etc.) and themathematics one is learning. It does not make these “real world” problems. No one seriouslyimagines that some Farmer Jones is really interested in maximizing the area of his necessarilyrectangular stream-side pasture with a fixed amount of fencing, or that your friend Sally justhappens to notice that the train passing her is moving at 54.6 mph. To my mind genuinelyinteresting “real world” problems require, in general, way too much background to fit comfortablyinto an already overstuffed calculus course. You will find in this collection just a very few seriousapplications, problem 15 in Chapter 29, for example, where the background is either minimal orlargely irrelevant to the solution of the problem.ix
xPREFACEI 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 oranother, through generations of calculus texts, making any serious attempt at proper attributionquite futile. If anyone feels slighted, please contact me.There will surely be errors. I will be delighted to receive corrections, suggestions, or criticismaterdman@pdx.eduAI have placed the the L TEX source files on my web page so that anyone who wishes can downloadthe material, edit it, add to it, and use it for any noncommercial purpose.
Part 1PRELIMINARY MATERIAL
CHAPTER 1INEQUALITIES AND ABSOLUTE VALUES1.1. BackgroundTopics: inequalities, absolute values.1.1.1. Definition. If x and a are two real numbers the distance between x and a is x a . Formost purposes in calculus it is better to think of an inequality like x 5 2 geometrically ratherthen algebraically. That is, think “The number x is within 2 units of 5,” rather than “The absolutevalue of x minus 5 is strictly less than 2.” The first formulation makes it clear that x is in the openinterval (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 symmetricneighborhood (or a δ-neighborhood) of a.1.1.3. Definition. A deleted (or punctured) neighborhood of a point a R is an openinterval around a from which a has been deleted. Thus, for example, the deleted δ-neighborhoodabout 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 neighborhood 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); andc(4) x A 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 inPart 6, the set Rn , Euclidean n-dimensional space.3
41. INEQUALITIES AND ABSOLUTE VALUES1.2. Exercises(1) The inequality x 2 6 can be expressed in the form a x b where a b .and(2) The inequality 15 x 7 can be expressed in the form x a b wherea 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 .3x1(5) Solve the inequality 2 . Express your answer in interval notation.x 2x 1Answer: [,) [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 b .and(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 .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 inequalityAnswer: [2x2 31 . Express your answer in interval notation.142,].(13) Solve the inequality x 3 6. Express your answer in interval notation.Answer: (,] [,).xx 3(14) Solve the inequality . Express your answer in interval notation.x 2x 4Answer: (,) [,).x 1x 2(15) In interval notation the solution set for the inequality x 2x 3is ( ,) [, 2 ).(16) Solve the inequalityAnswer: (4x2 x 19 1. Express your answer in interval notation.x3 x2 4x 4,].(17) Solve the equation 2 x 3 2 15 x 3 7 0.Answer: x ,x ,x , and x .2(18) Solve the inequality x 1 . Express your answer in interval notation.xAnswer: [,0) [,).and
1.3. PROBLEMS1.3. Problems(1) Let a, b R. Show that a b a b .(2) Let a, b R. Show that ab 12 (a2 b2 ).5
61. INEQUALITIES AND ABSOLUTE VALUES1.4. Answers to Odd-Numbered Exercises(1) 4, 87(3) 4(5) [ 12 , 1) [2, )(7) 3, 7(9) 11, 3 (11) 7, 7(13) ( , 3] [9, )(15) ( , 3) [ 47 , 2)75(17) 10, , , 422
CHAPTER 2LINES IN THE PLANE2.1. BackgroundTopics: 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 slopeof the (nonvertical straight) line L which passes through these points isy2 y1mL : .x2 x1The equation for L isy 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) itis common to say that it has infinite slope and write mL . The equation for a vertical line isx x0 where (x0 , y0 ) is any point lying on L.)Two nonvertical lines L and L 0 are parallel if their respective slopes mL and mL 0 are equal.(Any two vertical lines are parallel.) They are perpendicular if their respective slopes are negative1reciprocals; that is, if mL 0 . (Vertical lines are always perpendicular to horizontal lines.)mL7
82. LINES IN THE PLANE2.2. Exercises(1) The equation of the line passing through the points ( 7, 3) and (8, 2) is ay x b wherea 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 isand its y-intercept is.(4) The equation of the line which passes through the point (4, 2) and is perpendicular to theand b .line x 2y 1 is ax by 1 0 where a (5) The equation of the line which is parallel to the line x 32 y and b point ( 1, 3) is 2x ay b 0 where a 52and passes through the.
2.3. PROBLEMS92.3. Problems(1) The town of Plainfield is 4 miles east and 6 miles north of Burlington. Allentown is 8miles west and 1 mile north of Plainfield. A straight road passes through Plainfield andBurlington. A second straight road passes through Allentown and intersects the first roadat a point somewhere south and west of Burlington. The angle at which the roads intersectis π/4 radians. Explain how to find the location of the point of intersection and carry outthe computation you describe.(2) Prove that the line segment joining the midpoints of two sides of a triangle is half thelength of the third side and is parallel to it. Hint. Try not to make things any morecomplicated than they need to be. A thoughtful choice of a coordinate system may behelpful. One possibility: orient the triangle so that one side runs along the x-axis and onevertex is at the origin.
102. LINES IN THE PLANE2.4. Answers to Odd-Numbered Exercises(1) 3, 2(3) 8, 6(1) 3, 11
CHAPTER 3FUNCTIONS3.1. BackgroundTopics: functions, domain, codomain, range, bounded above, bounded below, composition offunctions.3.1.1. Definition. If S and T are sets we say that f is a function from S to T if for every xin S there corresponds one and only one element f (x) in T . The set S is called the domain of fand is denoted by dom f . The set T is called the codomain of f . The range of f is the set ofall f (x) such that x belongs to S. It is denoted by ran f . The words function, map, mapping, andtransformation are synonymous.A function f : A B is said to be real valued if B R and is called a function of a real variableif A R.The notation f : S T : x 7 f (x) indicates that f is a function whose domain is S, whosecodomain is T , and whose value at x is f (x). Thus, for example, f : R R : x 7 x2 defines thereal valued function whose value at each real number x is given by f (x) x2 . We use dom f todenote 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 forevery x S, It is bounded below by a number K if K f (x) for every x S. And it isbounded if it is bounded both above and below; that is, if there exists N 0 such that f (x) Nfor every x S.3.1.3. Definition. Let f and g be real valued functions of a real variable. Define the compositeof 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
123. FUNCTIONS3.2. Exercises1(1) Let f (x) 1 . Then:11 1x(a) f ( 12 ) .,(b) The domain of f is the set of all real numbers except 7 x2 9(2) Let f (x) . Then dom f (,] [,25 x2 (3) Find the domain and range of the function f (x) 2 4 x2 3.,] and ran f [,].Answer: dom f [, and.).(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) 1.(6) Let f (x) 21 11 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. x2 4 . Then, in interval notation, that part of the domain of f which(7) Let f (x) 5 36 x2and b .is to the right of the origin is [2, a) (a, b] where a (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 byg(x) (2x) 1 (f (1 x) f (1 x)). Then g(x) can be written in the form ax2 bx cwhere a ,b , and c .(10) The cost of making a widget is 75 cents. If they are sold for 1.95 each, 3000 widgets canbe sold. For every cent the price is lowered, 60 more widgets can be sold.(a) If
1.2. Exercises 4 1.3. Problems 5 1.4. Answers to Odd-Numbered Exercises6 Chapter 2. LINES IN THE PLANE7 2.1. Background 7 2.2. Exercises 8 2.3. Problems 9 2.4. Answers to Odd-Numbered Exercises10 Chapter 3. FUNCTIONS11 3.1. Background 11 3.2. Exercises 12 3.3. Problems 15 3.4. Answers to Odd-Numbered Exercises17 Part 2. LIMITS AND CONTINUITY 19 .
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