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DIFFERENTIALCALCULUSQUESTIONS

FOR MATHEMATICS100AND180Elyse YEAGER

Joel FELDMANAndrew RECHNITZERTHIS DOCUMENT WAS TYPESET ONTHURSDAY1STDECEMBER, 2016.

œLegal stuff

Copyrightc

2016 Elyse Yeager, Joel Feldman and Andrew Rechnitzer

In the near future this will be licensed under the Creative Commons Attribution- NonCommercial-ShareAlike 4.0 International License. You can view a copy of the license at

HOW TO USE THIS BOOKœIntroduction

First of all, welcome to Calculus!

This book is written as a companion to the

CLP notes

ŸŸHow to Work Questions

This book is organized into four sections: Questions, Hints, Answers, and Solutions. As you are working problems, resist the temptation to prematurely peek at the back! It"s important to allow yourself to struggle for a time with the material. Even professional mathematicians don"t always know right away how to solve a problem. The art is in gathering your thoughts and figuring out a strategy to use what you know to find out what you don"t. If you find yourself at a real impasse, go ahead and look for a hint in the Hints section. Think about it for a while, and don"t be afraid to read back in the notes to look for a key idea that will help you proceed. If you still can"t solve the problem, well, we included the Solutions section for a reason! As you"re reading the solutions, try hard to understand why we took the steps we did, instead of memorizing step-by-step how to solve that one particular problem. If you struggled with a question quite a lot, it"s probably a good idea to return to it in a few days. That might have been enough time for you to internalize the necessary ideas, and youmightfinditeasilyconquerable. Patyourselfontheback-sometimesmathmakesyou feel good! If you"re still having troubles, read over the solution again, with an emphasis on understanding why each step makes sense. One of the reasons so many students are required to study calculus is the hope that it will improve their problem-solving skills. In this class, you will learn lots of concepts, and be asked to apply them in a variety of situations. Often, this will involve answering one i

HOW TO USE THIS BOOKreally big problem by breaking it up into manageable chunks, solving those chunks, then

putting the pieces back together. When you see a particularly long question, remain calm and look for a way to break it into pieces you can handle.

ŸŸWorking with Friends

Study buddies are fantastic! If you don"t already have friends in your class, you can ask your neighbours in lecture to form a group. Often, a question that you might bang your Regular study times make sure you don"t procrastinate too much, and friends help you maintain a positive attitude when you might otherwise succumb to frustration. Struggle in mathematics is desirable, but suffering is not. When working in a group, make sure you try out problems on your own before coming together to discuss with others. Learning is a process, and getting answers to questions that you haven"t considered on your own can rob you of the practice you need to master skills and concepts, and the tenacity you need to develop to become a competent problem- solver.

ŸŸTypes of QuestionsQ[1]: Questions outlined in blue make up therepresentative question set. This set of

questions is intended to cover the most essential ideas in each section. These questions are usually highly typical of what you"d see on an exam, although some of them are atypical but carry an important moral. If you find yourself unconfident with the idea behind one of these, it"s probably a good idea to practice similar questions. This representative question set is our suggestion for a minimal selection of questions to

work on. You are highly encouraged to work on more.Q[2](): In addition to original problems, this book contains problems pulled from quizzes

and exams given at UBC for Math 100 and 180 (first-semester calculus) and Math 120 (honours first-semester calculus). These problems are marked with a star. The authors would like to acknowledge the contributions of the many people who collaborated to produce these exams over the years.

Instructions and other comments that are attached to more than one question are written in this font. The

questions are organized into Stage 1, Stage 2, and Stage 3.

ŸŸStage 1

The first category is meant to test and improve your understanding of basic underlying concepts. These often do not involve much calculation. They range in difficulty from very basic reviews of definitions to questions that require you to be thoughtful about the concepts covered in the section.ii

HOW TO USE THIS BOOKŸŸStage 2

Questionsinthiscategoryareforpracticingskills. It"snotenoughtounderstandthephilo- sophical grounding of an idea: you have to be able to apply it in appropriate situations.

This takes practice!

ŸŸStage 3

The last questions in each section go a little farther than Stage 2. Often they will combine more than one idea, incorporate review material, or ask you to apply your understanding of a concept to a new situation. In exams, as in life, you will encounter questions of varying difficulty. A good skill to practice is recognizing the level of difficulty a problem poses. Exams will have some easy question, some standard questions, and some harder questions.iii

HOW TO USE THIS BOOKiv

CONTENTSHow to use this booki

I The questions

1

1 Limits3

1.1 Drawing Tangents and a First Limit

3

1.2 Another Limit and Computing Velocity

4

1.3 The Limit of a Function

5

1.4 Calculating Limits with Limit Laws

7

1.5 Limits at Infinity

12

1.6 Continuity

14

2 Derivatives

17

2.1 Revisiting tangent lines

17

2.2 Definition of the derivative

20

2.3 Interpretations of the derivative

25

2.4 Arithmetic of derivatives - a differentiation toolbox

26

2.5 Proofs of the arithmetic of derivatives

28

2.6 Using the arithmetic of derivatives - examples

28

2.7 Derivatives of Exponential Functions

30

2.8 Derivatives of trigonometric functions

32

2.9 One more tool - the chain rule

35

2.10 The natural logarithm

37

2.11 Implicit Differentiation

40

2.12 Inverse Trigonometric Functions

42

2.13 The Mean Value Theorem

45

2.14 Higher Order Derivatives

48

3 Applications of derivatives

53

3.1 Velocity and acceleration

53
v

CONTENTS CONTENTS

3.2 Related Rates

55

3.3 Exponential Growth and Decay

63

3.3.1 Carbon Dating

63

3.3.2 Newton"s Law of Cooling

65

3.3.3 Population Growth

67

3.3.4 Further problems

68

3.4 Taylor polynomials

69

3.4.1 Zeroeth Approximation

69

3.4.2 First Approximation

71

3.4.3 Second Approximation

72

3.4.4 Still Better Approximations

74

3.4.5 Some Examples

75

3.4.6 Estimating Changes andDx,Dynotation. . . . . . . . . . . . . . . . 76

3.4.7 Further Examples

78

3.4.8 The error in Taylor polynomial approximation

79

3.4.9 Further problems

81

3.5 Optimization

84

3.5.1 Local and global maxima and minima

84

3.5.2 Finding global maxima and minima

86

3.5.3 Max/min examples

87

3.6 Sketching Graphs

89

3.6.1 Domain, intercepts and asymptotes

89

3.6.2 First derivative - increasing or decreasing

91

3.6.3 Second derivative - concavity

93

3.6.4 Symmetries

94

3.6.5 A checklist for sketching

97

3.6.6 Sketching examples

97

3.7 L"H

ˆopital"s Rule and indeterminate forms. . . . . . . . . . . . . . . . . . . . 101

4 Towards mathematics 101

105

4.1 Introduction to antiderivatives

105

5 Extra problems

109

5.1.4 Calculating Limits with Limit Laws

109

5.1.5 Limits at Infinity

110

5.1.6 Continuity

110

5.2.8 Derivatives of trigonometric functions

111

5.3.4 Taylor polynomials

111

II Hints to problems

113

1.1 Drawing Tangents and a First Limit

115

1.2 Another Limit and Computing Velocity

115

1.3 The Limit of a Function

115

1.4 Calculating Limits with Limit Laws

116

1.5 Limits at Infinity

117

1.6 Continuity

119 vi

CONTENTS CONTENTS

2.1 Revisiting tangent lines

119

2.2 Definition of the derivative

120

2.3 Interpretations of the derivative

121

2.4 Arithmetic of derivatives - a differentiation toolbox

121

2.5 Proofs of the arithmetic of derivatives

122

2.6 Using the arithmetic of derivatives - examples

122

2.7 Derivatives of Exponential Functions

123

2.8 Derivatives of trigonometric functions

123

2.9 One more tool - the chain rule

124

2.10 The natural logarithm

126

2.11 Implicit Differentiation

127

2.12 Inverse Trigonometric Functions

128

2.13 The Mean Value Theorem

129

2.14 Higher Order Derivatives

131

3.1 Velocity and acceleration

132

3.2 Related Rates

133

3.3.1 Carbon Dating

134

3.3.2 Newton"s Law of Cooling

135

3.3.3 Population Growth

136

3.3.4 Further problems

136

3.4.1 Zeroeth Approximation

137

3.4.2 First Approximation

137

3.4.3 Second Approximation

137

3.4.4 Still Better Approximations

138

3.4.5 Some Examples

139

3.4.6 Estimating Changes andDx,Dynotation. . . . . . . . . . . . . . . . . . . 139

3.4.7 Further Examples

140

3.4.8 The error in Taylor polynomial approximation

140

3.4.9 Further problems

141

3.5.1 Local and global maxima and minima

142

3.5.2 Finding global maxima and minima

143

3.5.3 Max/min examples

143

3.6.1 Domain, intercepts and asymptotes

145

3.6.2 First derivative - increasing or decreasing

145

3.6.3 Second derivative - concavity

145

3.6.4 Symmetries

146

3.6.5 A checklist for sketching

146

3.6.6 Sketching examples

146

3.7 L"H

ˆopital"s Rule and indeterminate forms. . . . . . . . . . . . . . . . . . . 147

4.1 Introduction to antiderivatives

148

5.1.4 Calculating Limits with Limit Laws

149

5.1.5 Limits at Infinity

150

5.1.6 Continuity

150

5.2.8 Derivatives of trigonometric functions

150

5.3.4 3.4.8 - The error in a Taylor polynomial approximation

151 vii

CONTENTS CONTENTS

III Answers to problems

153
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