Ambar N. Sengupta 17th November, 2011 - LSU Math

[Pages:330]Introductory Calculus Notes

Ambar N. Sengupta 17th November, 2011

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Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1 Sets: Language and Notation

13

1.1 Sets and Elements . . . . . . . . . . . . . . . . . . . . . . . . 13

1.2 Everything from nothing . . . . . . . . . . . . . . . . . . . . . 14

1.3 Subsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.4 Union, Intersections, Complements . . . . . . . . . . . . . . . 17

1.5 Integers and Rationals . . . . . . . . . . . . . . . . . . . . . . 17

1.6 Cartesian Products . . . . . . . . . . . . . . . . . . . . . . . . 18

1.7 Mappings and Functions . . . . . . . . . . . . . . . . . . . . . 19

1.8 Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2 The Extended Real Line

25

2.1 The Real Line . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.2 The Extended Real Line . . . . . . . . . . . . . . . . . . . . . 26

3 Suprema, Infima, Completeness

29

3.1 Upper Bounds and Lower Bounds . . . . . . . . . . . . . . . . 29

3.2 Sup and Inf: Completeness . . . . . . . . . . . . . . . . . . . . 30

3.3 More on Sup and Inf . . . . . . . . . . . . . . . . . . . . . . . 31

4 Neighborhoods, Open Sets and Closed Sets

33

4.1 Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.2 Neighborhoods . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.3 Types of points for a set . . . . . . . . . . . . . . . . . . . . . 35

4.4 Interior, Exterior, and Boundary of a Set . . . . . . . . . . . . 37

4.5 Open Sets and Topology . . . . . . . . . . . . . . . . . . . . . 38

4.6 Closed Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

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4.7 Open Sets and Closed Sets . . . . . . . . . . . . . . . . . . . . 40 4.8 Closed sets in R and in R . . . . . . . . . . . . . . . . . . . . 40

5 Magnitude and Distance

41

5.1 Absolute Value . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5.2 Inequalities and equalities . . . . . . . . . . . . . . . . . . . . 42

5.3 Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5.4 Neighborhoods and distance . . . . . . . . . . . . . . . . . . . 43

6 Limits

45

6.1 Limits, Sup and Inf . . . . . . . . . . . . . . . . . . . . . . . . 46

6.2 Limits for 1/x . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

6.3 A function with no limits . . . . . . . . . . . . . . . . . . . . . 51

6.4 Limits of sequences . . . . . . . . . . . . . . . . . . . . . . . . 52

6.5 Lim with sups and infs . . . . . . . . . . . . . . . . . . . . . . 54

7 Limits: Properties

57

7.1 Up and down with limits . . . . . . . . . . . . . . . . . . . . . 57

7.2 Limits: the standard definition . . . . . . . . . . . . . . . . . . 59

7.3 Limits: working rules . . . . . . . . . . . . . . . . . . . . . . . 61

7.4 Limits by comparing . . . . . . . . . . . . . . . . . . . . . . . 64

7.5 Limits of composite functions . . . . . . . . . . . . . . . . . . 66

8 Trigonometric Functions

69

8.1 Measuring angles . . . . . . . . . . . . . . . . . . . . . . . . . 69

8.2 Geometric specification of sin, cos and tan . . . . . . . . . . . 70

8.3 Reciprocals of sin, cos, and tan . . . . . . . . . . . . . . . . . 74

8.4 Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

8.5 Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

8.6 Limits for sin and cos . . . . . . . . . . . . . . . . . . . . . . . 78

8.7 Limits with sin(1/x) . . . . . . . . . . . . . . . . . . . . . . . 79

8.8 Graphs of trigonometric functions . . . . . . . . . . . . . . . . 80

8.9 Postcript on trigonometric functions . . . . . . . . . . . . . . 81

Exercises on Limits . . . . . . . . . . . . . . . . . . . . . . . . 82

9 Continuity

85

9.1 Continuity at a point . . . . . . . . . . . . . . . . . . . . . . . 85

9.2 Discontinuities . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

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9.3 Continuous functions . . . . . . . . . . . . . . . . . . . . . . . 86 9.4 Two examples using Q . . . . . . . . . . . . . . . . . . . . . . 87 9.5 Composites of continuous functions . . . . . . . . . . . . . . . 88 9.6 Continuity on R . . . . . . . . . . . . . . . . . . . . . . . . . 88

10 The Intermediate Value Theorem

91

10.1 Inequalities from limits . . . . . . . . . . . . . . . . . . . . . . 92

10.2 Intermediate Value Theorem . . . . . . . . . . . . . . . . . . . 93

10.3 Intermediate Value Theorem: a second formulation . . . . . . 94

10.4 Intermediate Value Theorem: an application . . . . . . . . . . 95

10.5 Locating roots . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

11 Inverse Functions

99

11.1 Inverse trigonometric functions . . . . . . . . . . . . . . . . . 99

11.2 Monotone functions: terminology . . . . . . . . . . . . . . . . 102

11.3 Inverse functions . . . . . . . . . . . . . . . . . . . . . . . . . 103

12 Maxima and Minima

107

12.1 Maxima and Minima . . . . . . . . . . . . . . . . . . . . . . . 107

12.2 Maxima/minima with infinities . . . . . . . . . . . . . . . . . 110

12.3 Closed and bounded sets . . . . . . . . . . . . . . . . . . . . . 111

13 Tangents, Slopes and Derivatives

113

13.1 Secants and tangents . . . . . . . . . . . . . . . . . . . . . . . 114

13.2 Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

13.3 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

13.4 The derivative of x2 . . . . . . . . . . . . . . . . . . . . . . . . 118

13.5 Derivative of x3 . . . . . . . . . . . . . . . . . . . . . . . . . . 120

13.6 Derivative of xn . . . . . . . . . . . . . . . . . . . . . . . . . . 121

13.7 Derivative of x-1 = 1/x . . . . . . . . . . . . . . . . . . . . . . 121

13.8 13.9

Derivative Derivative

of of

x-k x1/2

= =

1/xk x.

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122 123

13.10Derivatives of powers of x . . . . . . . . . . . . . . . . . . . . 125

13.11Derivatives with infinities . . . . . . . . . . . . . . . . . . . . 125

14 Derivatives of Trigonometric Functions

127

14.1 Derivative of sin is cos . . . . . . . . . . . . . . . . . . . . . . 127

14.2 Derivative of cos is - sin . . . . . . . . . . . . . . . . . . . . . 129

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14.3 Derivative of tan is sec2 . . . . . . . . . . . . . . . . . . . . . 129

15 Differentiability and Continuity

131

15.1 Differentiability implies continuity . . . . . . . . . . . . . . . . 131

16 Using the Algebra of Derivatives

133

16.1 Using the sum rule . . . . . . . . . . . . . . . . . . . . . . . . 134

16.2 Using the product rule . . . . . . . . . . . . . . . . . . . . . . 134

16.3 Using the quotient rule . . . . . . . . . . . . . . . . . . . . . . 135

17 Using the Chain Rule

137

17.1 Initiating examples . . . . . . . . . . . . . . . . . . . . . . . . 137

17.2 The chain rule . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

18 Proving the Algebra of Derivatives

141

18.1 Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

18.2 Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

18.3 Quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

19 Proving the Chain Rule

145

19.1 Why it works . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

19.2 Proof the chain rule . . . . . . . . . . . . . . . . . . . . . . . . 146

20 Using Derivatives for Extrema

151

20.1 Quadratics with calculus . . . . . . . . . . . . . . . . . . . . . 152

20.2 Quadratics by algebra . . . . . . . . . . . . . . . . . . . . . . 153

20.3 Distance to a line . . . . . . . . . . . . . . . . . . . . . . . . . 155

20.4 Other geometric examples . . . . . . . . . . . . . . . . . . . . 160

Exercises on Maxima and Minima . . . . . . . . . . . . . . . . 164

21 Local Extrema and Derivatives

167

21.1 Local Maxima and Minima . . . . . . . . . . . . . . . . . . . . 167

Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . 169

22 Mean Value Theorem

171

22.1 Rolle's Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 171

22.2 Mean Value Theorem . . . . . . . . . . . . . . . . . . . . . . . 172

22.3 Rolle's theorem on R . . . . . . . . . . . . . . . . . . . . . . 174

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23 The Sign of the Derivative

175

23.1 Positive derivative and increasing nature . . . . . . . . . . . . 175

23.2 Negative derivative and decreasing nature . . . . . . . . . . . 179

23.3 Zero slope and constant functions . . . . . . . . . . . . . . . . 179

24 Differentiating Inverse Functions

181

24.1 Inverses and Derivatives . . . . . . . . . . . . . . . . . . . . . 182

25 Analyzing local extrema with higher derivatives

185

25.1 Local extrema and slope behavior . . . . . . . . . . . . . . . . 185

25.2 The second derivative test . . . . . . . . . . . . . . . . . . . . 188

26 Exp and Log

191

26.1 Exp summarized . . . . . . . . . . . . . . . . . . . . . . . . . 191

26.2 Log summarized . . . . . . . . . . . . . . . . . . . . . . . . . . 193

26.3 Real Powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

26.4 Example Calculations . . . . . . . . . . . . . . . . . . . . . . . 197

26.5 Proofs for Exp and Log . . . . . . . . . . . . . . . . . . . . . . 198

27 Convexity

205

27.1 Convex and concave functions . . . . . . . . . . . . . . . . . . 205

27.2 Convexity and slope . . . . . . . . . . . . . . . . . . . . . . . 206

27.3 Checking convexity/concavity . . . . . . . . . . . . . . . . . . 208

27.4 Inequalities from convexity/concavity . . . . . . . . . . . . . . 209

27.5 Convexity and derivatives . . . . . . . . . . . . . . . . . . . . 213

27.6 Supporting Lines . . . . . . . . . . . . . . . . . . . . . . . . . 215

27.7 Convex combinations . . . . . . . . . . . . . . . . . . . . . . . 218

Exercises on Maxima/Minima , Mean Value Theorem, Convexity223

28 L'Hospital's Rule

225

28.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226

28.2 Proving l'Hospital's rule . . . . . . . . . . . . . . . . . . . . . 228

Exercises on l'Hosptal's rule . . . . . . . . . . . . . . . . . . . 232

29 Integration

233

29.1 From areas to integrals . . . . . . . . . . . . . . . . . . . . . . 233

29.2 The Riemann integral . . . . . . . . . . . . . . . . . . . . . . . 235

29.3 Refining partitions . . . . . . . . . . . . . . . . . . . . . . . . 237

29.4 Estimating approximation error . . . . . . . . . . . . . . . . . 239

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29.5 Continuous functions are integrable . . . . . . . . . . . . . . . 240 29.6 A function for which the integral does not exist . . . . . . . . 242 29.7 Basic properties of the integral . . . . . . . . . . . . . . . . . . 243

30 The Fundamental Theorem of Calculus

245

30.1 Fundamental theorem of calculus . . . . . . . . . . . . . . . . 245

30.2 Differentials and integrals . . . . . . . . . . . . . . . . . . . . 246

30.3 Using the fundamental theorem . . . . . . . . . . . . . . . . . 249

30.4 Indefinite integrals . . . . . . . . . . . . . . . . . . . . . . . . 252

30.5 Revisiting the exponential function . . . . . . . . . . . . . . . 254

31 Riemann Sum Examples

257

31.1 Riemann sums for

N dx 1 x2

. . . . . . . . . . . . . . . . . . . . . 257

31.2 Riemann sums for 1/x . . . . . . . . . . . . . . . . . . . . . . 260

31.3 Riemann sums for x . . . . . . . . . . . . . . . . . . . . . . . 261

31.4 Riemann sums for x2 . . . . . . . . . . . . . . . . . . . . . . . 264

31.5 Power sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266

32 Integration Techniques

269

32.1 Substitutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 269

32.2 Some trigonometric integrals . . . . . . . . . . . . . . . . . . . 276

32.3 Summary of basic trigonometric integrals . . . . . . . . . . . . 280

32.4 Using trigonometric substitutions . . . . . . . . . . . . . . . . 282

32.5 Integration by parts . . . . . . . . . . . . . . . . . . . . . . . . 286

Exercises on the Substitution Method . . . . . . . . . . . . . . 290

33 Paths and Length

291

33.1 Paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291

33.2 Lengths of paths . . . . . . . . . . . . . . . . . . . . . . . . . 294

33.3 Paths and Curves . . . . . . . . . . . . . . . . . . . . . . . . . 295

33.4 Lengths for graphs . . . . . . . . . . . . . . . . . . . . . . . . 297

34 Selected Solutions

301

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324

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