[PDF] Math 137 Calculus 1 for Honours Mathematics Course Notes

View - Download Math 137 Calculus 1 for Honours Mathematics Course Notes

Previous PDF

Next PDF

Math 137

Calculus 1

for Honours Mathematics

Course Notes

Barbara A. Forrest and Brian E. ForrestVersion 1.62

Copyright

c

Barbara A. Forrest and Brian E. Forrest.

All rights reserved.

August 1, 2023

All rights, including copyright and images in the content of these course notes, are owned by the course authors Barbara Forrest and Brian Forrest. By accessing these course notes, you agree that you may only use the content for your own personal, non-commercial use. You are not permitted to copy, transmit, adapt, or change in any way the content of these course notes for any other purpose whatsoever without the prior written permission of the course authors.

Author Contact Information:

Barbara Forrest (baforres@uwaterloo.ca)

Brian Forrest (beforres@uwaterloo.ca)

i

QUICK REFERENCE PAGE 1

Right Angle Trigonometry

sin=oppositehypotenuse cos=adjacenthypotenuse tan=oppositeadjacent csc=1sinsec=1coscot=1tanRadians

The anglein

radians equals the length of the directed arcBP, taken positive counter-clockwise and negative clockwise.

Thus,radians=180

or 1rad=180 .Definition of Sine and Cosine

For any, cosand sinare

defined to be thexandy coordinates of the pointPon the unit circle such that the radius

OPmakes an angle ofradians

with the positivexaxis. Thus sin=AP, and cos=OA.The Unit Circle ii

QUICK REFERENCE PAGE 2

Trigonometric IdentitiesPythagoreancos

2+sin2=1Identity

Range1cos11sin1Periodicitycos(2)=cossin(2)=sinSymmetrycos()=cossin()=sinSum and Dierence Identitiescos(A+B)=cosAcosBsinAsinBcos(AB)=cosAcosB+sinAsinBsin(A+B)=sinAcosB+cosAsinBsin(AB)=sinAcosBcosAsinBComplementary Angle Identities

cos( 2

A)=sinAsin(

2

2=1+cos22

Identitiessin

2=1cos22

Other1+tan2A=sec2Aiii

QUICK REFERENCE PAGE 3

f(x)=x2f(x)=x3f(x)=jxj f(x)=cos(x)f(x)=sin(x)f(x)=tan(x) f(x)=sec(x)f(x)=csc(x)f(x)=cot(x) f(x)=ex1 1 f(x)=ln(x) iv

QUICK REFERENCE PAGE 4

Differentiation RulesFunctionDerivative

f(x)=cxa,a,0,c2Rf

0(x)=caxa1f(x)=sin(x)f

0(x)=cos(x)f(x)=cos(x)f

0(x)=sin(x)f(x)=tan(x)f

0(x)=sec2(x)f(x)=sec(x)f

0(x)=sec(x)tan(x)f(x)=arcsin(x)f

0(x)=1p1x2f(x)=arccos(x)f

0(x)=1p1x2f(x)=arctan(x)f

0(x)=11+x2f(x)=exf

0(x)=exf(x)=axwitha>0f

0(x)=axln(a)f(x)=ln(x) forx>0f

0(x)=1xTable of AntiderivativesR

xndx=xn+1n+1+CR 1x dx=ln(jxj)+CR exdx=ex+CR sin(x)dx=cos(x)+CR cos(x)dx=sin(x)+CR sec2(x)dx=tan(x)+CR

11+x2dx=arctan(x)+CR

1p1x2dx=arcsin(x)+CR

1p1x2dx=arccos(x)+CR

sec(x)tan(x)dx=sec(x)+CR axdx=axln(a)+Cn-th degree Taylor polynomial forfcentered atx=aT n;a(x)=nP k=0f (k)(a)k!(xa)k=f(a)+f0(a)(xa)+f00(a)2! (xa)2++f(n)(a)n!(xa)nLinear Approximations (L0(x)) and Taylor Polynomials (Tn;0(x))f(x)=exL

2;0(x)=f(0)+f0(0)(x0)+f00(0)2!

(x0)2=e0+e0(x)+e02! (x0)2=1+x+x22 T

3;0(x)=1+x+x22

+x36 T

4;0(x)=1+x+x22

+x36 +x424 f(x)=sin(x)L

0(x)=T1;0(x)=xT

2;0(x)=xT

3;0(x)=xx36

T

4;0(x)=xx36

f(x)=cos(x)L

0(x)=T1;0(x)=1T

2;0(x)=1x22

T

3;0(x)=1x22

T

4;0(x)=1x22

+x424 v

Table of Contents

Page

1 Sequences and Convergence 1

1.1 Absolute Values. . . . . . . . . . . . . . . . . . . . . . . . . . .1

1.1.1 Inequalities Involving Absolute Values. . . . . . . . . . .3

1.2 Sequences and Their Limits. . . . . . . . . . . . . . . . . . . . .7

1.2.1 Introduction to Sequences. . . . . . . . . . . . . . . . .7

1.2.2 Recursively Defined Sequences. . . . . . . . . . . . . .10

1.2.3 Subsequences and Tails. . . . . . . . . . . . . . . . . .16

1.2.4 Limits of Sequences. . . . . . . . . . . . . . . . . . . . .17

1.2.5 Divergence to1. . . . . . . . . . . . . . . . . . . . . .27

1.2.6 Arithmetic for Limits of Sequences. . . . . . . . . . . . .28

1.3 Squeeze Theorem. . . . . . . . . . . . . . . . . . . . . . . . . .37

1.4 Monotone Convergence Theorem. . . . . . . . . . . . . . . . .39

1.5 Introduction to Series. . . . . . . . . . . . . . . . . . . . . . . .45

1.5.1 Geometric Series. . . . . . . . . . . . . . . . . . . . . .49

1.5.2 Divergence Test. . . . . . . . . . . . . . . . . . . . . . .51

2 Limits and Continuity 56

2.1 Introduction to Limits for Functions. . . . . . . . . . . . . . . . .56

2.2 Sequential Characterization of Limits. . . . . . . . . . . . . . . .66

2.3 Arithmetic Rules for Limits of Functions. . . . . . . . . . . . . .70

2.4 One-sided Limits. . . . . . . . . . . . . . . . . . . . . . . . . . .76

2.5 The Squeeze Theorem. . . . . . . . . . . . . . . . . . . . . . .78

2.6 The Fundamental Trigonometric Limit. . . . . . . . . . . . . . .82

2.7 Limits at Infinity and Asymptotes. . . . . . . . . . . . . . . . . .86

2.7.1 Asymptotes and Limits at Infinity. . . . . . . . . . . . . .87

2.7.2 Fundamental Log Limit. . . . . . . . . . . . . . . . . . .93

2.7.3 Vertical Asymptotes and Infinite Limits. . . . . . . . . . .98

2.8 Continuity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .103

2.8.1 Types of Discontinuities. . . . . . . . . . . . . . . . . . .105

2.8.2 Continuity of Polynomials,sin(x),cos(x),exandln(x) . . .107

2.8.3 Arithmetic Rules for Continuous Functions. . . . . . . .110

2.8.4 Continuity on an Interval. . . . . . . . . . . . . . . . . .113

2.9 Intermediate Value Theorem. . . . . . . . . . . . . . . . . . . .115

2.9.1 Approximate Solutions of Equations. . . . . . . . . . . .119

2.9.2 The Bisection Method. . . . . . . . . . . . . . . . . . . .123

2.10 Extreme Value Theorem. . . . . . . . . . . . . . . . . . . . . . .126

2.11 Curve Sketching: Part 1. . . . . . . . . . . . . . . . . . . . . . .130

vi

3 Derivatives 133

3.1 Instantaneous Velocity. . . . . . . . . . . . . . . . . . . . . . . .133

3.2 Definition of the Derivative. . . . . . . . . . . . . . . . . . . . .135

3.2.1 The Tangent Line. . . . . . . . . . . . . . . . . . . . . .137

3.2.2 Differentiability versus Continuity. . . . . . . . . . . . . .138

3.3 The Derivative Function. . . . . . . . . . . . . . . . . . . . . . .141

3.4 Derivatives of Elementary Functions. . . . . . . . . . . . . . . .143

3.4.1 The Derivative ofsin(x)andcos(x) . . . . . . . . . . . . .145

3.4.2 The Derivative ofex. . . . . . . . . . . . . . . . . . . . .147

3.5 Tangent Lines and Linear Approximation. . . . . . . . . . . . . .150

3.5.1 The Error in Linear Approximation. . . . . . . . . . . . .154

3.5.2 Applications of Linear Approximation. . . . . . . . . . . .157

3.6 Newton"s Method. . . . . . . . . . . . . . . . . . . . . . . . . . .161

3.7 Arithmetic Rules of Differentiation. . . . . . . . . . . . . . . . .166

3.8 The Chain Rule. . . . . . . . . . . . . . . . . . . . . . . . . . .170

3.9 Derivatives of Other Trigonometric Functions. . . . . . . . . . .175

3.10 Derivatives of Inverse Functions. . . . . . . . . . . . . . . . . .177

3.11 Derivatives of Inverse Trigonometric Functions. . . . . . . . . .183

3.12 Implicit Differentiation. . . . . . . . . . . . . . . . . . . . . . . .189

3.13 Local Extrema. . . . . . . . . . . . . . . . . . . . . . . . . . . .196

3.13.1 The Local Extrema Theorem. . . . . . . . . . . . . . . .199

3.14 Related Rates. . . . . . . . . . . . . . . . . . . . . . . . . . . .202

4 The Mean Value Theorem 208

4.1 The Mean Value Theorem. . . . . . . . . . . . . . . . . . . . . .208

4.2 Applications of the Mean Value Theorem. . . . . . . . . . . . .213

4.2.1 Antiderivatives. . . . . . . . . . . . . . . . . . . . . . . .213

4.2.2 Increasing Function Theorem. . . . . . . . . . . . . . . .218

4.2.3 Functions with Bounded Derivatives. . . . . . . . . . . .221

4.2.4 Comparing Functions Using Their Derivatives. . . . . . .223

4.2.5 Interpreting the Second Derivative. . . . . . . . . . . . .226

4.2.6 Formal Definition of Concavity. . . . . . . . . . . . . . .228

4.2.7 Classifying Critical Points:

The First and Second Derivative Tests. . . . . . . . . . .232

4.2.8 Finding Maxima and Minima on[a;b] . . . . . . . . . . .236

4.3 L"Hˆopital"s Rule. . . . . . . . . . . . . . . . . . . . . . . . . . . .239

4.4 Curve Sketching: Part 2. . . . . . . . . . . . . . . . . . . . . . .247

5 Taylor Polynomials and Taylor"s Theorem 259

5.1 Introduction to Taylor Polynomials and Approximation. . . . . . .259

5.2 Taylor"s Theorem and Errors in Approximations. . . . . . . . . .271

5.3 Big-O. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .279

5.3.1 Calculating Taylor Polynomials. . . . . . . . . . . . . . .286

vii

Chapter 1

Sequences and Convergence

It is often the case that in order to solve complex mathematical problems we must first replace the problem with a simpler version for which we have appropriate tools to find a solution. In doing so our solution to the simplified problem may not work for the original question, but it may be close enough to provide us with useful in- formation. Alternatively, we may be able to design an algorithm that will generate successive approximate solutions to the full problem in such a manner that if we ap- ply the process enough times, the result will eventually be as close as we would like to the actual solution. For example, there is no algebraic method to solve the equationquotesdbs_dbs47.pdfusesText_47

[PDF] maths casse tete

[PDF] maths chez alfred

[PDF] MATHS CNED Exercices, Pourcentages !

[PDF] MATHS CNED SECONDE

[PDF] maths collection diabolo maths 4e programme 2007 hachette education exercice 65 page 63

[PDF] maths college

[PDF] maths college 4eme

[PDF] maths collège aire

[PDF] Maths coordonnées fonction

[PDF] Maths courbe dm

[PDF] Maths cube et fonctions

[PDF] maths cycle 4 hachette

[PDF] Maths de 2°

[PDF] maths de 3eme

[PDF] maths de dm