The Calculus Lifesaver All the Tools You Need to Excel at Calculus 1st edition by Adrrian Banner ISBN 140083578X 9781400835782

1 Functions, Graphs, and Lines

1.1 Functions

1.1.1 Interval notation

1.1.2 Finding the domain

1.1.3 Finding the range using the graph

1.1.4 The vertical line test

1.2 Inverse Functions

1.2.1 The horizontal line test

1.2.2 Finding the inverse

1.2.3 Restricting the domain

1.2.4 Inverses of inverse functions

1.3 Composition of Functions

1.4 Odd and Even Functions

1.5 Graphs of Linear Functions

1.6 Common Functions and Graphs

2 Review of Trigonometry

2.1 The Basics

2.2 Extending the Domain of Trig Functions

2.2.1 The ASTC method

2.2.2 Trig functions outside [0; 2]

2.3 The Graphs of Trig Functions

2.4 Trig Identities

3 Introduction to Limits

3.1 Limits: The Basic Idea

3.2 Left-Hand and Right-Hand Limits

3.3 When the Limit Does Not Exist

3.4 Limits at 1 and 􀀀1

3.4.1 Large numbers and small numbers

3.5 Two Common Misconceptions about Asymptotes

3.6 The Sandwich Principle

3.7 Summary of Basic Types of Limits

4 How to Solve Limit Problems Involving Polynomials

4.1 Limits Involving Rational Functions as x ! a

4.2 Limits Involving Square Roots as x ! a

4.3 Limits Involving Rational Functions as x ! 1

4.3.1 Method and examples

4.4 Limits Involving Poly-type Functions as x ! 1

4.5 Limits Involving Rational Functions as x ! 􀀀1

4.6 Limits Involving Absolute Values

5 Continuity and Dierentiability

5.1 Continuity

5.1.1 Continuity at a point

5.1.2 Continuity on an interval

5.1.3 Examples of continuous functions

5.1.4 The Intermediate Value Theorem

5.1.5 A harder IVT example

5.1.6 Maxima and minima of continuous functions

5.2 Dierentiability

5.2.1 Average speed

5.2.2 Displacement and velocity

5.2.3 Instantaneous velocity

5.2.4 The graphical interpretation of velocity

5.2.5 Tangent lines

5.2.6 The derivative function

5.2.7 The derivative as a limiting ratio

5.2.8 The derivative of linear functions

5.2.9 Second and higher-order derivatives

5.2.10 When the derivative does not exist

5.2.11 Dierentiability and continuity

6 How to Solve Dierentiation Problems

6.1 Finding Derivatives Using the Denition

6.2 Finding Derivatives (the Nice Way)

6.2.1 Constant multiples of functions

6.2.2 Sums and dierences of functions

6.2.3 Products of functions via the product rule

6.2.4 Quotients of functions via the quotient rule

6.2.5 Composition of functions via the chain rule

6.2.6 A nasty example

6.2.7 Justication of the product rule and the chain rule

6.3 Finding the Equation of a Tangent Line

6.4 Velocity and Acceleration

6.4.1 Constant negative acceleration

6.5 Limits Which Are Derivatives in Disguise

6.6 Derivatives of Piecewise-Dened Functions

6.7 Sketching Derivative Graphs Directly

7 Trig Limits and Derivatives

7.1 Limits Involving Trig Functions

7.1.1 The small case

7.1.2 Solving problems|the small case

7.1.3 The large case

7.1.4 The other” case

7.1.5 Proof of an important limit

7.2 Derivatives Involving Trig Functions

7.2.1 Examples of dierentiating trig functions

7.2.2 Simple harmonic motion

7.2.3 A curious function

8 Implicit Dierentiation and Related Rates

8.1 Implicit Dierentiation

8.1.1 Techniques and examples

8.1.2 Finding the second derivative implicitly

8.2 Related Rates

8.2.1 A simple example

8.2.2 A slightly harder example

8.2.3 A much harder example

8.2.4 A really hard example

9 Exponentials and Logarithms

9.1 The Basics

9.1.1 Review of exponentials

9.1.2 Review of logarithms

9.1.3 Logarithms, exponentials, and inverses

9.1.4 Log rules

9.2 Denition of e

9.2.1 A question about compound interest

9.2.2 The answer to our question

9.2.3 More about e and logs

9.3 Dierentiation of Logs and Exponentials

9.3.1 Examples of dierentiating exponentials and logs

9.4 How to Solve Limit Problems Involving Exponentials or Logs

9.4.1 Limits involving the denition of e

9.4.2 Behavior of exponentials near 0

9.4.3 Behavior of logarithms near 1

9.4.4 Behavior of exponentials near 1 or 􀀀1

9.4.5 Behavior of logs near 1

9.4.6 Behavior of logs near 0

9.5 Logarithmic Dierentiation

9.5.1 The derivative of xa

9.6 Exponential Growth and Decay

9.6.1 Exponential growth

9.6.2 Exponential decay

9.7 Hyperbolic Functions

10 Inverse Functions and Inverse Trig Functions

10.1 The Derivative and Inverse Functions

10.1.1 Using the derivative to show that an inverse exists

10.1.2 Derivatives and inverse functions: what can go wrong

10.1.3 Finding the derivative of an inverse function

10.1.4 A big example

10.2 Inverse Trig Functions

10.2.1 Inverse sine

10.2.2 Inverse cosine

10.2.3 Inverse tangent

10.2.4 Inverse secant

10.2.5 Inverse cosecant and inverse cotangent

10.2.6 Computing inverse trig functions

10.3 Inverse Hyperbolic Functions

10.3.1 The rest of the inverse hyperbolic functions

11 The Derivative and Graphs

11.1 Extrema of Functions

11.1.1 Global and local extrema

11.1.2 The Extreme Value Theorem

11.1.3 How to nd global maxima and minima

11.2 Rolle’s Theorem

11.3 The Mean Value Theorem

11.3.1 Consequences of the Mean Value Theorem

11.4 The Second Derivative and Graphs

11.4.1 More about points of in ection

11.5 Classifying Points Where the Derivative Vanishes

11.5.1 Using the rst derivative

11.5.2 Using the second derivative

12 Sketching Graphs

12.1 How to Construct a Table of Signs

12.1.1 Making a table of signs for the derivative

12.1.2 Making a table of signs for the second derivative

12.2 The Big Method

12.3 Examples

12.3.1 An example without using derivatives

12.3.2 The full method: example 1

12.3.3 The full method: example 2

12.3.4 The full method: example 3

12.3.5 The full method: example 4

13 Optimization and Linearization

13.1 Optimization

13.1.1 An easy optimization example

13.1.2 Optimization problems: the general method

13.1.3 An optimization example

13.1.4 Another optimization example

13.1.5 Using implicit dierentiation in optimization

13.1.6 A dicult optimization example

13.2 Linearization

13.2.1 Linearization in general

13.2.2 The dierential

13.2.3 Linearization summary and examples

13.2.4 The error in our approximation

13.3 Newton’s Method

14 L’H^opital’s Rule and Overview of Limits

14.1 L’H^opital’s Rule

14.1.1 Type A: 0/0 case

14.1.2 Type A: 1=1 case

14.1.3 Type B1 (1􀀀1)

14.1.4 Type B2 (0 1)

14.1.5 Type C (11, 00, or 10)

14.1.6 Summary of l’H^opital’s Rule types

14.2 Overview of Limits

15 Introduction to Integration

15.1 Sigma Notation

15.1.1 A nice sum

15.1.2 Telescoping series

15.2 Displacement and Area

15.2.1 Three simple cases

15.2.2 A more general journey

15.2.3 Signed area

15.2.4 Continuous velocity

15.2.5 Two special approximations

16 Denite Integrals

16.1 The Basic Idea

16.1.1 Some easy examples

16.2 Denition of the Denite Integral

16.2.1 An example of using the denition

16.3 Properties of Denite Integrals

16.4 Finding Areas

16.4.1 Finding the unsigned area

16.4.2 Finding the area between two curves

16.4.3 Finding the area between a curve and the y-axis

16.5 Estimating Integrals

16.5.1 A simple type of estimation

16.6 Averages and the Mean Value Theorem for Integrals

16.6.1 The Mean Value Theorem for integrals

16.7 A Nonintegrable Function

17 The Fundamental Theorems of Calculus

17.1 Functions Based on Integrals of Other Functions

17.2 The First Fundamental Theorem

17.2.1 Introduction to antiderivatives

17.3 The Second Fundamental Theorem

17.4 Indenite Integrals

17.5 How to Solve Problems: The First Fundamental Theorem

17.5.1 Variation 1: variable left-hand limit of integration

17.5.2 Variation 2: one tricky limit of integration

17.5.3 Variation 3: two tricky limits of integration

17.5.4 Variation 4: limit is a derivative in disguise

17.6 How to Solve Problems: The Second Fundamental Theorem

17.6.1 Finding indenite integrals

17.6.2 Finding denite integrals

17.6.3 Unsigned areas and absolute values

17.7 A Technical Point

17.8 Proof of the First Fundamental Theorem

18 Techniques of Integration, Part One

18.1 Substitution

18.1.1 Substitution and denite integrals

18.1.2 How to decide what to substitute

18.1.3 Theoretical justication of the substitution method

18.2 Integration by Parts

18.2.1 Some variations

18.3 Partial Fractions

18.3.1 The algebra of partial fractions

18.3.2 Integrating the pieces

18.3.3 The method and a big example

19 Techniques of Integration, Part Two

19.1 Integrals Involving Trig Identities

19.2 Integrals Involving Powers of Trig Functions

19.2.1 Powers of sin and/or cos

19.2.2 Powers of tan

19.2.3 Powers of sec

19.2.4 Powers of cot

19.2.5 Powers of csc

19.2.6 Reduction formulas

19.3 Integrals Involving Trig Substitutions

19.3.1 Type 1: p a2 􀀀 x2

19.3.2 Type 2: p x2 + a2

19.3.3 Type 3: p x2 􀀀 a2

19.3.4 Completing the square and trig substitutions

19.3.5 Summary of trig substitutions

19.3.6 Technicalities of square roots and trig substitutions

19.4 Overview of Techniques of Integration

20 Improper Integrals: Basic Concepts

20.1 Convergence and Divergence

20.1.1 Some examples of improper integrals

20.1.2 Other blow-up points

20.2 Integrals over Unbounded Regions

20.3 The Comparison Test (Theory)

20.4 The Limit Comparison Test (Theory)

20.4.1 Functions asymptotic to each other

20.4.2 The statement of the test

20.5 The p-test (Theory)

20.6 The Absolute Convergence Test

21 Improper Integrals: How to Solve Problems

21.1 How to Get Started

21.1.1 Splitting up the integral

21.1.2 How to deal with negative function values

21.2 Summary of Integral Tests

21.3 Behavior of Common Functions near 1 and 􀀀1

21.3.1 Polynomials and poly-type functions near 1 and 􀀀1

21.3.2 Trig functions near 1 and 􀀀1

21.3.3 Exponentials near 1 and 􀀀1

21.3.4 Logarithms near 1

21.4 Behavior of Common Functions near 0

21.4.1 Polynomials and poly-type functions near 0

21.4.2 Trig functions near 0

21.4.3 Exponentials near 0

21.4.4 Logarithms near 0

21.4.5 The behavior of more general functions near 0

21.5 How to Deal with Problem Spots Not at 0 or 1

22 Sequences and Series: Basic Concepts

22.1 Convergence and Divergence of Sequences

22.1.1 The connection between sequences and functions

22.1.2 Two important sequences

22.2 Convergence and Divergence of Series

22.2.1 Geometric series (theory)

22.3 The nth Term Test (Theory)

22.4 Properties of Both Innite Series and Improper Integrals

22.4.1 The comparison test (theory)

22.4.2 The limit comparison test (theory)

22.4.3 The p-test (theory)

22.4.4 The absolute convergence test

22.5 New Tests for Series

22.5.1 The ratio test (theory)

22.5.2 The root test (theory)

22.5.3 The integral test (theory)

22.5.4 The alternating series test (theory)

23 How to Solve Series Problems

23.1 How to Evaluate Geometric Series

23.2 How to Use the nth Term Test

23.3 How to Use the Ratio Test

23.4 How to Use the Root Test

23.5 How to Use the Integral Test

23.6 Comparison Test, Limit Comparison Test, and p-test

23.7 How to Deal with Series with Negative Terms

24 Taylor Polynomials, Taylor Series, and Power Series

24.1 Approximations and Taylor Polynomials

24.1.1 Linearization revisited

24.1.2 Quadratic approximations

24.1.3 Higher-degree approximations

24.1.4 Taylor’s Theorem

24.2 Power Series and Taylor Series

24.2.1 Power series in general

24.2.2 Taylor series and Maclaurin series

24.2.3 Convergence of Taylor series

24.3 A Useful Limit

25 How to Solve Estimation Problems

25.1 Summary of Taylor Polynomials and Series

25.2 Finding Taylor Polynomials and Series

25.3 Estimation Problems Using the Error Term

25.3.1 First example

25.3.2 Second example

25.3.3 Third example

25.3.4 Fourth example

25.3.5 Fifth example

25.3.6 General techniques for estimating the error term

25.4 Another Technique for Estimating the Error

26 Taylor and Power Series: How to Solve Problems

26.1 Convergence of Power Series

26.1.1 Radius of convergence

26.1.2 How to nd the radius and region of convergence

26.2 Getting New Taylor Series from Old Ones

26.2.1 Substitution and Taylor series

26.2.2 Dierentiating Taylor series

26.2.3 Integrating Taylor series

26.2.4 Adding and subtracting Taylor series

26.2.5 Multiplying Taylor series

26.2.6 Dividing Taylor series

26.3 Using Power and Taylor Series to Find Derivatives

26.4 Using Maclaurin Series to Find Limits

27 Parametric Equations and Polar Coordinates

27.1 Parametric Equations

27.1.1 Derivatives of parametric equations

27.2 Polar Coordinates

27.2.1 Converting to and from polar coordinates

27.2.2 Sketching curves in polar coordinates

27.2.3 Finding tangents to polar curves

27.2.4 Finding areas enclosed by polar curves

28 Complex Numbers

28.1 The Basics

28.1.1 Complex exponentials

28.2 The Complex Plane

28.2.1 Converting to and from polar form

28.3 Taking Large Powers of Complex Numbers

28.4 Solving zn = w

28.4.1 Some variations

28.5 Solving ez = w

28.6 Some Trigonometric Series

28.7 Euler’s Identity and Power Series

29 Volumes, Arc Lengths, and Surface Areas

29.1 Volumes of Solids of Revolution

29.1.1 The disc method

29.1.2 The shell method

29.1.3 Summary

29.1.4 Variation 1: regions between a curve and the y-axis

29.1.5 Variation 2: regions between two curves

29.1.6 Variation 3: axes parallel to the coordinate axes

29.2 Volumes of General Solids

29.3 Arc Lengths

29.3.1 Parametrization and speed

29.4 Surface Areas of Solids of Revolution

30 Dierential Equations

30.1 Introduction to Dierential Equations

30.2 Separable First-order Dierential Equations

30.3 First-order Linear Equations

30.3.1 Why the integrating factor works

30.4 Constant-coecient Dierential Equations

30.4.1 Solving rst-order homogeneous equations

30.4.2 Solving second-order homogeneous equations

30.4.3 Why the characteristic quadratic method works

30.4.4 Nonhomogeneous equations and particular solutions

30.4.5 Finding a particular solution

30.4.6 Examples of nding particular solutions

30.4.7 Resolving con icts between yP and yH

30.4.8 Initial value problems (constant-coecient linear)

30.5 Modeling Using Dierential Equations

Appendix A Limits and Proofs

A.1 Formal Denition of a Limit

A.1.1 A little game

A.1.2 The actual denition

A.1.3 Examples of using the denition

A.2 Making New Limits from Old Ones

A.2.1 Sums and dierences of limits|proofs

A.2.2 Products of limits|proof

A.2.3 Quotients of limits|proof

A.2.4 The sandwich principle|proof

A.3 Other Varieties of Limits

A.3.1 Innite limits

A.3.2 Left-hand and right-hand limits

A.3.3 Limits at 1 and 􀀀1

A.3.4 Two examples involving trig

A.4 Continuity and Limits

A.4.1 Composition of continuous functions

A.4.2 Proof of the Intermediate Value Theorem

A.4.3 Proof of the Max-Min Theorem

A.5 Exponentials and Logarithms Revisited

A.6 Dierentiation and Limits

A.6.1 Constant multiples of functions

A.6.2 Sums and dierences of functions

A.6.3 Proof of the product rule

A.6.4 Proof of the quotient rule

A.6.5 Proof of the chain rule

A.6.6 Proof of the Extreme Value Theorem

A.6.7 Proof of Rolle’s Theorem

A.6.8 Proof of the Mean Value Theorem

A.6.9 The error in linearization

A.6.10 Derivatives of piecewise-dened functions

A.6.11 Proof of l’H^opital’s Rule

A.7 Proof of the Taylor Approximation Theorem

Appendix B Estimating Integrals

B.1 Estimating Integrals Using Strips

B.1.1 Evenly spaced partitions

B.2 The Trapezoidal Rule

B.3 Simpson’s Rule

B.3.1 Proof of Simpson’s rule

B.4 The Error in Our Approximations

B.4.1 Examples of estimating the error

B.4.2 Proof of an error term inequality

List of Symbols

Index