Princeton lifesaver study guide.

Includes index

Welcome --

How to use this book to study for an exam --

Two all-purpose study tips --

Key sections for exam review (by topic) --

Acknowledgments --

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[pi]] --

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 [infinity] and -[infinity] --

3.4.1.

Large number 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[alpha] --

4.2.

Limits involving square roots as x -> a[alpha] --

4.3.

Limits involving rational functions as x -> [infinity] --

4.3.1.

Method and examples --

4.4.

Limits involving poly-type functions as x -> [infinity] --

4.5.

Limits involving rational functions as x -> -[infinity] --

4.6.

Limits involving absolute values --

5.

Continuity and differentiability --

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.

Differentiability --

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 ration --

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.

Differentiability and continuity --

6.

How to solve differentiation problems --

6.1.

Finding derivatives using the definition --

6.2.

Finding derivatives (the nice way) --

6.2.1.

Constant multiples of functions --

6.2.2.

Sums and differences 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.

Justification 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-defined 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 differentiating trig functions --

7.2.2.

Simple harmonic motion --

7.2.3. A

curious function --

8.

Implicit differentiation and related rates --

8.1.

Implicit differentiation --

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.

Definition 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.

Differentiation of logs and exponentials --

9.3.1.

Examples of differentiating exponentials and logs --

9.4.

How to solve limit problems involving exponentials or logs --

9.4.1.

Limits involving the definition 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 [infinity] or -[infinity] --

9.4.5.

Behavior of logs near [infinity] --

9.4.6.

Behavior of logs near 0 --

9.5.

Logarithmic differentiation --

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 find global maxima and minima --

11.2.

Rolle's Theorem --

11.3. The

mean value theorem --

11.3.1.

Consequence of the man value theorem --

11.4. The

second derivative and graphs --

11.4.1.

More about points of inflection --

11.5.

Classifying points where the derivative vanishes --

11.5.1.

Using the first 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 differentiation in optimization --

13.1.6. A

difficult optimization example --

13.2.

Linearization --

13.2.1.

Linearization in general --

13.2.2. The

differential --

13.2.3.

Linearization summary and example --

13.2.4. The

error in our approximation --

13.3.

Newton's method --

14.

L'Hôpital's rule and overview of limits --

14.1.

L'Hôpital's rule --

14.1.1.

Type A : 0/0 case --

14.1.2.

Type A : ±[infinity]/±[infinity] case --

14.1.3.

Type B1 ([infinity] - [infinity]) --

14.1.4.

Type B2 (0 x ± [infinity]) --

14.1.5.

Type C (1 ± [infinity], 0⁰, or [infinity]⁰) --

14.1.6.

Summary of l'Hôpital'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.

Definite integrals --

16.1. The

basic idea --

6.1.1.

Some easy example --

16.2.

Definition of the definite integral --

16.2.1. An

example of using the definition --

16.3.

Properties of definite 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.

Indefinite integrals --

17.5.

How to solve problems : the first fundamental theorem --

17.5.1.

Variation 1 : variable left-hand limit on 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 indefinite integrals --

17.6.2.

Finding definite 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 definite integrals --

18.1.2.

How to decide what to substitute --

18.1.3.

Theoretical justification 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 : [square root] a² - x² --

19.3.2.

Type 2 : [square root] x² + a² --

19.3.3.

Type 3 : [square root] 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 [infinity] and -[infinity] --

21.3.1.

Polynomials and poly-type functions near [infinity] and -[infinity] --

21.3.2.

Trig function near [infinity] and -[infinity] --

21.3.3.

Exponentials near [infinity] and -[infinity] --

21.3.4.

Logarithms near [infinity] --

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 [infinity] --

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 infinite 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.

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 find 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.

Differentiating 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.

Find 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... and variations --

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.

Differential equations --

30.1.

Introduction to differential equations --

30.2.

Separable first-order differential equations --

30.3.

First-order linear equations --

30.3.1.

Why the integrating factor works --

30.4.

Constant-coefficient differential equations --

30.4.1.

Solving first-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.

Funding a particular solution --

30.4.6.

Examples of finding particular solutions --

30.4.7.

Resolving conflicts between yP and yH --

30.4.8.

Initial value problems (constant-coefficient linear) --

30.5.

Modeling using differential equations -- Appendix A : Limits and proofs --

A.1.

Formal definition of a limit --

A.1.1. A

little game --

A.1.2. The

actual definition --

A.1.3.

Examples of using the definition --

A.2.

Making new limits from old ones --

A.2.1.

Sums and differences 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.

Infinite limits --

A.3.2.

Left-hand and right-hand limits --

A.3.3.

Limits at [infinity] and -[infinity] --

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.

Differentiation and limits --

A.6.1.

Constant multiples of functions --

A.6.2.

Sums and differences 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-defined functions --

A.6.11.

Proof of l'Hôspital'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