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The calculus lifesaver : all the tools you need to excel at calculus

Author: Adrian D Banner
Publisher: Princeton, New Jersey : Princeton University Press, [2007]
Series: Princeton lifesaver study guide.
Edition/Format:   Print book : EnglishView all editions and formats
Summary:
"Finally, a calculus book you can pick up and actually read! Developed especially for students who are motivated to earn an A but only score average grades on exams, The Calculus Lifesaver has all the essentials you need to master calculus."--Jacket.
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Material Type: Internet resource
Document Type: Book, Internet Resource
All Authors / Contributors: Adrian D Banner
ISBN: 0691131538 9780691131535 0691130884 9780691130880
OCLC Number: 100000310
Notes: Includes index.
Description: xxi, 728 pages : illustrations ; 26 cm.
Contents: 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 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[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], 0°, 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. 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 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.
Series Title: Princeton lifesaver study guide.
Responsibility: Adrian Banner.
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