The calculus lifesaver : all the tools you need to excel at calculus
 Author/Creator
 Banner, Adrian D., 1975
 Language
 English.
 Imprint
 Princeton, N.J. : Princeton University Press, c2007.
 Physical description
 xxi, 728 p. : ill ; 26 cm.
 Series
 Princeton lifesaver study guide.
Access
Available online

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QA303.2 .B36 2007

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QA303.2 .B36 2007
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Contents/Summary
 Contents

 Welcome
 How to use this book to study for an exam
 Two allpurpose 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. Lefthand and righthand 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 polytype 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 higherorder 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 piecewisedefined 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 yaxis
 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 lefthand 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 blowup 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 ptest (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 polytype 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 polytype 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 ptest (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 ptest
 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. Higherdegree 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 yaxis
 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 firstorder differential equations
 30.3. Firstorder linear equations
 30.3.1. Why the integrating factor works
 30.4. Constantcoefficient differential equations
 30.4.1. Solving firstorder homogeneous equations
 30.4.2. Solving secondorder 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 (constantcoefficient 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. Lefthand and righthand 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 maxmin 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 piecewisedefined 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.
 Publisher's Summary
 For many students, calculus can be the most mystifying and frustrating course they will ever take. "The Calculus Lifesaver" provides students with the essential tools they need not only to learn calculus, but to excel at it. All of the material in this userfriendly study guide has been proven to get results. The book arose from Adrian Banner's popular calculus review course at Princeton University, which he developed especially for students who are motivated to earn A's but get only average grades on exams. The complete course will be available for free on the Web in a series of videotaped lectures. This study guide works as a supplement to any singlevariable calculus course or textbook. Coupled with a selection of exercises, the book can also be used as a textbook in its own right. The style is informal, nonintimidating, and even entertaining, without sacrificing comprehensiveness. The author elaborates standard course material with scores of detailed examples that treat the reader to an 'inner monologue'  the train of thought students should be following in order to solve the problem  providing the necessary reasoning as well as the solution. The book's emphasis is on building problemsolving skills. Examples range from easy to difficult and illustrate the indepth presentation of theory. "The Calculus Lifesaver" combines ease of use and readability with the depth of content and mathematical rigor of the best calculus textbooks. It is an indispensable volume for any student seeking to master calculus. It serves as a companion to any singlevariable calculus textbook. It is informal, entertaining, and not intimidating. Informative videos that follow the book  a full fortyeight hours of Banner's Princeton calculusreview course  is available at Adrian Banner lectures. More than 475 examples (ranging from easy to hard) provide stepbystep reasoning. It has theorems and methods justified and connections made to actual practice. Difficult topics such as improper integrals and infinite series are covered in detail. It is tried and tested by students taking freshman calculus.
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Table of contents
Contributor biographical information
Publisher description
Subjects
Bibliographic information
 Publication date
 2007
 Responsibility
 Adrian Banner.
 Series
 A Princeton lifesaver study guide
 Note
 Includes index.
 ISBN
 0691131538 (cloth)
 9780691131535 (cloth)
 0691130884 (pbk.)
 9780691130880 (pbk.)