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The calculus lifesaver : all the tools you need to excel at calculus / Adrian Banner.

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Author/Creator:
Banner, Adrian D., 1975-
Language:
English.
Publication date:
2007
Imprint:
Princeton, N.J. : Princeton University Press, c2007.
Format:
  • Book
  • xxi, 728 p. : ill ; 26 cm.
Note:
Includes index.
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 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], 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
  • 19.3.2. Type 2 : [square root] x² + a²
  • 19.3.3. Type 3 : [square root] x²
  • 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.
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.
Series:
A Princeton lifesaver study guide
Princeton lifesaver study guide.
Subjects:
ISBN:
0691131538
9780691131535
0691130884
9780691130880

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