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.

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], 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. 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.