The Essential Calculus Workbook: Limits and Derivatives

Ready to step up your game in calculus? This workbook isn't the usual parade of repetitive questions and answers. Author Tim Hill's approach lets you work on problems you enjoy, rather than through exercises and drills you fear, without the speed pressure, timed testing, and rote memorization that damage your experience of mathematics. Working through varied problems in this anxiety-free way helps you develop an understanding of numerical relations apart from the catalog of mathematical facts that's often stressed in classrooms and households. This number sense, common in high-achieving students, lets you apply and combine concepts, methods, and numbers flexibly, without relying on distant memories.


• Solutions to basic problems are steeped in the fundamentals, including notation, terminology, definitions, theories, proofs, physical laws, and related concepts.

• Advanced problems explore variations, tricks, subtleties, and real-world applications.

• Problems build gradually in difficulty with little repetition. If you get stuck, then flip back a few pages for a hint or to jog your memory.

• Numerous pictures depicting mathematical facts help you connect visual and symbolic representations of numbers and concepts.

• Treats calculus as a problem-solving art requiring insight and intuitive understanding, not as a branch of logic requiring careful deductive reasoning.

• Discards the common and damaging misconception that fast students are strong students. Good students aren't particularly fast with numbers because they think deeply and carefully about mathematics.

• Detailed solutions and capsule reviews greatly reduce the need to cross reference a comprehensive calculus textbook.


Topics covered: The tangent line. Delta notation. The derivative of a function. Differentiable functions. Leibniz notation. Average and instantaneous velocity. Speed. Projectile paths. Rates of change. Acceleration. Marginal cost. Limits. Epsilon-delta definition. Limit laws. Trigonometric limits. Continuity. Continuous functions. The Mean Value Theorem. The Extreme Value Theorem. The Intermediate Value Theorem. Fermat's theorem.


Prerequisite mathematics: Elementary algebra. Real numbers. Functions. Graphs. Trigonometry.


Contents

1. The Slope of the Tangent Line

2. The Definition of the Derivative

3. Velocity and Rates of Change

4. Limits

5. Continuous Functions


About the Author

Tim Hill is a statistician living in Boulder, Colorado. He holds degrees in mathematics and statistics from Stanford University and the University of Colorado. Tim has written guides for calculus, trigonometry, algebra, geometry, precalculus, permutations and combinations, debt, mortgages, and Excel pivot tables. When he's not crunching numbers, Tim climbs rocks, hikes canyons, and avoids malls.