Calculus : concepts and contexts
著者
書誌事項
Calculus : concepts and contexts
Brooks/Cole Pub. Co., c1998
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注記
Includes index
内容説明・目次
内容説明
This text embodies the broad principles of calculus reform-conceptual understanding motivated by real-world applications and the application of the "Rule of Four" in numerical, visual, algebraic, and verbal interpretations. At the same time, this book retains traditional calculus. The author emphasizes visualization and problem solving.
目次
- 1. Functions and models: Four Ways to Represent a Function. New Functions from Old Functions. Graphing Calculators and Computers. Parametric Curves. Laboratory Project: Families of Hypocycloids. Exponential Functions. Inverse Functions and Logarithms. Models and Curve Fitting. 2. Limits and derivatives: The Tangent and Velocity Problems. The Limit of a Function. Calculating Limits Using the Limit Laws. Continuity. Limits Involving Infinity. Tangents, Velocities, and Other Rates of Change. Derivatives. Writing Project: Early Methods for Finding Tangents. The Derivative as a Function. Linear Approximations. What does f' say about f? 3. Differentiation rules: Derivatives of Polynomials and Exponential Functions. The Product and Quotient Rules. Rates of Change in the Natural and Social Sciences. Derivatives of Trigonometric Functions. The Chain Rule. Laboratory Project: Bezier Curves. Applied Project: Where Should a Pilot Start Descent? Implicit Differentiation. Derivatives of Logarithmic Functions. Discovery Project: Hyperbolic Functions. Linear Approximations and Differentials. Laboratory Project: Taylor Polynomials. 4. Applications of differentiation: Related Rates. Maximum and Minimum Values. Applied Project: The Calculus of Rainbows. Derivatives and the Shapes of Curves. Graphing with Calculus and Calculators. Indeterminate Forms and l'Hospital's Rule. Writing Project: The Origins of l'Hospital's Rule. Optimization Problems. Applied Project: The Shape of a Can. Applications to Economics. Newton's Method. Antiderivatives. 5. Integrals: Areas and Distances. The Definite Integral. Evaluating Definite Integrals. Discovery Project: Area Functions. The Fundamental Theorem of Calculus. Writing Project: Newton, Leibniz, and the Invention of Calculus. The Substitution Rule. Integration by Parts. Integration Using Tables and Computer Algebra Systems. Discovery Project: Patterns in Integrals. Approximate Integration. Improper Integrals. 6. Applications of integration: More about Areas. Volumes. Arc Length. Average Value of a Function. Applied Project: Where to Sit at the Movies. Applications to Physics. Applications to Economics and Biology. Probability. 7. Differential equations: Modelling with Differential Equations. Direction Fields. Euler's Method. Separable Equations. Applied Project: Which is Faster, Going Up or Coming Down? Exponential Growth and Decay. Applied Project: Calculus and Baseball. The Logistic Equation. Predator-Prey Systems. Some Special Second Order Equations. 8. Infinite sequences and series: Sequences. Laboratory Project: Logistic Sequences. Series. The Integral and Comparison Tests
- Estimating Sums. Other Convergence Tests. Power Series. Representation of Functions as Power Series. Taylor and Maclaurin Series. Writing Project: Newton, Gregory, Taylor, and Maclaurin. The Binomial Series. Applications of Taylor Polynomials. Applied Project: Radiation from the Stars. Using Series to Solve Differential Equations. (Part contents).
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