Calculus: Concepts and Contexts

Emphasizing conceptual understanding, this calculus text includes real world applications and stresses multiple representations - numerical, visual, algebraic, and verbal interpretations. The concept of derivative is covered before the rules of differentiation are developed. This approach gives students a conceptual understanding of and an opportunity to work with the derivative and before they are introduced to the "rules'" or formulas. The problems maintain a balance between algebraic skills and conceptual understanding.

Table of Contents:
A Preview of Calculus. Part I: Functions and Models. 1.1 Four Ways to Represent a Function. 1.2 New Functions from Old Functions. 1.3 Graphing Calculators and Computers. 1.4 Parametric Curves. 1.5 Exponential Functions. 1.6 Inverse Functions and Logarithms 1.7 Models and Curve Fitting. Review. Principles of Problem Solving. Part II: Limits and Derivatives. 2.1 The Tangent and Velocity Problems. 2.2 The Limit of a Function. 2.3 Calculating Limits Using the Limit Laws. 2.4 Continuity. 2.5 Limits Involving Infinity. 2.6 Tangents, Velocities, and Other Rates of Change. 2.7 Derivatives. 2.8 The Derivative as a Function. 2.9 Linear Approximations. 2.10 What does f' say about f? Review. Focus on Problem Solving. Part III: Differentiation Rules. 3.1 Derivatives of Polynomials and Exponential Functions. 3.2 The Product and Quotient Rules. 3.3 Rates of Change in the Natural and Social Sciences. 3.4 Derivatives of Trigonometric Functions. 3.5 The Chain Rule. 3.6 Implicit Differentiation 3.7 Derivatives of Logarithmic Functions 3.8 Linear and Quadratic Approximations; Taylor Polynomials. Review. Focus on Problem Solving. Part IV: Applications of Differentiation. 4.1 Related Rates. 4.2 Maximum and Minimum Values. 4.3 Derivatives and the Shapes of Curves. 4.4 Graphing with Calculus and Calculators. 4.5 Indeterminate Forms and l'Hospital's Rule. 4.6 Optimization Problems. 4.7 Applications to Economics. 4.8 Newton's Method. 4.9 Antiderivatives. Review. Focus on Problem Solving. Part V: Integrals. 5.1 Areas and Distances. 5.2 The Definite Integral. 5.3 Evaluating Definite Integrals 5.4 The Fundamental Theorem of Calculus. 5.5 The Substitution Rule. 5.6 Integration by Parts. 5.7 Integration Using Tables and Computer Algebra Systems. 5.8 Approximate Integration. 5.9 Improper Integrals. Review. Focus on Problem Solving. Part VI: Applications OF Integration. 6.1 Areas Between Curves. 6.2 Volumes. 6.3 Arc Length. 6.4 Average Value of a Function. 6.5 Applications to Physics. 6.6 Applications to Economics and Biology. 6.7 Probability. Review. Focus on Problem Solving. PART VII: Differential Equations. 7.1 Basic Concepts; Direction Fields. 7.2 Euler's Method. 7.3 Separable Equations. 7.4 Exponential Growth and Decay. 7.5 The Logistic Equation. 7.6 First Order Linear Equations. 7.7 Second Order Linear Equations. Review. Focus on Problem Solving. Part VIII: Polar Coordinates. 8.1 Curves in Polar Coordinates. 8.2 Tangents to Polar Curves. 8.3 Areas and Lengths in Polar Coordinates. 8.4 Conic Sections in Polar Coordinates. Review. Focus on Problem Solving. Part IX: Infinite Sequences and Series. 9.1 Sequences. 9.2 Series. 9.3 The Integral and Comparison Tests; Estimating Sums. 9.4 Other Convergence Tests. 9.5 Power Series. 9.6 Representation of Functions as Power Series. 9.7 Taylor and Maclaurin Series. 9.8 The Binomial Series. 9.9 Applications of Taylor Polynomials. 9.10 Using Series to Solve Differential Equations. Review. Focus on Problem Solving.