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Exploring Functions and Calculus: A Comprehensive Overview for Pre-Calculus and Calculus Students

This comprehensive guide explores different ways to represent functions, from elementary algebraic functions to transcendental functions. It covers the concept of limits, derivatives, calculating derivatives, and applications of derivatives in various fields. The text delves into definite integrals, antiderivatives, Riemann sums, and the Fundamental Theorem of Calculus. Ideal for pre-calculus and calculus students.

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Exploring Functions and Calculus: A Comprehensive Overview for Pre-Calculus and Calculus Students

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  1. Outline of MA 111 (4/29/09) • Pre-Calculus – Functions • Different ways to represent functions • New functions from old • Elementary functions: algebraic (power functions, polynomials, rational functions) and transcendental functions (exponential and log functions, trig functions, inverse trig functions)

  2. Outline continued • Calculus – Limits and the Derivative • Concept of limit; limits of functions • Definition of the derivative as a limit • Calculating Derivatives: The Facts and the Rules • 2nd derivatives and concavity • Differentiability • Implicit differentiation

  3. Outline continued • Applications of the Derivative: • Rates of change in natural & social science • Exponential growth and decay • Critical points, extrema, inflection points • Optimization problems • Related rate problems

  4. Outline continued • Definite Integrals and Antiderivatives • Concept and calculation of antiderivatives (most general and specific, given an initial condition) • The definite integral of a function on an interval and connection to areas • Approximating definite integrals via Riemann sums (rectangles and trapezoids) • Finding exact definite integrals via the Fundamental Theorem of Calculus Part 1 • Manufacturing antiderivatives via FTC Part 2 • Average value (height) of a function on an interval

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