IG Antiderivatives_4ae635e448202691504ce1f370787219

Chapter 5: Integration

Section 1: Antiderivatives and Indefinite Integrals

  • Antiderivative Concept

    • A function F is an antiderivative of a function f on interval I if on this interval F'(x) = f(x).

Antidifferentiation

  • Antidifferentiation of a given function does not yield a unique function but a family of functions.

Theorem 1: Antiderivatives

  • If the derivatives of two functions are equal on an open interval (a, b), they differ by at most a constant.

    • Symbolically: If F and G are differentiable functions on (a, b), and F'(x) = G'(x) for all x in (a, b), then F(x) = G(x) + k for some constant k.

Example 1: A Family of Antiderivatives

  • (A) Find all antiderivatives of f(x) = x.

  • (B) Graph the antiderivative of f(x) that contains the points (0,0), (0,1), and (0,2).

  • (C) The relation between the graphs of these three antiderivatives: they are vertical translations of each other.

Indefinite Integrals: Formulas and Properties

  • The symbol dx indicates that antidifferentiation is performed with respect to the variable x.

  • The arbitrary constant C is known as the constant of integration.

Example 2: Using Indefinite Integration Notation

  • Indefinite integration and differentiation are inverse operations with the exception of the constant of integration.

    • This relationship can be expressed symbolically.

Formulas: Indefinite Integrals of Basic Functions

Properties: Indefinite Integrals

Example 2: Using Indefinite Integral Properties and Formulas

Example 3: Application of Indefinite Integral Properties and Formulas

Example 4: Application - Particular Antiderivative

  • Problem: Find the equation of the curve passing through (3, 5) if the slope is given by dy/dx = 2x at any point x.

Example 5: Cost Function

  • Given C'(x) = 0.3x^2 + 2x with fixed costs of $3,000, find the cost function C(x) and the cost of producing 30 units.

Example 6: Advertising

  • Scenario: A satellite radio station has 27,000 daily listeners and wants to increase to 41,000.

  • Determine how many more days the advertising campaign should last if the growth rate is given.

    • Result: The campaign should last an additional 40 days.