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Antiderivatives

Antiderivatives. Indefinite Integrals. Definition. A function F is an antiderivative of f on an interval I if F’(x) = f(x) for all x in I . Example: F(x) = x 3 when f(x) = 3x 2 In other words, we are going backwards – given the derivative, what was the original function?.

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Antiderivatives

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  1. Antiderivatives Indefinite Integrals

  2. Definition • A function F is an antiderivative of f on an interval I if F’(x) = f(x) for all x in I. • Example: F(x) = x3 when f(x) = 3x2 In other words, we are going backwards – given the derivative, what was the original function?

  3. Representation of Antiderivatives • If F is an antiderivative of f on an interval I, then G is an antiderivative of f iff • G(x) = F(x) + C, for all x in I • where C is a constant

  4. Representation of Antiderivatives • The constant C is called the constant of integration. • The family of functions represented by G is the general antiderivative of f and the answer is the general solution of the differential equation.

  5. Differential Equation • A differential equation in x and y is an equation that involves x, y, and derivatives of y. • Find the general solution of the differential equation y’ = 2.

  6. Differential Equations • Find the general solution of the differential equation y’ = x2 • Find the general solution of the differential equation

  7. Notation for Antiderivatives • When solving a differential equation of the form • The operation of finding all solutions of this equation is called antidifferentiation (or indefinite integration)

  8. Notation for Antiderivatives The general solution is denoted by Where f(x) is the integrand, dx is the variable of integration and C is the constant of integration. This is an antiderivative of f with respect to x.

  9. Basic Integration Rules Differentiation and antidifferentiation are inverse functions (one “undoes” the other) The basic integration rules come from the basic derivative rules

  10. Basic Integration Rules Rules:

  11. Applying the Basic Integration Rules Evaluate the indefinite integral

  12. Rewriting Before Integrating

  13. Integrating Polynomial Functions

  14. Integrating Polynomials

  15. Rewriting Before Integrating We have no division rules at this point When dividing by a monomial, put the monomial under each of the monomials in the numerator

  16. Rewriting Before Integrating

  17. Rewriting Before Integrating

  18. Rewriting Before Integrating

  19. Initial Conditions and Particular Solutions A particular solution allows you to determine the C in a differential equation. In order to do this, initial conditions must be given.

  20. Finding a Particular Solution

  21. Finding a Particular Solution

  22. Solving a Vertical Motion Problem A ball is thrown upward with an initial velocity of 64 feet per second from an initial height of 80 feet a. Find the position function giving the height s as a function of the time t. b. When does the ball hit the ground?

  23. On-line Help Tutorial Video Help

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