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Miss Battaglia AP Calculus AB/BC

4.1 Antiderivatives and Indefinite Integration Objective: Use indefinite integral notation for antiderivatives and use basic integration rules to find antiderivatives . Miss Battaglia AP Calculus AB/BC. Definition of Antiderivative

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Miss Battaglia AP Calculus AB/BC

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  1. 4.1 Antiderivatives and Indefinite IntegrationObjective: Use indefinite integral notation for antiderivatives and use basic integration rules to find antiderivatives. Miss Battaglia AP Calculus AB/BC

  2. Definition of Antiderivative A function F is an antiderivative of f on an interval I if F’(x)=f(x) for all x in I. Representation of Antiderivatives If F is an antiderivative of f on an interval I, then G is an antiderivative of f on the interval I if and only if G is of the form G(x)=f(x)+C, for all x in I where C is a constant.

  3. Solving a 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

  4. Notation for Antiderivatives When solving a differential equation of the form it is convenient to write it in the equivalent differential form The general solution is denoted by

  5. Basic Integration Rules *Refer to handout

  6. Applying the Basic Integration Rules Describe the antiderivatives of 3x. Original Integrate Rewrite Simplify

  7. Rewriting Before Integrating

  8. Integrating Polynomial Functions a. b. c.

  9. Rewriting Before Integrating

  10. Rewriting Before Integrating

  11. Finding a Particular Solution Find the general solution of and find the particular solution that satisfies the initial condition F(1)=0.

  12. Solving a Vertical Motion Problem A ball is thrown upward with an initial velocity of 64 ft/sec from an initial height of 80 ft. • Find the position function giving the height s as a function of the time t. • When does the ball hit the ground?

  13. Rewriting Before Integrating

  14. Classwork/Homework • Read 4.1 Page 255 #5,6,10,11,12,14,27,29,33,35,37,38,49,60,61,64,84

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