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calc notes lecture 23

calc notes lecture 23 - 2 dt x e dx x 2 f 7 1 dx x x g 29 x...

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Lecture Section Objectives Assignment 24 5.1 Antiderivatives and Indefinite Integrals 5.1: 1-19 odd, 23, 25, 33-41 odd, 49-61 odd, 71, 75, 77 Understanding Goals: 1. Understand the definition of antiderivative. 2. Understand how to use integral sign for antiderivatives. 3. Understand how to use basic integration rules to find antiderivatives of functions. 4. Understand how to fine particular solutions of indefinite integrals using initial conditions. 5. Understand how to use antiderivatives to solve real-life problems. Question to think about: We know f x x ( ) = 2 has derivative f x x '( ) = 2 . Is x 2 the only function whose derivative is 2 x ? Integration is the “inverse” of differentiation '( ) ( ) F x dx F x C = + Differentiation is the “inverse” of integration ( ) ( ) d f x dx f x dx � � = � � ° 1
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Example 1 : Find a) 5 dx x b) 2 x dx x c) x dx
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Unformatted text preview: 2 dt x e) dx x 2 f) 7 1 dx x x g) ( 29 x x dx + h) ( 1) x dx + i) 3 2 (2 5 1) x x x dx-+ + j) 1 x dx x + 3 Differential equation : an equation involves x , y and derivatives of y . Particular solution of differential equation: 4 Example 2 : Find the general solution of 2 1 '( ) f x x = , x and find the particular solution that satisfies the initial condition (1) f = . Example 3 : Solve the differential equation a) ( 29 ( 29 '( ) 2 3 2 3 , (3) f x x x f =-+ = b) 2 "( ) , '(0) 6, (0) 3 f x x f f = = = 5 Example 4 : A ball is thrown upward with an initial velocity of 64 ft/s 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? 6...
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