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4.1 Antiderivatives and Indefinite Integration

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Suppose you were asked to find a function F whose derivative is From your knowledge of derivatives, you would probably say The function F is an antiderivative of f. In general, a function F is an antiderivative of f (x) if Note that F is an antiderivative, not the antiderivative Ex:

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In general: is the antiderivative of Example 1 Example 2: Solving a Differential Equation Gives the entire family of antiderivatives

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Notation for Antiderivatives: When solving a differential equation it is convenient to write the differential form The operation of finding all solutions of this equation is antidifferentiation or indefinite integration Integrand Variable of integration Constant of integration

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Practice

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Initial Conditions and Particular Solutions Solve the differential equation: Solve the differential equation above if the curve passes through (2,4)—called an initial condition.

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Initial Conditions and Particular Solutions Find the general solution of: and find the particular solution that satisfies the initial condition F(1)=0

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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, as shown in the figure. 1.Find the position function giving the height s as a function of time t 2.When does the ball hit the ground?

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