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# L29 - 9 Rectilinear Motion – Suppose a particle moves in...

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L29 – Antiderivatives Def – A function F is called an of f on an interval I if for all x on I . f ( x ) = 2 x f ( x ) = x 4 f ( x ) = sec 2 ( x ) f ( x ) = cos(3 x ) f ( x ) = e 1 2 x 1

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Thm – If F is an antiderivative of f on I , then is the most general antiderivative of f on I , where C represents any constant. If f ( x ) = x n , then F ( x ) = If n 0, then x If n < 0, then x Find F ( x ) if f ( x ) = x - 5 2
If f ( x ) = 1 x , ﬁnd F ( x ). Antidiﬀerentiation Formulas Function Antiderivative cf ( x ) f ( x ) ± g ( x ) x n ( n 6 = - 1) 1 x e x sin( x ) cos( x ) sec 2 ( x ) sec( x )tan( x ) csc 2 ( x ) csc( x )cot( x ) 1 1 - x 2 1 1 + x 2 3

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Find all functions f ( x ) such that f 0 ( x ) = x 4 - 3 x + 2 - 1 1 + x 2 4
Find all functions g ( x ) such that g 0 ( x ) = sec 2 ( x ) - 1 x + 3 x 2 - 2 x 3 2 5

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Determine f ( x ) if f 0 ( x ) = sin( x ) and f ( π ) = - 1. 6
Suppose that the slope of the tangent line to the curve y = f ( x ) at any point is x 4 + x 2 3 . If the curve passes through (1 , 4), ﬁnd f ( x ). 7

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The graph of a function f is given. Make a rough sketch of an antiderivative F if F (0) = - 1. 8
Sketch the graph of the antiderivative F that satisﬁes the initial condition F (0) = 2 if f ( x ) = x 2 - 4 x . Use a directional ﬁeld.

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Unformatted text preview: 9 Rectilinear Motion – Suppose a particle moves in a straight line and has accel-eration function a ( t ) = 3 t-4. If the initial velocity is 8 ft/sec and the initial displacement is 3 ft, ﬁnd its position function s ( t ). 10 An object is projected upward from a 96-foot high bridge and with an initial velocity of 16 ft/sec. a) Find its height at any time t . b) What is its maximum height? c) When will it hit the ground? 11 A car is moving at 48 ft/sec when the brakes are applied. It decelerates at a constant rate of 6 ft/sec until it stops. If t = 0 represents the time at which the brakes are applied, how far will it go before stopping? a ( t ) = v (0) = 12...
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L29 - 9 Rectilinear Motion – Suppose a particle moves in...

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