math111_2006spring_prelim3sol

# math111_2006spring_prelim3sol - Math 111 Prelim 3 Name...

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Math 111 (Spring 2006) Prelim 3 (4/25/2006) 2 1. [16pts (4pts each)] (a) Evaluate integraldisplay [2 e x + 4 cos x ] dx = 2 e x + 4 sin x + C (b) Evaluate integraldisplay 2 1 4 + u 2 u 3 du = integraldisplay 2 1 4 u 3 + 1 u du = - 2 u 2 + ln | u | vextendsingle vextendsingle vextendsingle vextendsingle 2 1 = - 1 2 + ln 2 - ( - 2 + 0) = 3 2 + ln 2 CONTINUE TO NEXT PAGE
Math 111 (Spring 2006) Prelim 3 (4/25/2006) 3 (c) Evaluate integraldisplay sin( θ ) - 1 cos 2 ( θ ) = integraldisplay sin( θ ) cos( θ ) 1 cos( θ ) - 1 cos 2 ( θ ) = integraldisplay tan( θ ) sec( θ ) - sec 2 ( θ ) = sec( θ ) - tan( θ ) + C (d) Find the most general antiderivative of f ( x ) = 1 x cos( x ) - ln( x ) sin( x ) . F ( x ) = cos x ln x + C CONTINUE TO NEXT PAGE

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Math 111 (Spring 2006) Prelim 3 (4/25/2006) 4 2. [8 pts] Find the derivative of h ( x ) = integraldisplay 0 e x sin 3 t dt = - integraldisplay e x 0 sin 3 t dt h ( x ) = - e x sin 3 ( e x ) 3. [8 pts] Given that integraldisplay x 1 2 f ( t ) dt = x 2 cos 2 x find f ( x ). d dx integraldisplay x 1 2 f ( t ) dt = d dx x 2 cos 2 x 2 f ( x ) = - 2 x 2 cos x sin x + 2 x cos 2 x 2 f ( x ) = 2( x cos 2 x - x 2 cos x sin x ) f ( x ) = x cos 2 x - x 2 cos x sin x CONTINUE TO NEXT PAGE
Math 111 (Spring 2006) Prelim 3 (4/25/2006) 5 4. [7 pts] If w ( t ) is the rate of growth of a child in pounds per year, what does integraldisplay 10 5 w ( t ) dt represent? Since integraldisplay 10 5 w ( t ) dt = w (10) - w (5), where w ( t ) is the weight of the child at time t , then the integral represents the amount of pounds gained between age 5 and age 10.

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