16soln.practice1 - 16B Calculus Francisco Santos Practice...

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16B Calculus Francisco Santos October 15, 2007 Practice Midterm 1 SOLUTIONS Name: ID: 1. (9 points) Determine whether each statement is true (T) or false (F), then CIRCLE the appropriate response. You do not need to show work. Assume that x and y are positive. ( a ) ln( x + y ) = (ln x )(ln y ) F ( b ) ln( x 3 ) = 3(ln x ) T ( c ) e x e y = e x + y T 2. (20 points) Carlos and Jeff are playing catch by the math building. Carlos is on the ground, and Jeff is on top of MSB, 49 feet off the ground. a. With what initial velocity must Carlos toss the ball up so that Jeff can (just barely) catch it? Solution: Note, part a and b have the SAME solution up to sign. Think about why. That gives the easiest soln to part a. A longer soln is here: a ( t ) = - 32 = v ( t ) = s ( t ) so v ( t ) = v ( t ) dt = - 32 dt = - 32 t + v 0 where v 0 = v (0) the initial velocity which is unknown. Likewise s ( t ) = ( - 32 t + v 0 ) = - 16 t 2 + v 0 t + s 0 = - 16 t 2 + v 0 t because s (0) = 0 if we start on the ground. That the ball just reaches Jeff means its maximum height is 49 . So at the time t 1 that v ( t 1 ) = 0 we also need s ( t 1 ) = 49 . That gives us 2 equations in 2 unknowns and we can solve for v 0 . - 32 t 1 + v 0 = 0 (so t 1 = v 0 / 32 ) - 16 t 2 1 + v 0 t 1 = 49 that is - 16( v 0 / 32) 2 + v 0 ( v 0 / 32) = 49 - v 2 0 / 64 + v 2 0 / 32 = 49 v 2 0 / 64 = 49 v 0 = ± 49 · 64 = ± 7 · 8 = ± 56 Since we require positive (upward) velocity, v 0 = 56 feet/sec . b. If Jeff then drops the ball off the roof, with what velocity will it hit the ground?
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Name: ID: 2 Solution: Now at t = 0 we start on the roof with 0 velocity, so v ( t ) = - 32 t + 0 = - 32 t s ( t
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