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prelim spring2006 solutions

# prelim spring2006 solutions - Math 111 Prelim 1 Feb 21 2006...

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Math 111 (Spring 2006) Prelim 1 Solutions (02/21/2006) 2 1. [12pts (3pts each)] Evaluate the following limits. You must show all your work. Note that simply plotting points is not su ffi cient to receive full credit. (a) lim s 3 s 2 - s s 2 - 2 = 3 2 - 3 3 2 - 2 = 6 7 (direct substitution property) (b) lim x →- 3 x 2 - 9 x 2 - x - 12 = lim x →- 3 ( x - 3)( x + 3) ( x - 4)( x + 3) = lim x →- 3 x - 3 x - 4 = lim x →- 3 - 3 - 3 - 3 - 4 = 6 7 (c) lim h 0 e h - 1 h = 1 Since this is the formula for the definition of derivative of f ( x ) = e x at the point x = 0. Note that: f (0) = lim h 0 f (0 + h ) - f (0) h = lim h 0 e h - 1 h Since f ( x ) = e x , then f (0) = 1. (d) lim x →∞ x 3 + 2 x + 1 7 - x = lim x →∞ x 3 x + 2 x x + 1 x 7 x - x x = lim x →∞ x 2 + 2 + 1 x 7 x - 1 = -∞ (because ( x 2 + 2 + 1 x ) → ∞ and ( 7 x - 1) → - 1 as x → ∞ ) CONTINUE TO NEXT PAGE
Math 111 (Spring 2006) Prelim 1 Solutions (02/21/2006) 3 2. [15pts (3pts each)] The women’s luge final for the 2006 Winter Olympics in Torino, Italy was held last Wednesday. The fastest luge run was completed in 46.820 seconds by Sylke Otto, an athlete from Germany. The table below shows the data from her fastest run. time (s) distance (m) 0 0 5 . 832 60 20 . 370 420 33 . 596 870 40 . 375 1140 46 . 820 1435 (a) Using the data from the table, compute her average velocity (in meters per second) over the last 565 meters of the run. average velocity = 1435 - 870 46 . 820 - 33 . 596 = 42 . 72 m/s (b) The function s ( t ) = 0 . 19 t 2 + 21 . 32 t models this run reasonable well. Using this function, give the formula for her average velocity over the interval [ t, t + h ]. average velocity = . 19( t + h ) 2 + 21 . 32( t + h ) - ( . 19 t 2 + 21 . 32 t ) t + h - t CONTINUE TO NEXT PAGE

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Math 111 (Spring 2006) Prelim 1 Solutions (02/21/2006) 4 (c) Using the function, s ( t ) = 0 . 19 t 2 + 21 . 32 t given in (b), set up the limit as h 0 to find her instantaneous velocity at time t .
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