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Unformatted text preview: MAC 2233/0001006 Business Calculus, Fall 2011 TEST II Name: Date: 10/20/2011 Time allowed: 50 minutes Section: Show ALL steps. Fifty points equal 100% (1 point) Question 1 . (4+4+2 points) Mr. Rose started, 4 years ago, a computer networking company serving the local businesses. The quarterly revenue R (in thousand dollars) is given by the graph below, where t is the number of quarters since June 2007. (a) Estimate the average rate of change of R from t = 0 to t = 16. Solution. The average rate of change equals the slope of the secant line between t = 0 and t = 16. From the graph, we see that R = 2 when t = 0, and R = 19 . 5 when t = 16. The slope of the secant line (and thus the average rate of change of R from t = 0 through t = 16) is R t = 19 . 5 2 16 1 . 10 thousand dollars per quarter . (b) Estimate the rate of change of R at t = 11. Solution. The rate of change of R at t = 11 equals the slope of the tangent line at t = 11. Thus, we draw the tangent line (shown in blue) at t = 11, and find two convenient points on the tangent line: (5 , 6)) and (11 , 15). The rate of change of R at t = 11 therefore equals (the slope of the tangent line at t = 11, which is) R t = 15 6 11 5 = 1 . 5 thousand dollars per quarter . (c) Estimate the value(s) of t at which R is increasing the fastest. Solution. These are values of t where the slope of the tangent line is greatest. Thus, the answer is at t = 9 . (The exact answer is actually t = 8 . 6.) Keeping Scores : Question Bonus 1 2 3 4 5 6 Total Score Out of 1 10 9 7 8 11 13 59 1 Question 2 . (3+3+3 points) Find the derivatives of the following functions. You do not need to simplify your answers. (a) y = ( t 2 2)( t 3 3 t + 1). Solution. Using the product rule, we get y = (2 t )( t 3 3 t + 1) + ( t 2 2)(3 t 2 3) = 5 t 4 15 t 2 + 2 t + 6 . (b) y = x 3 1 x 3 +1 . Solution. Use the quotient rule to obtain y = (3 x 2 )( x 3 + 1) ( x 3 1)(3 x 2 ) ( x 3 + 1) 2 = 6 x 2 ( x 3 + 1) 2 ....
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 Fall '08
 DANIELYAN
 Calculus

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