ex1solutionsfall05

# ex1solutionsfall05 - Math 116 First Midterm Exam SOLUTIONS...

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2 1. (8 points) Consider the functions f and g deFned below. Assume a is a nonzero constant. f ( x ) = ( x a ) 2 x , g ( x ) = x cos( ax ) , (a) (4 pts.) ±ind the family of antiderivatives of f ( x ). Show step-by-step work. We have f ( x ) = x 2 2 ax + a 2 x = x 3 / 2 2 ax 1 / 2 + a 2 x 1 / 2 . So, i f ( x ) dx = 2 5 x 5 / 2 4 3 ax 3 / 2 + 2 a 2 x 1 / 2 + C. (b) (4 pts.) ±ind the family of antiderivatives of g ( x ). Show step-by-step work. Using integration by parts, u = x u = 1 v = cos( ax ) v = sin( ax ) a . Then, i x cos( ax ) dx = x a sin( ax ) i sin( ax ) a dx = x a sin( ax ) + cos( ax ) a 2 + C.
3 2. (13 points) In the magical tale of Harry Potter and the Half-blood Prince, Harry and professor Dumbledore go in search of a horcrux, a dark magic device created by the dark wizard Voldemort in order to hide and preserve a piece of his soul. To get to the horcrux, Dumbledore must drink a lethal green elixir Flled with dark power. Harry’s task is to scoop some of the magic liquid into a goblet, feed it to Dumbledore, then turn around and scoop some more liquid into the goblet, repeating these steps until the elixir is gone. Harry notices that, after the Frst glass, Dumbledore’s drinking speed increases at a decreasing rate. By the end, Dumbledore is drinking as slowly as possible. The graph below represents Dubledore’s drinking speed, s (in ±uid ounces per second) against time, t (in seconds.) s t 1 2 3 4 4 5 6 8 12 16 20 24 28 32 (a) (5 pts.) Sketch the amount a of the elixir that Dumbledore has drunk as a function of time t during the Frst 32 seconds after he started drinking. Include scales on each axis, and label them appropriately. Clearly indicate on your sketch the heights of the graph of

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ex1solutionsfall05 - Math 116 First Midterm Exam SOLUTIONS...

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