140Bquiz4csols

140Bquiz4csols - 1 MATH 035 QUIZ#4 Solutions Date: Sept 21,...

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Unformatted text preview: 1 MATH 035 QUIZ#4 Solutions Date: Sept 21, 2007 Duration: 15 minutes [ 6 ] 1. First we compute the simplified difference quotient: a) f (x + h) - f (x) 1 = h h 1 = h 1 = h 1 = h x+h x - (x + h) - 3 x - 3 (x + h)(x - 3) - x(x + h - 3) (x + h - 3)(x - 3) 2 x - 3x + xh - 3h - x2 - xh + 3x (x + h - 3)(x - 3) -3 (x + h - 3)(x - 3) b) Computing the limit to get the derivative: f (x) = lim h0 -3 -3 = . (x + h - 3)(x - 3) (x - 3)2 -3 3 =- . 2 (1 - 3) 4 c) Evaluating: f (1) = [ 2 ] 2. By simply matching the given limit with f (x) = lim you can check that f (x) = f (x + h) - f (x) h0 h x and x = a = 9 are correct choices. There are actually others. [ 2 ] 3. All of these expressions involve rates of change of y with respect to x between x = 2 and x = 4. The graph gets less steep as x increases. It is therefore steepest at x = 2 and least steep at x = 4, so f (2) is the largest and f (4) is the smallest. Next f (3) - f (2) = and similarly, f (3) - f (2) = (average rate over [2,3]) 3-2 The slope of the tangent to the graph of f is smaller everywhere on (3, 4) than on (2, 3), so the latter expression above is smaller. Therefore the numbers in decreasing order are: 1 f (2) > f (3) - f (2) > [f (4) - f (2)] > f (4). 2 1 f (4) - f (2) [f (4) - f (2)] = = (average rate over [2,4]) 2 4-2 ...
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This note was uploaded on 03/30/2008 for the course MATH 140B taught by Professor Fabbri,marc during the Fall '07 term at Penn State.

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