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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]) 32 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 42 ...
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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.
 Fall '07
 FABBRI,MARC
 Math

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