1205_T3_soln - F07 Math 1205 -- Test 3 28 Nov 2007...

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Unformatted text preview: F07 Math 1205 -- Test 3 28 Nov 2007 SOLUTIONS 1. [12] Find the absolute extrema of f H x L = 3 x 1 3 H 4- x L on the interval @- 1, 8 D . f ' H x L = x- 2 3 H 4- x L + 3 x 1 3 H- 1 L = 4- x- 3 x x 2 3 = 4 I 1- x M x 2 3 CRITICAL POINTS: f ' H x L = 0 when x = 1 f ' H x L DNE when x = EVALUATE AT CRITICAL POINTS AND ENDPOINTS: f H- 1 L = - 3 H 5 L = - 15 f H L = f H 1 L = 3 H 3 L = 9 f H 8 L = 6 H- 4 L = - 24 ABSOLUTE MINIMUM: -24 ABSOLUTE MAXIMUM: 9 2. [12] Let f H x L = x 3 + 2 x 2- 6 x + 2. You wish to estimate a root of this function using Newton's Method. Starting with the initial guess x = 1, calculate x 1 and x 2 . f ' H x L = 3 x 2 + 4 x- 6 f H 1 L = - 1 f ' H 1 L = 1 f H 2 L = 6 f ' H 2 L = 14 x 1 = x- f I x M f ' I x M = 1- I- 1 M 1 = 2 x 2 = x 1- f H x 1 L f ' H x 1 L = 2- 6 14 = 11 7 3. Let h H x L be the function shown in the graph below. a b c q d e f g i. [6] Label statements (a-c) as True/False (a) ___ F ___ h H x L satisifies the hypotheses of the Mean Value Theorem on the interval @ a , g D . h H x L is NOT differentiable over the entire open inteval H a , g L . (b) ___ F ___ h H x L satisfies the hypotheses of Rolle's Theorem on the interval @ c , f D . h H c L h H f L (c) ___ T ___ There are exactly 2 values in the interval H a , d L which satisfy the conclusion of Rolle's Theorem....
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1205_T3_soln - F07 Math 1205 -- Test 3 28 Nov 2007...

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