1205_T3_soln - F07 Math 1205 Test 3 H4 x L 3 x 3 H-1L = 1...

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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 = 0 EVALUATE AT CRITICAL POINTS AND ENDPOINTS: f H - 1 L = - 3 H 5 L = - 15 f H 0 L = 0 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 0 = 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 0 - f I x 0 M f ' I x 0 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. ii. [6] Create and label a point q on the graph that satisfies the conclusion of the MVT on the interval H b , e L . Explain or show on the graph how q satisfies the conclusion of the MVT.
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