mat320mid2sols

mat320mid2sols - Stony Brook University - MAT 320 Midterm...

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Stony Brook University - MAT 320 Midterm II Solutions 1. (25 points) The functions f and g are continuous on the interval [ a, b ] and diFerentiable on ( a, b ). If f ( a ) = g ( a ) and f ( b ) = g ( b ), prove that there is a point c , with a < c < b , where f 0 ( c ) = g 0 ( c ). Apply Rolle’s Theorem to f - g . 2. (25 points) The function sin x is in±nitely diFerentiable on R , and its derivatives cycle through cos x, - sin x, - cos x, sin x . Let P k ( x ) be the k -th Taylor polynomial, based at 0, for sin x . Prove that lim k →∞ P k ( x ) = sin x for every x R . Choose x . By Taylor’s Theorem, there exists c between 0 and x such that sin x - P k ( x ) = f ( k +1) ( c ) x k +1 ( k + 1)! . Since | f ( k +1) ( c ) | ≤ 1, we have | sin x - P k ( x ) | ≤ | x | k +1 / ( k +1)!. The proof then follows from lim n →∞ a n /n ! = 0, true for any a R . 3. (25 points) The function f is continuous and twice diFerentiable on the interval [ a, b ], with both derivatives positive there:
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This note was uploaded on 01/31/2011 for the course AMS 319 taught by Professor Mark during the Fall '10 term at SUNY Stony Brook.

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mat320mid2sols - Stony Brook University - MAT 320 Midterm...

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