{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

mat320mid2sols

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

This preview shows pages 1–2. Sign up to view the full content.

Stony Brook University - MAT 320 Midterm II Solutions 1. (25 points) The functions f and g are continuous on the interval [ a, b ] and differentiable 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 infinitely differentiable 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 differentiable on the interval [ a, b ], with both derivatives positive there: f 0 ( x ) > 0 and f 00 ( x ) > 0 for every a x b . Suppose that f ( a ) < 0 and f ( b ) > 0.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}