MIT18_014F10_ex2_sols

# MIT18_014F10_ex2_sols - Exam 2 Solutions Total 60 points...

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

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.

Unformatted text preview: Exam 2 Solutions October 29, 2010 Total: 60 points Problem 1: Find the derivative of each of the following functions • g ( x ) = log(cos( x 2 )) √ x sin x h ( x ) = e • Solution For the first problem, we invoke the chain rule twice. Thus, d 1 dx (log(cos( x 2 ))) = cos( x 2 ) · ( − sin( x 2 )) · 2 x = − 2 x tan( x 2 ) . The second problem requires the chain rule and the product rule. We imme- diately get, d ( e √ x sin x ) = e √ x sin x 1 sin x + √ x cos x . dx 2 √ x Problem 2: Consider the function log x g ( x ) = . 2 x Determine the behavior of g in a neighborhood of x = 1. Specifically, is the function increasing or decreasing? Is it convex or concave? Justify your answers. d 1 d 2 Solution We determined dx log x = x for x > and dx x = 2 x = for x > 0. Thus, we can use the quotient rule to determine g ( x ) ,g ( x ) away from x = 0. A quick calculation yields 1 − 2 log x g ( x ) = 3 x and g ( x ) = − 5 + 6 log x ....
View Full Document

{[ snackBarMessage ]}

### Page1 / 4

MIT18_014F10_ex2_sols - Exam 2 Solutions Total 60 points...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online