MIT18_014F10_ex2_sols

MIT18_014F10_ex2_sols - Exam 2 Solutions October 29, 2010...

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

View Full Document Right Arrow Icon

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

View Full DocumentRight Arrow Icon
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

This note was uploaded on 01/18/2012 for the course MATH 18.014 taught by Professor Christinebreiner during the Fall '10 term at MIT.

Page1 / 4

MIT18_014F10_ex2_sols - Exam 2 Solutions October 29, 2010...

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

View Full Document Right Arrow Icon
Ask a homework question - tutors are online