MIT18_014F10_ex2

MIT18_014F10_ex2 - x for every irrational x . (a) Prove f (...

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon
EXAM 2 - OCTOBER 29, 2010 (1) (5 points each) Find the derivative of each of the following functions g ( x ) = log(cos( x 2 )) h ( x ) = e x sin x (2) (10 points) Consider the function log x g ( x ) = x 2 . Determine the behavior of g in a neighborhood of x = 1. Speci±cally, is the function increasing or decreasing? Is it convex or concave? Justify your answers. 1
Background image of page 1

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

View Full DocumentRight Arrow Icon
2 EXAM 2 - OCTOBER 29, 2010 (3) (10 points) Consider the functions f ( x ) = x sin x and g ( x ) = ( x + 5) cos x . Prove there exists c (0 ,π/ 2) such that f ( c ) = g ( c ). (If you are using a theorem, make sure you explain why the function or functions you are considering satisfy the hypotheses of the theorem.) (4) (15 points) DeFne f ( x ) such that f ( x ) = x for every rational value of x and f ( x ) =
Background image of page 2
Background image of page 3

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

View Full DocumentRight Arrow Icon
Background image of page 4
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: x for every irrational x . (a) Prove f ( x ) is continuous at x = 0. (b) Set a = 0. Prove that f ( x ) is not continuous at x = a . EXAM 2- OCTOBER 29, 2010 3 (5) (15 points) Let f be continuous. Prove that x x t f ( t )( x t ) dt = f ( u ) du dt. MIT OpenCourseWare http://ocw.mit.edu 18.014 Calculus with Theory Fall 2010 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms ....
View Full Document

Page1 / 4

MIT18_014F10_ex2 - x for every irrational x . (a) Prove f (...

This preview shows document pages 1 - 4. Sign up to view the full document.

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