2009 Calculus Test 2 - Math 1151. Calculus 2009. S1. Test 2...

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: Math 1151. Calculus 2009. S1. Test 2 Version 1b. (1) Determine all a, b R such that the function f ( x ) = ax + b if x 6 1 ln x if x > 1 is differentiable at x = 1. Solution. For f to be differentiable at x = 1, we require lim x 1- f ( x ) = lim x 1 + f ( x ) lim h - f (1 + h )- f (1) h = lim h + f (1 + h )- f (1) h . The first equation gives a + b = 0 and the second equation gives lim h - f (1 + h )- f (1) h = lim h - a (1 + h ) + b- ( a + b ) h = a = lim h + f (1 + h )- f (1) h = lim h + ln(1 + h ) h = lim h + 1 1 + h = 1 so ( a, b ) = (1 ,- 1). (2) State the mean value theorem and find a point which satisfies the conclu- sions of the MVT for f ( x ) = ( x- 1) 3 on [1 , 4]. Solution. The mean value theorem states that if f is a continuous function on [ a, b ] which is differentiable on ( a, b ), then there exists c ( a, b ) such that f ( c ) = f ( b )- f ( a ) b- a ....
View Full Document

This document was uploaded on 03/19/2012.

Page1 / 2

2009 Calculus Test 2 - Math 1151. Calculus 2009. S1. Test 2...

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