1110-fa2011-PRELIM2-solutions

# 1110-fa2011-PRELIM2-solutions - Math 1110 Prelim 2...

This preview shows pages 1–3. Sign up to view the full content.

Math 1110 Prelim 2 (10/27/2011) 1 Question 1. (16 points) Let f ( x ) and g ( x ) be differentiable functions, with f ( x ) a one-to-one func- tion. We know the following values: x f ( x ) g ( x ) f 0 ( x ) g 0 ( x ) - 2 - 2 1 2 8 - 1 1 7 4 1 0 3 9 9 - 3 1 4 - 3 2 6 2 6 - 2 8 - 3 . (a) Let h ( x ) = ln ( x + 2 ) g ( x ) . Then h 0 (- 1 ) = g ( x ) 1 x + 2 - ln ( x + 2 ) g 0 ( x ) ( g ( x )) 2 ± ± ± ± ± x =- 1 = 7 · 1 1 - 0 · 1 7 2 = 1 7 . (b) Let j ( x ) = g ( x ) tan - 1 ( 3x ) . Then j 0 ( 0 ) = ² g 0 ( x ) tan - 1 ( 3x ) + g ( x ) 3 1 + ( 3x ) 2 ³± ± ± ± x = 0 = - 3 · 0 + 9 · 3 1 + 0 = 27. (c) Let ( x ) = f g ( x ) . Then 0 ( 2 ) = f 0 ( g ( x )) g 0 ( x ) ± ± x = 2 = f 0 (- 2 ) · (- 3 ) = 2 · (- 3 ) = - 6. (d) Let p ( x ) = ´ f - 1 ( x ) + 2x 2 µ 4 . The ﬁrst thing to note is that f (- 1 ) = 1 , so f - 1 ( 1 ) = - 1 . Then p 0 ( 1 ) = 4 h f - 1 ( x ) + 2x 2 i 3 · ² 1 f 0 ( f - 1 ( x )) + 4x ³± ± ± ± x = 1 = 4 · [(- 1 ) + 2 ] 3 · ² 1 f 0 (- 1 ) + 4 ³ = 4 · [ 1 ] 3 · ² 1 4 + 4 ³ = 4 · 17 4 = 17.

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

View Full Document
Math 1110 Prelim 2 (10/27/2011) 2 Question 2. (20 points) Evaluate the following expressions. (a) cot ( sin - 1 ( - 1 3 )) e -1 3 Ƽ 8 We ﬁrst see that sin - 1 (- 1 3 ) is the angle θ given in the ﬁgure to the left. Then, taking the cotangent of that angle, we get cot ± sin - 1 ± - 1 3 ²² = 8 - 1 = - 8. (b) d dx h e ln ( x 2 + e ) i = d dx h x 2 + e i = 2x (c) The velocity of a particle is given by s 0 ( t ) = v ( t ) = 3t 4 - 20t 3 + 17t + tan - 1 ( 20 ) . Determine the acceleration a ( t ) . It is a ( t ) = v 0 ( t ) = 12t 3 - 60t 2 + 17 . (d) d dx h x x i for y = x x with x > 0 We use the inverse property of the natural exponential function and the natural log function, then use the chain rule along with the product rule to differentiate.
This is the end of the preview. Sign up to access the rest of the document.

## This note was uploaded on 03/01/2012 for the course MATH 1110 at Cornell University (Engineering School).

### Page1 / 6

1110-fa2011-PRELIM2-solutions - Math 1110 Prelim 2...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online