Gilbert_Exam01sol

Gilbert_Exam01sol - Version 017 – Exam01 – Gilbert –...

Info iconThis preview shows pages 1–3. 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: Version 017 – Exam01 – Gilbert – (56380) 1 This print-out should have 12 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 10.0 points Determine if lim x → parenleftBig 4 x- 12 e 3 x- 1 parenrightBig exists, and if it does, find its value. 1. limit = 4 2. limit = 0 3. limit does not exist 4. limit = 3 5. limit = 12 6. limit = 6 correct Explanation: Now 4 x- 12 e 3 x- 1 = 4 parenleftBig e 3 x- 1- 3 x x ( e 3 x- 1) parenrightBig = f ( x ) g ( x ) where f, g are everywhere differentiable func- tions such that lim x → f ( x ) = 0 , lim x → g ( x ) = 0 . Thus L’Hospital’s rule can be applied. But f ′ ( x ) = 12 e 3 x- 12 , while g ′ ( x ) = ( e 3 x- 1) + 3 xe 3 x , so lim x → f ( x ) g ( x ) = lim x → f ′ ( x ) g ′ ( x ) = lim x → parenleftBig 12( e 3 x- 1) e 3 x + 3 xe 3 x- 1 parenrightBig . As f ′ and g ′ are differentiable functions such that lim x → f ′ ( x ) = 0 , lim x → g ′ ( x ) = 0 we have to apply L’Hospital’s rule again. But f ′′ ( x ) = 36 e 3 x , g ′′ ( x ) = 6 e 3 x + 9 xe 3 x , from which it follows that lim x → f ′′ ( x ) = 36 , lim x → g ′′ ( x ) = 6 . Consequently, the limit exists and limit = 6 . 002 10.0 points The region R is bounded by the x-axis and the graphs of y = 4 x , x = 2 . A part of R is shown as the shaded region in x 2 y Compute the volume of the solid of revolution obtained by rotating R around the x-axis. 1. volume = 12 π 2. volume = 11 π 3. volume = 10 π 4. volume = 9 π 5. volume infinite Version 017 – Exam01 – Gilbert – (56380) 2 6. volume = 8 π correct Explanation: Since R extends all the way to x = ∞ , the volume of the solid of revolution obtained by rotating R around the x-axis is given by the improper integral V = π integraldisplay ∞ 2 y 2 dx = π integraldisplay ∞ 2 16 x 2 dx . Now integraldisplay t 2 16 x 2 dx = bracketleftBig- 16 x bracketrightBig t 2 = 16 parenleftBig 1 2- 1 t parenrightBig . But lim t →∞ 1 t = 0 . Consequently, V = lim t →∞ π integraldisplay t 2 16 x 2 dx = 8 π . 003 10.0 points Find the n th term, a n , of an infinite series ∑ ∞ n =1 a n when the n th partial sum, S n , of the series is given by S n = n n + 1 . 1. a n = 1 2 n 2 2. a n = 1 n ( n + 1) correct 3. a n = 5 n ( n + 1) 4. a n = 5 2 n 5. a n = 1 2 n 6. a n = 5 2 n 2 Explanation: Since S n = a 1 + a 2 + ··· + a n , we see that a 1 = S 1 , a n = S n- S n − 1 ( n > 1) ....
View Full Document

This note was uploaded on 04/12/2011 for the course M 408d taught by Professor Sadler during the Spring '07 term at University of Texas.

Page1 / 7

Gilbert_Exam01sol - Version 017 – Exam01 – Gilbert –...

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

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