This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Miller, Kierste Exam 2 Due: Oct 31 2007, 5:00 pm Inst: JEGilbert 1 This printout should have 18 questions. Multiplechoice questions may continue on the next column or page find all choices before answering. The due time is Central time. 001 (part 1 of 1) 10 points Find a vector function that represents the curve of intersection of the cone z = p x 2 + y 2 and the plane z y = 13. 1. r ( t ) = t i + t 2 169 26 j + t 2 + 169 26 k correct 2. r ( t ) = p 13(2 t + 13) i + ( t 13) j + t k 3. r ( t ) = t 2 169 26 i + t j + t 2 + 169 26 k 4. r ( t ) = t 2 169 26 i + t j + t 2 + 13 26 k 5. r ( t ) = p 13(2 t 13) i + t j + ( t + 13) k Explanation: The vector function r ( t ) = x ( t ) i + y ( t ) j + z ( t ) k intersects the cone z = p x 2 + y 2 and the plane z y = 13 when z ( t ) = q x ( t ) 2 + y ( t ) 2 , z ( t ) y ( t ) = 13 . Now this last condition already eliminates two of the answer choices, leaving only r ( t ) = t i + t 2 169 26 j + t 2 + 169 26 k , r ( t ) = p 13(2 t + 13) i + ( t 13) j + t k , and r ( t ) = p 13(2 t 13) i + t j + ( t + 13) k . But only the first of these equations defines a curve lying on the cone z = p x 2 + y 2 . Consequently, only r ( t ) = t i + t 2 169 26 j + t 2 + 169 26 k lies on both the cone and the plane. keywords: 002 (part 1 of 1) 10 points Determine f xy when f ( x, y ) = 2 xy ln( xy ) 3 xy . 1. f xy = 2(ln( xy xy ) + 1 2. f xy = 4(ln( xy ) + xy ) 1 3. f xy = 4(ln( xy ) xy ) + 1 4. f xy = 2ln( xy ) + 1 correct 5. f xy = 4ln( xy ) 1 6. f xy = 2ln( xy ) 1 Explanation: Since ln( xy ) = ln x + ln y , we see that f ( x, y ) = 2 xy (ln x + ln y ) 3 xy . But then, f x = 2 y (ln x + ln y ) + 2 xy x 3 y = 2 y (ln x + ln y ) y , in which case f xy = 2(ln x + ln y ) + 2 y y 1 = 2(ln x + ln y ) + 1 , Miller, Kierste Exam 2 Due: Oct 31 2007, 5:00 pm Inst: JEGilbert 2 after differentiating with respect to y . Conse quently, f xy = 2ln( xy ) + 1 . keywords: partial derivative, mixed partial derivative, log function, 003 (part 1 of 1) 10 points Find the arc length of the curve r ( t ) = (1 2 t ) i + ln(2 t ) j + (3 t 2 ) k between r (1) and r (3). 1. arc length = 8 + ln3 correct 2. arc length = 3 + ln6 3. arc length = 9 + 2ln3 4. arc length = 8 2ln3 5. arc length = 6 ln3 6. arc length = 8 ln3 Explanation: The length of a curve r ( t ) between r ( t ) and r ( t 1 ) is given by the integral L = Z t 1 t  r ( t )  dt . Now when r ( t ) = (1 2 t ) i + ln(2 t ) j + (3 t 2 ) k we see that r ( t ) = 2 i + 1 t j + 2 t k ....
View
Full
Document
 Fall '07
 Gilbert

Click to edit the document details