This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Math 116 — Final Exam December 17, 2009 Name: Instructor: Section: 1. Do not open this exam until you are told to do so. 2. This exam has 11 pages including this cover. There are 9 problems. Note that the problems are not of equal difficulty, so you may want to skip over and return to a problem on which you are stuck. 3. Do not separate the pages of this exam. If they do become separated, write your name on every page and point this out to your instructor when you hand in the exam. 4. Please read the instructions for each individual problem carefully. One of the skills being tested on this exam is your ability to interpret mathematical questions, so instructors will not answer questions about exam problems during the exam. 5. Show an appropriate amount of work (including appropriate explanation) for each problem, so that graders can see not only your answer but how you obtained it. Include units in your answer where that is appropriate. 6. You may use any calculator except a TI92 (or other calculator with a full alphanumeric keypad). However, you must show work for any calculation which we have learned how to do in this course. You are also allowed two sides of a 3 00 × 5 00 note card. 7. If you use graphs or tables to find an answer, be sure to include an explanation and sketch of the graph, and to write out the entries of the table that you use. 8. Turn off all cell phones and pagers , and remove all headphones. Problem Points Score 1 10 2 8 3 6 4 12 5 14 6 12 7 14 8 12 9 12 Total 100 Math 116 / Final (December 17, 2009) page 2 You may find the following expressions useful. And you may not. But you may use them if they prove useful. “Known” Taylor series (all around x = 0): sin( x ) = x x 3 3! + ··· + ( 1) n x 2 n +1 (2 n + 1)! + ··· cos( x ) = 1 x 2 2! + ··· + ( 1) n x 2 n (2 n )! + ··· e x = 1 + x + x 2 2! + ··· + x n n ! + ··· (1 + x ) p = 1 + px + p ( p 1) 2! x 2 + p ( p 1)( p 2) 3! x 3 + ··· “Known” integral expressions: Z x n ln xdx = 1 n + 1 x n +1 ln x 1 ( n + 1) 2 x n +1 + C Z e ax sin( bx ) dx = 1 a 2 + b 2 e ax ( a sin( bx ) b cos( bx )) + C Z e ax cos( bx ) dx = 1 a 2 + b 2 e ax ( a cos( bx ) + b sin( bx )) + C Z sin( ax )sin( bx ) dx = 1 b 2 a 2 ( a cos( ax )sin( bx ) b sin( ax )cos( bx )) + C, a 6 = b Z cos( ax )cos( bx ) dx = 1 b 2 a 2 ( b cos( ax )sin( bx ) a sin( ax )cos( bx )) + C, a 6 = b Z sin( ax )cos( bx ) dx = 1 b 2 a 2 ( b sin( ax )sin( bx ) + a cos( ax )cos( bx )) + C, a 6 = b “Known” equations from geometry: Volume of a sphere: V = 4 3 π r 3 Surface area of a sphere: A = 4 π r 2 Volume of a cylinder: V = π r 2 h Volume of a cone: V = 1 3 π r 2 h Math 116 / Final (December 17, 2009) page 3 1 . [10 points] For each statement below, circle TRUE if the statement is always true; otherwise, circle FALSE. No partial credit on this page....
View
Full
Document
This note was uploaded on 01/28/2012 for the course MATH 116 taught by Professor Irena during the Fall '07 term at University of Michigan.
 Fall '07
 Irena
 Math, Calculus

Click to edit the document details