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Unformatted text preview: M ATH 115 –F IRST M IDTERM February 5, 2008 NAME: ****SOLUTIONS**** INSTRUCTOR: SECTION NUMBER: 1. Do not open this exam until you are told to begin. 2. This exam has 9 pages including this cover. There are ?? questions. 3. Do not separate the pages of the exam. If any pages do become separated, write your name on them and point them out to your instructor when you turn in the exam. 4. Please read the instructions for each individual exercise carefully. One of the skills being tested on this exam is your ability to interpret questions, so instructors will not answer questions about exam problems during the exam. 5. Show an appropriate amount of work for each exercise so that the graders can see not only the answer but also how you obtained it. Include units in your answers where appropriate. 6. You may use your calculator. You are also allowed two sides of a 3 by 5 notecard. 7. If you use graphs or tables to obtain an answer, be certain to provide an explanation and sketch of the graph to show how you arrived at your solution. 8. Please turn off all cell phones and pagers and remove all headphones. PROBLEM POINTS SCORE 1 10 2 12 3 14 4 6 5 8 6 14 7 12 8 12 9 12 TOTAL 100 2 1. (2 points each) For each of the following, circle all statements which MUST be true. (a) Let f be a nondecreasing differentiable function defined for all x . • f ′ ( x ) ≥ for all x . • f ′′ ( x ) ≥ for all x . • f ( x ) = 0 for some x . (b) Let f and g be continuous at x = 1 , with f ( 1) = 0 and g ( 1) = 3 . • f · g is continuous at x = 1 . • g f is continuous at x = 1 . • f g is continuous at x = 1 . (c) Let f be differentiable at x = 2 , with f (2) = 17 . • lim x → 2 f ( x ) = 17 . • lim h → f (2 + h ) f (2) h = 17 . • lim h → f (2 + h ) f (2) h exists. (d) Let f be defined on [ a,b ] and differentiable on ( a,b ) , with f ′ ( x ) < for all x in ( a,b ) . • If a < c < d < b , then f ( c ) > f ( d ) . • f ′′ ( x ) > for some x in ( a,b ) . • f is continuous on ( a,b ) . (e) Let f be a twicedifferentiable function that is concaveup on ( a,b ) , with f ( a ) = 4 and f ( b ) = 1 . • For some x in ( a,b ) , f ( x ) = 2 . 5 . • For all x in ( a,b ) , f ′′ ( x ) ≥ . • f ′ ( a ) ≤ f ′ ( b ) . 3 2. If you pluck a guitar string, a point P on the string vibrates. The motion of the point P is given by g ( t ) = A cos(220 π t ) , where g ( t ) is the displacement (in mm) of P from its position before the string was plucked, t is the number of seconds after the string was plucked, and A is a positive constant. (a) (6 points) Sketch a graph of g ( t ) , for ≤ t ≤ 1 / 55 , on the axes below. Be sure to indicate A on your sketch....
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This note was uploaded on 04/01/2008 for the course MATH 115 taught by Professor Blakelock during the Winter '08 term at University of Michigan.
 Winter '08
 BLAKELOCK

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