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Unformatted text preview: Math 115 – Final Exam Solutions Name: 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 8 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 14 2 12 3 9 4 12 5 12 6 12 7 12 8 17 Total 100 2 1. (14 points) Problems (a), (b) and (c) below are independent of each other. (a) (5 pts.) Compute the linear approximation to g ( x ) = 3ln( x 2 ) near x = 1. • g ′ ( x ) = 3 1 x 2 (2 x ) = 6 x ; • g ′ (1) = 6; • g (1) = 3ln(1) = 0. So, g ( x ) ≃ 6( x − 1) near x = 1. (b) (3 pts.) Write the limit definition of the derivative of the function f ( x ) = e x − e − x at the point x = a . You do not need to simplify or attempt to compute the limit. f ′ ( a ) = lim h → e a + h − e − ( a + h ) − e a + e − a h . (c) (6 pts.) Assuming the following table accurately represents the behavior of the continuous function s ( x ) over the interval [0 , 12], approximate the following: [ NOTE : the values in the table are for s ′ ( x ), not s ( x )]. x 3 6 10 12 s ′ ( x ) − 6 − 3 1.2 17 (i) s ′′ (3) ≃ 1 2 parenleftbigg 0 + 3 3 + − 3 + 6 3 parenrightbigg = (1 + 1) / 2 = 1 (ii) All intervals in [0 , 12] (if any) over which s is decreasing. If s is decreasing, then s ′ < 0. So, s is decreasing over (0 , 6). (iii) All intervals in [0 , 12] (if any) over which s is concave down. If s is concave down, then s ′ is decreasing. Since s ′ is increasing over all of [0 , 12], there are NO intervals over which s is concave down. 3 2. (12 points) Problems (a) and (b) below are independent of each other. (a) (7 pts.) In each case, calculate the value of the given integral expression. Where appropriate, you may assume that f is a differentiable function. Your final answer should not contain any integral symbols and should be simplified as much as possible. You may assume the symbols a , b and c represent constants. Show your work!Show your work!...
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 Fall '08
 BLAKELOCK
 Math, Calculus

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