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Unformatted text preview: M ATH 115 –S ECOND M IDTERM November 18, 2008 NAME: SOLUTIONS INSTRUCTOR: SECTION NUMBER: 1. Do not open this exam until you are told to begin. 2. This exam has 8 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. If you need extra room, you may use the back of a page but be sure to clearly indicate and label your work. 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 18 4 8 5 6 6 16 7 16 8 14 TOTAL 100 2 1. For the following questions select true if the statement is always true, and false otherwise. Each question is worth 1 point. (a) If f is differentiable and f ′ ( p ) = 0 or f ′ ( p ) is undefined, then f ( p ) is either a local maximum or a local minimum. True False (b) For f a twice differentiable function, if f ′ is increasing, then f is concave up and increasing. True False (c) The global maximum of f ( x ) = x 2 on every closed interval is at one of the endpoints of the interval. True False (d) If f ( x ) has an inverse function g ( x ) , then g ′ ( x ) = 1 /f ′ ( x ) . True False (e) If a function is periodic with period c , then so is its derivative. True False (f) If C ( q ) represents the cost of producing a quantity q of goods, then C ′ (0) represents the fixed costs. True False (g) If a differentiable function f ( x ) has a global maximum on the interval ≤ x ≤ 10 at x = 0 , then f ′ ( x ) ≤ for ≤ x ≤ 10 . True False (h) If f ( x ) is differentiable and concave up, then f ′ ( a ) < f ( b ) − f ( a ) b − a for a < b . True False (i) If you zoom in with your calculator on the graph of y = f ( x ) in a small interval around x = 10 and see a straight line, then the slope of that line equals the derivative f ′ (10) . True False (j) If f ′ ( x ) ≥ for all x , then f ( a ) ≤ f ( b ) whenever a ≤ b . True False 3 2. In 1956, Marion Hubbert began a series of papers predicting that the United States’ oil production would peak and then decline. Although he was criticized at the time, Hubbert’s prediction was remarkably accurate. He modeled the annual oil production P ( t ) , in billions of barrels of oil, over time t , in years, as the derivative...
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 Fall '08
 BLAKELOCK
 Math, Calculus, Derivative, Mathematical analysis, oil production

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