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Unformatted text preview: Math 115 — Final Exam December 15, 2011 Name: EXAM SOLUTIONS Instructor: Section: 1. Do not open this exam until you are told to do so. 2. This exam has 14 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 ′′ × 5 ′′ 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. 9. You must use the methods learned in this course to solve all problems. Problem Points Score 1 14 2 6 3 16 4 10 5 8 6 10 7 14 8 12 9 10 Total 100 Math 115 / Final (December 15, 2011) page 2 1 . [14 points] You are online playing the Facebookbased game, FarmVille, and you receive land with 5 stalks of corn on it. You decide that you would like to model the corn population on this patch of land using your calculus skills, so you recall that a good model for population growth is the logistic model P ( t ) = L 1 + Ae − kt L > , A > , k > . a . [5 points] Using the limit definition of the derivative , write an explicit expression for the derivative of the function P ( t ) at t = 1. Do not evaluate this expression. Solution: P ′ (1) = lim h → L 1+ Ae k (1+ h ) L 1+ Ae k (1) h b . [5 points] Using the definition of the logistic model above, compute the following in terms of L, k, and A , showing your work or providing an explanation for each part: i. [1 points] lim t →∞ P ( t ) = L Since the exponential piece is decreasing, it tends to 0 as t → ∞ . Then, the denomi nator goes to 1, and the limit of P ( t ) as t → ∞ is L . ii. [1 points] lim t →−∞ P ( t ) = 0 Since the exponential factor goes to infinity at t → ∞ , the entire denominator goes to infinity, and the function tends to 0....
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 Fall '08
 BLAKELOCK
 Math, Derivative, Pythagorean Theorem, Convex function

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