Homework_9_Solutions - Math 1110 Name: Homework 9 Due 11/3...

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Unformatted text preview: Math 1110 Name: Homework 9 Due 11/3 or 11/4 in class Please print out these pages. Write your answers to the “text exercises” on separate paper and staple it to these pages. You should include computational details. These problems will be assessed for completeness. Always write neatly and legibly. Please answer the “presentation problems” in the spaces provided. Include full explanations and write your answers in complete, mathematically and grammatically correct sentences. Your answers will be assessed for style and accuracy, and you will be given written feedback on these problems. GRADES Text exercises / 20 Pres probs / 20 Staple Text exercises. Please do the following problems from the book. §3.11 #1, 4, 6ad, 11, 15, 17, 19, 30, 35, 40, 46 §4.1 #1, 4, 11, 12, 13, 14, 19, 30, 33, 36, 50, 64, 74, 78, 79, 80 §4.2 #1, 6, 12, 15, 22, 29, 32, 38, 40, 45, 47, 52, 57, 67 §4.3 #2, 5, 16, 22, 35, 44, 52, 66, 68, 72, 73, 78 /1 Math 1110 (Fall 2011) HW5 Presentation Problems Presentation problem 1. Suppose that a function f satisfies f(1) = 3 and its DERIVATIVE is ￿ d￿ 2 f(x) = e−x . dx Please show your work when you answer the questions below. (a) Find the linear approximation to f about x = 1. The linear approximation, L(x) of f(x) about x = 1 is given by the equation L(x) = f(1) + f ￿ (1)(x − 1) Substituting in the values we have, this gives L(x) = 3 + e−1(x − 1). (b) Use the linear approximation to estimate the value of f(1.5). We want to compute L(1.5). Substituting into our equation from part (a), this gives L(1.5) = 3 + e−1 · 1 2 2 Math 1 110 Sample T rue/Fals o g i s a lso o ne-to-one. t (b) I f f ( x) a nd g (x) b oth o ne-to-one f unctions d efined o n a ll o f l R., hen f Presentation p roblem 1 . D etermine w het Math 1110 (Fall 2011) HW5 Presentation sometimes) f alse, a nd c ircle y our r espons Problems Tnus I F elss 3 - a r eason w hy i t's t rue, o r a n e xample w Presentation problem 2. True/False. Determine whether the following statements arestruee venf unction,t hen s o i (a) I f f (x) i a n or false, and circle your response. Please give a brief explanation (in a complete sentence!). (a) If a differentiable function f is defined at c, and c is a critical point of f, then f has either a local minimum or a local maximum at c. The statement is false. Consider f(x) = x3. Note that f ￿ (x) = 3x2. Since f ￿ is always nonnegative, and is only 0 when x = 0, f is increasing on the entire real line. However, since f ￿ (0) = 0, we know that f has a critical point at x = 0. This critical point is neither a local maximum, nor a local minimum. (b) I f f ( x) a nd g (x) b oth o ne-to-one f un (b) On a recent trip to Boston, Professor Holm drove 350 miles in 7 hours, including a stop for gasoline. At some instant during the trip, her speedometer displayed exactly 50 miles per hour. We will assume, for the purposes of this problem, that distance traveled is a differentiable, non-decreasing function with respect to time. Let f(t) denote distance traveled in miles as a function of time in hours, where f(0) = 0 miles, and f(7) = 350 miles. Then, by the Mean ) Value Theorem, there is at least one point c in the interval [0, 7] such that f ￿ (c) = f(77−f(0) = 50 −0 miels per hour. Remember that the derivative of distance traveled with respect to time is speed, so there is at least one time at which Professor Holm was traveling at 50 miles per hour. Homework 9 Book Problem Answers: Section 3.11: Section 4.1: Section 4.2: Section 4.3: ...
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This note was uploaded on 03/01/2012 for the course MATH 1110 at Cornell University (Engineering School).

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