Exam3 FA16 - Exam sample PDF

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Math 140–01 EXAM 3 100 points possible November, 4 2016 Instructions: Write your name, section number and TA’s name on each answer sheet. Answer each numbered question on a separate answer sheet. You must show all appropriate work in order to receive credit for an answer. At the end of the exam, arrange the answer sheets in order and write and sign the honor pledge on the first one (only on the first one). Show all work – No calculators – Good luck. 1. (15 points) Find the function f given that f ′ (x) = sec2 x − xπ2 + sin(2x) and f (π) = 1. ,1 for x in the interval [ −1 2. (15 points) Let g(x) = x2/3 − 2x 8 8 ]. Find and classify all absolute extreme 3 values of g in the interval [ −1 , 1 ]. Determine at which number in the interval they are assumed. 8 8 3. (20 points) Find the area of the largest rectangle that has two sides on the positive x axis and the positive y axis, one vertex at the origin, and one vertex on the curve y = e−x . To get full credit you must draw a labelled picture describing the situation. 4. (20 points) a. The half-life time of a radioactive material is 1590 years. How long would it take for 2/3 of it to disappear? Do not simplify your answer which may include logs. b. Can you find a function f that is continuous on [0, 2], and differentiable on (0, 2) such that f (0) = −1, f(2) = 7, and f ′ (x) ≤ −2 for all x in (0, 2). Justify your answer by providing and example of such a function if it exists, or by giving reasons why such function cannot exist. 4x −1 5. (20 points) Let f (x) = ln( x−1 , and f ′′ (x) = x(2x−1) ) for x ∈ (1, ∞). Then, f ′ (x) = x(x−1) 2 (x−1)2 . a. Evaluate limx→1+ f (x) and find the vertical asymptotes (if any) of the graph of f . b. Evaluate limx→∞ f (x) and find the horizontal asymptotes (if any) of the graph of f . c. Find the interval(s) on which f is increasing and the interval(s) on which f is decreasing. d. Find the interval(s) on which the graph of f is concave upward. e. Sketch the graph of f on (1, ∞) including all pertinent information.

6. (10 points) Determine if the following limits exist as a number, as ∞ , as −∞, or does not exist. If the limit is a number, evaluate it. Justify your answers. √ 2 −ln 2 (ii) lim ln(x +1) (i) lim (x + 9x2 − 1) . x+1 x→−∞

x→−1

Bonus question (5 points) Evaluate the following limit. Justify your answer. (No partial credit.) sin3 x . 2 x→0 x tan 3x

lim

TAs: Emily Deboy (0111); Jianlong Liu (0112, 0122); John Padgett (0113); Jacob Renn (0121); Mark Magsino (0123, 0132); Michael Dworken ( 0131)....