M408 C Fall 2011Dr. Jeffrey DancigerExam 2November 3, 2011NAMEEIDSection time (circle one): 11:00am 1:00pm 2:00pmNo books, notes, or calculators. Show all your work.Do NOT open this exam booklet until instructed to do so.1Please leave this page blank.Question Points Score1 152 103 104 105 166 107 98 (bonus) 5Total 8021. [15 pts.] The following chart contains partial information about the continuous functionf(x) (defined for all numbers) and its derivatives.f(x) f0(x) f00(x)x < −2 − +x = −2 −10 DNE−2 < x < 0 + +x = 0 −2 6 00 < x < 4 + −x = 4 5 0 −4x > 4 ?? −(a): Find all local maxima and local minima. Justify your answer.(b): On what intervals is the graph of f(x) concave up? concave down? Locate any inflectionpoints.(c): Identify the intervals of increase and decrease for f(x).32. [10 pts.] Two cars start moving from the same point. One travels east at 15 mi/h and theother travels north at 20 mi/h. At what rate is the distance between the cars increasing twohours later?43. [10 pts.] The conditionsy = −x + 2−1 ≤ x ≤ 4define a line segment in the x-y plane. Find the points on the line segment with maximumand minimum distance to the origin.54. [10 pts.] Find the limit. You may use any method. Please show all steps clearly.(a): limθ→π21 − sin θ1 + cos 2θ(b): limx→0+x ln x65. [16 pts] Sketch the graph of y = xe−x. Label all important features of the graph. You mayuse this page and the next one to show your work.7Problem 5 extra page86. [10 pts](a): Let f (x) be continuous on [1, 5] and differentiable on (1, 5). What does the Mean ValueTheorem say about f(x)?(b): Suppose that f0(x) > 3 for all x, and that f(1) = 1. Show that f(5) > 13.97. [9 pts.](a): Find the linearization L(x) of f (x) = x1/3at x = 8.(b): Use your answer form part (a) to estimate the value of3√8.2.(c): What is the concavity of f(x) near x = 8? Use this to explain whether your answerfrom part (b) is an over-estimate or an under-estimate.108. [Bonus - 5pts] Let f (x) be de defined for all x. Suppose f(x) is continuous and differentiablewhenever x 6= 0 and satisfiesf(0) = 1 limx→0+f(x) = 1limx→0−f(x) = a limx→0+f0(x) = b.Give conditions on the constants a, b that guarantee f(0) is a local
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