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Unformatted text preview: Name: TA: Math 20A Midterm Exam 2 V1 November 19, 2009 Sec. No: PID: Sec. Time: Turn oﬀ and put away your cell phone. No calculators or any other electronic devices are allowed during this exam. You may use one page of notes, but no books or other assistance during this exam. Read each question carefully, and answer each question completely. Show all of your work; no credit will be given for unsupported answers. Write your solutions clearly and legibly; no credit will be given for illegible solutions. If any question is not clear, ask for clariﬁcation. # 1 2 3 4 5 Σ Points Score 6 8 6 6 6 32 1. (6 points) Diﬀerentiate the following functions; you need not simplify. (a) f (x) = x 2 + sin(x) (b) g (x) = e3x cos(4x) (c) h(x) = [cos (x)]x 2. (8 points) Let f (x) = x + 1 . The ﬁrst and second derivatives of f are given by: x−3 1 (x − 3)2 and f ′′ (x) = 2 . (x − 3)3 f ′ (x) = 1 − (a) Find the interval(s) on which f is increasing and the interval(s) on which f is decreasing. (b) Find the local maximum and local minimum values of f . (c) Find the interval(s) on which the graph of f is concave upward and the interval(s) on which the graph of f is concave downward. (d) Find the vertical asymptote(s), if any. 3. (6 points) Use the fact that 27 3 = 3 to ﬁnd a linear approximation for (27.03) 3 . 1 1 4. (6 points) Randall Cohn has a pool with the shape of an inverted cone which is 5 meters deep with a radius of 5 meters at the top (base). Randall ﬁlls the pool with his garden hose at a rate of 0.1 cubic meters per minute. At what rate is the water depth increasing when the depth is 3 meters? (Note: The volume of a cone of height h and 1 radius r is given by V = πr 2 h.) 3
r=5 d=5 5. (6 points) Find the point(s) on the ellipse x2 + xy + y 2 = 12 at which the corresponding tangent line is horizontal. ...
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 Fall '08
 Eggers
 Math, Derivative, Mathematical analysis, Convex function, 20A Midterm Exam

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