mt02fall04

mt02fall04 - MIDTERM 2 MATH 19 A 11/19/2004 Instructor:...

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Unformatted text preview: MIDTERM 2 MATH 19 A 11/19/2004 Instructor: Frank Béiuerle, Ph.D. Your Name: _____________________________ __ Your TA: _____________________________ __ Max Your score Problem 1: 20 Problem 2: 10 Problem 3: 10 Problem 4: 10 Problem 5: 10 Problem 6: 20 Problem 7: 20 TOTAL: 100 Good Luck! 1. (20 points) Compute the derivatives 3:1 of the following functions: (a) y = arctanfi/E) (c) y = ln(r + sin I) + tan(ta,n ac) + 6”“ (d) y 2 mm”, where a: > 0. 2. (10 points) Find the equation of the tangent line to the curve e‘flzxy at the point (e, l). 3. (10 points) Show that tans: m a: for a: VERY cloee to 0. 4. (10 points) Compute the derivatives f’(x), f”(:r) and of the function f(:r)_= sinh 21" 5. (10 points) (a) State the Extreme Value Theorem. 1 (b) The function f(3:) = - does not have an absolute max nor an absolute min on [~l,1]. I Why does this not contradict the extreme value theorem? (c) Suppose we know that f”(:c) < D for 1 S I S 3, f’[‘2) : D, and = 3. Give a rough sketch of of the graph of f locally around a: z 2. 6. (20 points) After heavy rainfall it is observed that the depth of water in a conical reservoir of radius 10 meters and height 30 meters is increasing at 5 meters/ hour when the depth is 5 meters. How fast is the water filling the reservoir at this instant? Recall that the volume 1 . of a cone is given by V = Ewrzh. Draw a picture and show all your work. 7. (20 points) Let f(z) = 3:55 - 53:3 + 1. Find the following: (a) Find the critical numbers of f. (b) Classify the critical numbers of f (i.e. determine whether they correspond to a local min, local max or neither.) (c) Give the interval(s) where f is increasing. (d) Give the interval(s) Where f is decreasing. (e) Give the absolute max and absolute min of f on the interval [—1, 2] ...
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mt02fall04 - MIDTERM 2 MATH 19 A 11/19/2004 Instructor:...

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