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# MAT190 Calculus I Name ________________ Exam 3 (Chapter 3) Spring 2012 INSTRUCTIONS: Show all your work on the pages provided. You may use the...

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MAT190 Calculus I Name ________________ Exam 3 (Chapter 3) Spring 2012 INSTRUCTIONS: Show all your work on the pages provided. You may use the graphing calculator for any problem that you feel it is appropriate. If you use the grapher on the calculator, please include a sketch of the required graphs. There are 9 questions on 6 pages. Make sure you have all of the pages. Good Luck !!!!! [ 10 pts.] Consider the function f(x) xtan(x) 2  1(a) Find the Linearization, L(x) , of the given function at x = . Show all steps! Round the coefficients to 3 decimal places. (b) Find the x-intercept of L(x) as a way of estimating the x-intercept of f(x). Round the coefficients to 3 decimal places. [10 pts.] 2. Use the derivative of the function f(x) xtan(x) 2  from problem #1 above, and use your TI to graph this derivative on the interval [-1.5, 1.5 ]. Accurately graph and label your derivative. Use the graph to find critical points. Label these points on your graph. Apply the extreme value theorem to find the absolute extrema of f(x) on the closed interval , [-1.5, 1.5].
[10 pts.] 3. (a) State the Mean Value Theorem IN YOUR OWN WORDS. What does it say in practical terms ? (b) Opie Timization is slowing down as he approaches a stop light. As he reaches the stop light, he has slowed to a speed of 4 ft/sec and the light turns green and Opie begins to accelerate at a rate of 27 ft/sec. for the first eleven seconds after the light turns green. Based on these facts, Opie's position function can be shown to be 2 s(t) 13.5t 4t  on the t interval [0,11]. Use the MEAN VALUE THEOREM to prove whether Opie should get a speeding ticket or not. Assume a 35 mph (51.3 ft/sec) speed limit. NOTE: You must specifically apply the THEOREM (and check the conditions) and make the appropriate conclusion as stated in the theorem. [10 pts.] 4. For some unknown function f(x), the first derivative of the function is given by f (x) 2cos(3x) 1  . Show ALL the algebra/trig needed to find all intervals (in interval notation) where f(x) is increasing and all intervals where f(x) is decreasing on [0,2 ) . Include a sign graph of + and - signs to support your answer.
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1a)
To find the linearization of the function at x=4
f ( x) = x tan x − 2, f (a) = 4 tan 229.18 − 2 = 2.631 f ' ( x) = tan x + x sec2 x , f ' ( 4) = tan 4 + 4 sec 2 4 = 10.52...

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