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# Exam 3 - and is a Lower Estimate Leave your answer as an...

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MATH 2010 EXAM 3, April 15, 2011. Write your solutions in a bluebook. 1. (10 points) Find the absolute maximum and minimum if they exist, of the function f ( x ) = x 2 e - 3 x over the interval (0 , ). Show the appropriate table with limit and critical point information. 2. (10 points) Evaluate the following limit: lim t 0 1 - cos(4 t ) t 2 . 3. (10 points) I want to find two positive numbers x and y whose product is 200 with the smallest possible sum. I will have to find the minimum of a certain function f ( x ) where x is in a certain interval. Find f ( x ) and the interval. Do nothing else. 4. (10 points) Find f ( x ) if f 0 ( x ) = x 1 3 + e 8 x + cos(5 x ) and f (0) = 4. 5. (10 points) A car is decelerating at a constant rate of - 2 fps per second. If its initial velocity was 40 fps, how many feet does it travel before it stops? 6. (10 points) Water pours into a large container at the rate of 2 - t cubic feet per minute after t minutes. How much water is added between t = 1 and t = 4 minutes? 7. (10 points) Approximate R 3 1 e x 2 dx by a Riemann sum which uses 4 equal subintervals
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Unformatted text preview: and is a Lower Estimate. Leave your answer as an unsimpliﬁed sum of numbers. Note that f ( x ) = e x 2 is increasing on [1,3]. 8. (10 points) Depicted in ﬁgure 8 is the velocity (in units of feet per second) of an object moving on a line. The numbers inside the bumps indicate areas. Suppose that when t = 0 its position is s = 2 a. What is its position when t = 10? b. What is the total distance traveled by the object between times t = 0 and t = 10? 9. (15 points) Depicted in ﬁgure 9 is the graph of a function f . Let g ( x ) = R x 1 f ( t ) dt . a) Evaluate g (1), g (5) and g (10). b). Find a sign chart for g . c. Use the sign chart above to sketch the graph of g . Show the intervals of increase and decrease and local maximum and minimum. (Do not worry about concavity.) 10. (5 points) Find the exact value of the following integral. Z 4 1 ( t 2 + 3 t ) dt....
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