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Math 131 Fall 2015 Test 3B Name: On my honor, as an Aggie, I have neither given nor received unauthorized aid on this academic work. Signature: You may use your calculator for all problems on the test but be sure to read each problem carefully. Some problems ask you to show some algebraic steps. 1. (2 points) .25 ∗ (f (1) + f (1.25) + f (1.5) + f (1.75)) approximates endpoints. True 2. 3. R2 −1 f (t)dt f (x)dx using right = 9 and F (2) = 12, where F 0 (x) = f (x), then F (−1) = −3 False (2 points) If f (x) = True 1 False (2 points) If True R2 3−x x2 −9 on the interval [0, 4], then x = 3 is a critical number. False 4. (6 points) Above is the graph of f (x).R If the area of region A is 10.3, region B is 11.5, 7 region C is 8.7, and region D is 2.3, what is −3 f (x)dx? (a) 15.4 (b) 13.1 (c) 32.8 (d) 30.5 5. (8 points) Find the absolute maximum and minimum values of f (x) = 27x − x3 on [0, 5] 6. (10 points) Find the antiderivative, F (x), of f (x) = x(3 − x)2 if F (1) = 3. 7. (10 points) Water flows from a full tank that holds 5000 liters at a rate of r(t) = 200 − 4t liters per minute, where 0 ≤ t ≤ 50. Find the amount of water left in the tank after 10 minutes. 8. (10 points) You have a rectangular piece of cardboard with length 1ft and width 4ft. You want to create the (open top) box with the largest volume by cutting squares out of the corner and folding the sides up. What are the dimensions of the box you create (rounded to 3 decimal places) Rx (8 points) Above is the graph of f (t). If g(x) = 0 f (t)dt, then (a) At what values of x do the local maximum and minimum values of g occur? (b) Where does g attain its absolute maximum value? (c) On what intervals is g concave downward? (d) Sketch a graph of g 9. R4 √ 10. (7 points) Evaluate: 11. (10 points) Find the derivative of g(x) if g(x) = 12. (10 points) Evaluate the indefinite integral, −4 16 − x2 dx (give an exact answer, no decimals) R R5 t x2 cos(t) dt (ln(x))2 dx x 2 8(x+1) x −4x 0 00 (15 points) Given that f (x) = (x−2) 2 , f (x) = (x−2)3 , and f (x) = (x−2)4 : (a) Find all critical numbers of f (x). (b) Find all local extrema (give the point(s) and state whether it’s a local max or a local min). (c) Give the intervals on which f is increasing/decreasing and concave up/concave down. (d) Find all asymptotes. (e) Use the information above to sketch a graph of f . (Hint: It is possible to cross horizontal asymptotes) 13. Scratch Paper