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(4 points) The velocity of an ant running along the edge ofa shelf is modeled by the function (t) 5t , OSt<1 v : 6d}, 1932 where t is in seconds and v is in centimeters per second. Estimate the time at which the ant is 4 cm from its starting position. (16 p oints) Calculate the inde?nite integrals listed below a. J- 3x?9 ?dx sz ?6x+1 I3?tan6 2 d6 cos 6 (2?x+x2)2 C. IT dx d. Icos2(3x) dx (4 points) Use the Mean Value Theorem to show that for any real numbers a, b Icosa ?cosb| S Ia ?b| (4 points) Let f(x)3x3 +54. a. Find an interval where the function f has one root. b. Use Rolle?s theorem to show that the function f has exactly one root. Hint. See Example 2 on page 283 of the textbook. (4 points) Use the identity cos2 x +sin2 x 1 to integrate J-cos3x sin2 x dx.

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