PRACTICE EXAM 2 FOR THIRD MIDTERM EXAM ANDREW J BLUMBERG 1 Notes This practice exam has more problems than the real exam will To make most effective use of this document take the exam under conditions simulating the real exam no book no calculator 1 Short answer questions a Suppose that f 0 x g 0 x What can we say about f and g Rb b Define the definite integral a f x dx c Explain what the second derivative tells us about maxima and minima d Riemann sums can be computed using the left endpoint the midpoint and the right endpoint Which is better Explain ex 2 Sketch the curve f x 1 e x by finding maxima minima concavity inflection points and so forth x 3 Define the function f x e x 1 for x 6 0 and f 0 1 Compute f 0 0 using L Hopital s rule Hint use the definition of the derivative 4 Find the derivative of Z x2 ln sin x dx x2 Hint split this up 5 Solve the following optimization problems a Find the area of the rectange of maximal area which has two vertices on the x axis and two vertices with x 0 and lying on the curve 9 x2 b I want to make a cylindrical barrel to hold 10 cubic meters of molasses Assume that the sides cost 36 per square meter and the circular ends cost 18 per square meter What should the dimensions of the cheapest barrel with the required volume be 6 Let f x x1 R5 a Approximate the definite integral 1 f x dx using a Riemann sum with 2 intervals and using the lefthandR side of the rectangle 5 b Approximate the definite integral 1 f x dx using a Riemann sum with 4 intervals and using the lefthand side of the interval c Compute the definite integral using the fundamental theorem of calculus 7 Compute the following definite integrals R 100 3 a 0 8x2 ex 5 dx R1 b 1 ex sin ex dx Date November 7 2011 1 2 ANDREW J BLUMBERG E mail address blumberg math utexas edu
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