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Exam 4 Math 115 Fall 2011 Name: ___________________ This is a closed book exam. You may use a calculator and the formulas handed out with the exam. You may find that your calculator can do some of the problems. If this is so, you still need to show how to do the problem by hand. In other words, show all work and explain any reasoning which is not clear from the computations. (This is particularly important if I am to be able to give part credit.) Turn in this exam along with your answers. However, don't write your answers on the exam itself; leave them on the pages with your work. Also turn in the formulas; put them on the formula pile. 1. (20 points) A cylindrical can with circular base is to be made out of 10 square feet of material. Thus the combined surface area of the bottom, top and sides is to be 10 ft 2. Find the dimensions that give the largest volume of the can. 2. (15 points) Water is leaking out of a container at a rate of ( t - sin t) gallons per hour where t is time in hours since you start your observations. Suppose at time t = 0 there is l0 gallons of water in the container. How much water is left in the container after 3 hours?. 3. a. (7 points) Write the following sum using sigma notation. 2 4 8 16 512 1024 + + + + + + 20 18 16 14 4 2 5 b. (8 points) Find the value of the sum j [ sin(j) ] . I would like an actual number for your j=3 answer. Assume sin(j) means sine of j radians. 4 4. (20 points) Find 2 (x – x ) dx using the definition of the integral as a limit of approximating sums. The 2 formula sheet should be helpful. NOTE: You can not do this by finding an antiderivative. π/3 5. (15 points) Find ( sec2(x) - 1 ) dx. Use the fundamental theorem of calculus. You shouldn't use the x /4 method of problem 4. x 2 sin(t ) dt 6. (15 points) Let f(x) = 1 1 - x4 x df 2 . Find dx . Note: Your answer will involve sin(t ) dt. 1