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Unformatted text preview: MATH 152 FALL 2009 CALCULUS II FOR THE MATHEMATICAL AND PHYSICAL SCIENCES PRACTICE FINAL EXAM: HINTS AND SOLUTIONS Note that your final will contain questions that are DIFFERENT from the ones below. This is simply my best guess what your exam might look like. You will benefit most from it by attempt- ing to take it as a practice exam AFTER you have reviewed all the topics in this course, under exam conditions (that is, during an uninterrupted time interval of 3 hours, without using your textbook or notes, without a calculator, but with the departments formula sheet). Here are some hints and final answers (note that these are *not* fully worked solutions). Question 1. Determine whether n = 2 1 n ( log n ) 2 converges or diverges. Answer 1. It converges by the integral test. Set f ( x ) = 1 x ln ( x ) 2 , and consider lim R R R 2 1 x ln ( x ) 2 dx. (We should check that f is positive, decreasing and continuous.) This integral requires u-substitution, so set u = ln x, du = 1 x dx and compute. Question 2. Find the volume of liquid needed to fill a sphere of radius R = 10 m to height h = 5 m. Answer 2. Use the method of disks. The radius r at any height y is given by r = p R 2- ( R- y ) 2 . Thus, the volume of the filled portion of the sphere is Z h r 2 dy = Z h ( R 2- ( R- y ) 2 ) dy = ( Rh 2- h 3 3 ) . Question 3. Determine the values of x for which n = x n n 4 + 2 converges....
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- Fall '08