midterm1-practice - x y z that lie inside of the sphere of...

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MATH 31B SECTION 2 FIRST PRACTICE MIDTERM Please note: Show your work. Correct answers not accompanied by sufFcent explanations will receive little or no credit. Please call one of the proctors if you have any questions about a problem. No calculators, computers, PDAs, cell phones, or other devices will be permitted. If you have a question about the grading or believe that a problem has been graded incorrectly, you must bring it to the attention of the professor within 2 weeks of the exam. Problem 1. Let S be the cone obtained by rotating the line triangle bounded by the lines y x , x 1, y 0 about the x -axis. Set up and then evaluate the integral expressing the volume of S using (a) the method of cross-sections and (b) the method of cylindrical shells. Problem 2. Let S be the region consisting of those points
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Unformatted text preview: x y z that lie inside of the sphere of radius 1, and which satisfy 0 x 3. ±ind the volume of S . Problem 3. Give an example of a function f x for which the trapezoidal rule approximation with n 2 to the integral 1 f x dx is exact (i.e., the error is zero), while the trapezoidal rule approximation to the same integral with n 4 gives a nonzero error. (No formula is necessary; a graph of the function, with explanations, will sufFce). Problem 4. Compute the integral e x e x dx . Problem 5. Let f x x 2 1, x 0. (a) Show that f is one-to-one on ∞ ; (b) compute the inverse of f x ; (c) Fnd the derivative of the inverse of f x at the point 2. Problem 6. Graph the function f x e x e x . 1...
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