P12 - B. Find the general antiderivative of f ( x ) = | x |...

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Math 141, Problem Set #12 (due in class Mon., 12/5/11) Note: To get full credit for a problem, it is not enough to give the right answer; you must explain your reasoning. Stewart, section 4.5, problems 18, 26, 30, and 50. Hint: For problem 50, express θ as a difference of two angles, each of which can be expressed as the arctangent of some simple expression. Stewart, section 4.6, problems 4, 6, 10, 22. Stewart, section 4.7, problems 6, 10 (specify the domain!), 16, 20, 26, 28, 30, 32, 36, 40, 43, 46. Also, do the following additional problems. A. (a) Find the point on the graph of y = | x | that is closest to the point (2 , 4). (b) Find the point on the graph of y = | x | that is closest to the point (2 , - 4). (Hint: to minimize the distance, minimize the square of the distance.) For both (a) and (b), use the First Derivative Test to confirm that the relevant critical point is indeed a local minimum.
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Unformatted text preview: B. Find the general antiderivative of f ( x ) = | x | 3 . C. Does the greatest-integer function f ( x ) = [ x ] (where [ x ] is dened as the greatest integer n satisfying n x ) have an antiderivative? Explain. D. (a) Show that if f is odd, then every antiderivative of f is even. (Hint: We must show that if f ( x )+ f (-x ) = 0 for all x and F ( x ) = f ( x ) for all x , then F (-x ) = F ( x ) for all x .) (b) Show that if f is even, and f has an antiderivative, then f has exactly one antiderivative that is odd. Please dont forget to write down on your assignment who you worked on the assignment with (if nobody, then write I worked alone), and write down on your time-sheet how many minutes you spent on each problem (this doesnt need to be exact)....
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