P6 - f ( a ) 6 = 0 cannot be dropped in part (a) by giving...

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Math 141, Problem Set #6 (due in class Weds., 10/26/11) Note: To get full credit for a non-routine problem, it is not enough to give the right answer; you must explain your reasoning. Stewart, section 2.4, problems 16, 34, 40, 44, 46, 55. Stewart, section 2.5, problems 30, 42, 52, 54, 58, 60, 64, 66, 67, 70. Stewart, section 2.6, problems 14, 16, 18, 20, 24, 34, 38, 42, 43 (hint: differ- entiate the differential equation), 44. Also: A. Derive the Quotient Rule from the Product Rule, the basic Power Rule (the version on page 97, not the one on page 116!), and the Chain Rule. (Hint: f ( x ) /g ( x ) can be written as f ( x ) times 1 /g ( x ), and 1 /g ( x ) can be written as h ( g ( x )) where h ( x ) = 1 /x = x - 1 .) B. (a) Let F ( x ) = | f ( x ) | where the function f ( x ) is differentiable at a . Assume f ( a ) 6 = 0. Show that F ( x ) is differentiable at a , and express F 0 ( a ) in terms of f ( a ) and f 0 ( a ). (Hint: You can do this with the Chain Rule.) (b) Show that the assumption
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Unformatted text preview: f ( a ) 6 = 0 cannot be dropped in part (a) by giving an example of a dierentiable function f ( x ) such that f ( a ) = 0 and the function F ( x ) = | f ( x ) | is NOT dierentiable at x = a . (c) Does there exist a dierentiable function f ( x ) such that f ( a ) = 0 and the function F ( x ) = | f ( x ) | IS dierentiable at x = a ? C. Sketch the curve y 2 = x 2-x 4 using implicit dierentiation to identify all horizontal and vertical tangent-lines. At what angle does the curve cross itself at the origin? 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). 1...
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