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

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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 diﬀerentiable function f ( x ) such that f ( a ) = 0 and the function F ( x ) = | f ( x ) | is NOT diﬀerentiable at x = a . (c) Does there exist a diﬀerentiable function f ( x ) such that f ( a ) = 0 and the function F ( x ) = | f ( x ) | IS diﬀerentiable at x = a ? C. Sketch the curve y 2 = x 2-x 4 using implicit diﬀerentiation to identify all horizontal and vertical tangent-lines. At what angle does the curve cross itself at the origin? Please don’t 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 doesn’t need to be exact). 1...
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## This note was uploaded on 02/13/2012 for the course MATH 141 taught by Professor Staff during the Fall '11 term at UMass Lowell.

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