Graphs and GradientsDate: 05/13/2003 at 14:24:38 From: Kim Subject: Derivative as a function The problem asks us to sketch graphs of f and f' for f(x) = sin x and to guess the formula for f'(x). Could you explain how you would work this problem out? Thank you. Date: 05/14/2003 at 13:46:37 From: Doctor Dotty Subject: Re: Derivative as a function Hi Kim, Thanks for the question, The derivative of a function at a point is the gradient of the curve at that point. If f(x) = Cos x, then the graph of f(x) is: | ..1.. .' | '. .' | '. -3Pi/2 -Pi -Pi/2.' | '. Pi/2 Pi 3Pi/2 '.''''''''''''''''':''''''''|'''''''''.''''''''''''''''. '. .' | '. : '. .' | '. .' '.. ..' | '.. ..' '''' -1 '''' | So what is the graph of f'(x)? We look at the gradient. Well, at -3Pi/2, it is negative but increasing. It goes through zero at -Pi, but starts decreasing again at -Pi/2. It goes through zero at zero, but starts increasing again at Pi/2. It goes through zero at Pi, but starts decreasing again at 3Pi/2, and so forth. So it looks like this: | | ....... 1 ... .'' '. | .'' -3Pi/2 -Pi.' -Pi/2 ''.| Pi/2 Pi.' 3Pi/2 ''''''''.'''''''''''''''''':'''''''''''''''''.''''''''' .' |'.. .' ..' | '. ..' ''' -1 ''''''' | | | This is a reflection of the graph of y = sin x about the y-axis; so at a guess, I would say that f'(x) = -sin x. Can you sketch the derivative of sin x in a similar manner? Write back if I can be of any more help on this or anything else. - Doctor Dotty, The Math Forum http://mathforum.org/dr.math/ |
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