Stewart6e4_5 - intervals of increase and decrease ....

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MATH 191, Sections 1 and 3 Calculus I Fall 2007 Section 4.5: Sketchin’! With the help of calculus, we’ve now got several tools that’ll help us understand the nature of the graph of a function f ( x ). Let’s summarize our findings in an 8-step process which ends with the sketch of the graph. A. Find the domain of the function: this’ll tell you where to look for the graph in the first place! B. Find any intercepts the function may have: finding f (0) (if defined) tells you the y -intercept, if there is one; solving for f ( x ) = 0 finds any x -intercepts. C. Does the graph have any symmetry ? That is, is the function odd, or even, or does it have a period , some number n such that f ( x + n ) = f ( x ) for any x ? D. What about asymptotes ? You can find all horizontal and vertical asymp- totes by evaluating lim x →±∞ f ( x ) and by finding where the function grows without bound, respectively. E. Use the I/D Test to find
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Unformatted text preview: intervals of increase and decrease . Remember this means nding the derivative and then using this to nd critical points, where the sign of the derivative might change. F. Find local maxima and minima using either the First Derivative Test or the Second Derivative Test. G. Using the second derivative and the Concavity Test, nd the intervals of upward and downward concavity . Where are the inection points, if any? H. Finally, put it all together to get a sketch of the graph! Examples. Try out your newfound wisdom by sketching graphs of the following functions: f ( x ) = xe x , h ( t ) = 1 1+ e-t , and m ( ) = cos 2 ( )-2 sin( ). f ( x ) = e-( x-4) 2 . Homework. Do exercises 12, 29, 32, and 47 from Section 4.5 (pages 314-315) of your text. This is due on Friday, November 16th ....
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This note was uploaded on 04/08/2008 for the course MATH 191 taught by Professor Bahls during the Fall '07 term at UNC Asheville.

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