P23Ans - both increasing AND concave down • both...

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Project 22/23 MAC 2233 1. Examine the function h ( x ) = 27 x 2 x ( x + 1) 3 . What is its domain ? x 6 = 0 , 1 Determine the horizontal asymptotes and vertical asymptotes, if any, for the graph of h ( x ) . Does the function have any removable discontinuities? hole at x = 0 ... graph 27 x ( x + 1) 3 and remember to remove the hole asymptotes: x = - 1 , y = 0 What are the intercepts of the graph of h ( x ) ? none (once hole is removed) Calculate h 0 ( x ) , and use a number line to determine the interval(s) on which the function is increasing/decreasing. What are the local extrema? Are they cusps or horizontal tangent lines? inc ( -∞ , - 1) , ( - 1 , 0) , (0 , 1 2 ) (if hole at x = 0 considered) dec ( 1 2 , ) local max at x = 1 2 Calculate h 00 ( x ) . Use a number line to determine the interval(s) on which the function is concave up/down. What are the points of inflection? inc ( -∞ , - 1) , (1 , ) dec ( - 1 , 0) , (0 , 1) (if hole at x = 0 considered) Separate a number line into the intervals on which h ( x ) is: both increasing AND concave up
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Unformatted text preview: both increasing AND concave down • both decreasing AND concave up • both decreasing AND concave down and draw the basic shape of the curve on each interval. Now piece your shapes together to draw the function h ( x ) below. Label all intercepts, local extrema, inflection points, removable discontinuities, and asymptotes. 2. Find the absolute maximum and minimum value of the function f ( x ) = 3( x 2-2 x ) 1 3 on the interval [0 , 4] absmax is 6 at x = 4 absmin is 0 at both x = 0 and x = 2 Try to sketch the function on [0 , 4] . Use intercepts and a number line for the first derivative, to decide where the function is increasing/decreasing and also the special features at the critical points (vertical tangent, cusp, etc.). inc (1 , 2) , (2 , 4) dec (0 , 1) VTL at x = 0 intercept at x = 0 , 2...
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P23Ans - both increasing AND concave down • both...

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