P24 - (There are four possibilities each would be the left...

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Project 24 MAC 2311 YOU MUST SHOW YOUR WORK TO RECEIVE FULL CREDIT!! 1. Note that the information learned in this lecture should prepare you for curve sketching!! Here’s a function you probably can’t visualize: f ( x ) = 3 x (7 - x 2 ) 2 3 . Verify that f 0 ( x ) = 7(3 - x 2 ) (7 - x 2 ) 1 3 and f 00 ( x ) = - 28 x (9 - x 2 ) 3(7 - x 2 ) 4 3 . Use Product Rule, Chain Rule, and factoring of negative exponents. On what intervals is f ( x ) increasing? decreasing? (Include a number line.) What are the critical numbers for f ( x )? Determine the local extrema using the ﬁrst derivative test. For each one that you ﬁnd, label it as a cusp or a horizontal tangent and draw its shape. On what intervals is f ( x ) concave up? concave down? (Include a number line.) What are the inﬂection points for f ( x )?

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On what intervals is the function f ( x ): both increasing AND concave up both increasing AND concave down both decreasing AND concave up both decreasing AND concave down Next to each interval above, draw the basic shape of the curve on the interval.
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Unformatted text preview: (There are four possibilities: each would be the left or right half of the shapes ∪ and ∩ .) Now try to piece your shapes together to make a rough free-hand sketch of the function f ( x ) below. Label your cusps, horizontal tangents, and inﬂec-tion points. As one last experiment, use the second derivative test to verify the local extrema that you found. Do you have any critical numbers at which second derivative test cannot be used? 2. Suppose a particle moves in a straight line so that its position in feet at time t seconds is given by s ( t ) = t 2 e-t . How would you ﬁnd where the particle has its greatest speed (absolute value of velocity)? Find the greatest speed on [0 , 2]; on [2 , 4]. To what special type of point on the graph of s ( t ) do these correspond? Explain. .....
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P24 - (There are four possibilities each would be the left...

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