# P13 - at the graphs of y = e x and y = x 2 on the same axes...

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Project 13 MAC 2311 1. Examine the function f ( x ) = x 4 + x x 3 . Calculate f 0 ( x ) using quotient rule, expressing the answer as a sum of powers of x (and constants). Calculate f 0 ( x ) once again, but this time simplify ﬁrst, by dividing each term of the numerator by x 3 . Which of the two methods above is more eﬃcient? What horizontal tangent lines does f ( x ) have? Note that f 0 ( x ) has an inﬁnite limit as x approaches zero. Does this alone mean it has a vertical tangent line at x = 0? Explain. 2. Calculate the derivative of the function g ( x ) = xe x x + e x using a combination of product and quotient rules. Does g ( x ) have horizontal tangent lines? Justify your response. 1

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Now calculate the derivative of the function k ( x ) = xe x x - e x using a combi- nation of product and quotient rules. Does k ( x ) have horizontal tangent lines? Justify your response (Hint: look
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Unformatted text preview: at the graphs of y = e x and y = x 2 on the same axes or use Intermediate Value Theorem). By comparing the graphs of y = e x , y =-x , and y = x on the same axes (using the idea of the hint above), decide whether either of the functions g ( x ) or k ( x ) has vertical asymptotes. 3. Examine functions of the form h ( x ) = e ax for diﬀerent numbers a . Calculate the derivatives of h 1 ( x ) = e 2 x and h 2 ( x ) = e-x by using the fact that e 2 x = e x · e x and e-x = 1 e x . Do these agree with the general formula d dx e ax = ae ax ? Use the formula above and the simpliﬁcation ideas of this section to calculate the derivative of the function f ( x ) = ( e x + e-x ) 2 e 2 x . 2...
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P13 - at the graphs of y = e x and y = x 2 on the same axes...

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