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Form A

DUE DATE:

MATH $171-$ CALCULUS I PROJECT $3$

Note: Show all supporting work. Answer every part of each question.

Use methods of algebra and calculus to analyze the graph of the function

$f(x)=\frac{-x^3-x5}{2x^{3}3x^2-7}$

Include all the following in your analysis:

a$)$ the domain

b$)$ intercepts

c$)$ equations of asymptotes $($both vertical and horizontal$)$

d$)$ relative extrema $($be sure to provide all derivatives, identify critical numbers, and show the test for those values $-$ naming the test you are using.$)$

e$)$ intervals where the function is increasing or decreasing

f$)$ inflection points $($be sure to identify possible inflection points and be sure to show the test for those values.$)$

g$)$ intervals where the graph is concave up and where it is concave down $($be sure to include the vertical asymptotes in your intervals.$)$

h$)$ provide computer$-$generated graphs of your function that illustrate all of the key points that you have listed. You may need to provide more than one graph.

Note: You may use Maple to graph the function, determine the domain and intercepts, compute derivatives, and solve equations $($follow the directions in the Classroom Supplement, Section $4\mathrm{.}3.)$ However, do not submit pages of printed results from Maple. You must present a logical argument for all your conclusions and present your work so that your methods are clear.

$2.$ An industrial tank is formed by adjoining two hemispheres to the ends of a right circular cylinder. The tank must have a volume of $4000$ cubic feet. The material for the lateral surface costs $$5$ per square foot, and the material for the hemispherical ends costs $$10$ per square foot. Find the dimensions of the tank that will minimize cost$\mathrm{.}10$:$418$

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$2.$ An industrial tank is formed by adjoining two hemispheres to the ends of a right circular cylinder. The tank must have a volume of $4000$ cubic feet. The material for the lateral surface costs $$5$ per square foot, and the material for the hemispherical ends costs $$10$ per square foot. Find the dimensions of the tank that will minimize cost.

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