Question

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

$$ f(x)=\frac{-x^{3}-x+5}{2 x^{3}+3 x^{2}-7} $$

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 4.3.) 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.

Please show all steps and complete all subquestions             

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     

Answer 1

Answer 2

Answer 3