- Use methods of algebra and calculus to analyze the graph of the function
.
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 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.
- An architect has been asked to design a Norman window for a small office space, to add interest and light. A Norman Window is constructed by adjoining a semicircle to the top of an ordinary rectangle (see below). This window will have special decorative trim purchased by linear foot, so it will pay to get the most light for the least perimeter of window. If 10 feet of trim is allotted for the Norman window (including the segment between the semicircle and the rectangle), what dimensions will maximize the area?
Sample Solution