As it is always the case that correct answers without sufficient mathematical justification may not receive full credit, make sure that you show all your work Please circle, draw a box around, highlight, or otherwise clearly indicate your final answer for each question. By signing your name above, you attest to the fact that the work you are presenting is wholly your own. This work is due in class as assigned by the instructor and noted in the course syllabus.
1. Locate the absolute extrema of the function f (x) = 2x/(x2+1) on the closed interval [-2, 2].
2. The surface area of a cell in a honeycomb is given by
S = 6hs + 3s2/2 - ((√3 - cosθ)/sinθ)
where h and s are positive constants and θ is the angle at which the upper faces meet the altitude of the cell. Find the angle θ(Π/6 ≤ θ ≤ Π/2) the minimizes the surface area S.
3. Determine whether Rolle's Theorem can be applied to f(x) = (x - 3)(x + 1)2 on the closed interval [-1,3]. If Rolle Theorem can be applied, find all values of c in the open interval (-1.3) such that f '(c) = 0. If Rolle's Theorem cannot be applied. explain why not.
4. Let f(x) = x3 - 6x2+ 15
a) Find the critical numbers of f (if any).
b) Find the open interval(s) on which the function is increasing or decreasing.
c) Apply the First Derivative Test to identify any relative extrema.
d) Use a graphing utility to confirm your results. (You need not graph the function )
5. The graph of f is shown in the figure to the right. Sketch a possible graph of the derivative of f.
