Exam 2 - Graphing and Optimization with the Derivative

On this exam you may use your calculator to check yourself, to find derivatives, and to solve equations. But you must show work, especially with the limits in (1).

(1) Find the following limits:

(a)

$$\lim_{x \to \infty} \frac{3x^2-7x+6}{x^2-2x+1}$$

(b)

$$\lim_{x \to \infty} \frac{\cos x}{x^2}$$

(c)

$$\lim_{x \to -\infty} \frac{x}{\sqrt{4x^2+7}}$$

(2) Find the maximum and minimum values of the following functions on their given intervals:

(a) f(x) = x³ + 12x² + 45x + 3 on [-6,0]

(b) $h(t) = \frac{t}{t-3}$ on [-1,1]

(c) $g(x) = \csc x$ on $[\frac{\pi}{6},\frac{\pi}{3}]$

(3) Apply the Mean Value Theorem to the following function on the given interval. State why you can use the theorem (what necessary hypotheses are met) and then give the value(s) of c in the interval with

$$f' (c) = \frac{f(b)-f(a)}{b-a}.$$

$$f(x) = x^3 \ \ [-1,1]$$

(4) Let f(x) = x³+6x²+9x

(a) Find the x- and y-intercepts of f(x).

(b) Where is the graph of y=f(x) increasing/decreasing?

(c) Find all the critical points of f(x).

(d) Find the relative extrema of f(x).

(e) Where is the graph of y=f(x) concave up/concave down?

(f) What are the inflection points of f(x)?

(g) Graph the function, being sure to label all the information from (a) through (f).

(5) Same as (4) except $f(x) = \frac{x^2+4}{x^2-4}$ and you should also find all the asymptotes for the graph.

(6) Same as (5) except $f(x) = \cos (2x)$.

(7) A cistern with a square base is to hold 16,000 square feet of water. The metal top costs three times as much per square foot as the concrete sides and base. What are the dimensions of the most economical cistern? (That is find the dimensions that minimize the cost function)

(8) Find the numbers so that the sum of the first and three times the second is 18, and their product is a maximum.