Mathematics 133 – Intensive Calculus for Science 1 Makeup Exam II – November 14, 2005

Directions: Do all work in the blue exam booklet. You may use the supplied calculator on any of these problems.

I. The following graph shows y = f(x).

A) (5) Identify all points where f appears not to be differentiable. B) (15) Using the information in the graph, construct a “qualitative” plot of y = f′(x)

for x ≠ your points from part A.

II. (20) Using the limit definition, find f′(x) for f(x) = 3x^{2} + 2x − 1.

III. Find the derivatives of each of the following functions by applying the appropriate “shortcut” derivative rules:

A) (10)

f(x) = e^{x}(x^{1.3} − \frac{4}{\sqrt{x}})

B) (10)

g(x) = (3^{2x} + 2)^{3}

C) (10)

h(x) = \frac{x + 1}{3x^{4} − 1}

IV. (15) The value of a 1995 model car at time t years after it was made is approximated by P (t) = 20(.85)^{t} (in units of $1000). What was the value in 2000, and how fast was the price changing then? Was it increasing or decreasing?

V. (15) It costs C(q) = 23q^{2/3} + 1000 dollars for an publishing company to manufacture q copies of its star author’s latest novel. Compute C′′(q). Is the graph of the cost function concave up or down? What does your answer mean in everyday terms?