Mathematics 133 – Intensive Calculus for Science 1 Exam II – November 4, 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) = x^{2} − x + 4.

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

A) (10)

f(x) = 2^{x}(x^{4} + \frac{3}{\sqrt[3]{x}})

B) (10)

g(x) = (e^{πx} + 2)^{3}

C) (10)

h(x) = \frac{x^{2} − 3}{2x^{2} + 3x}

IV. (15) The price of an antique rocking chair at time t years after 1950 is approximated by P (t) = 100(1.05)^{t}. What was the price in 2000, and how fast was the price changing then? Was it increasing or decreasing?

V. (15) It costs C(q) = 50q^{1/4} + 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?