Mathematics 133 – Intensive Calculus for Science 1 Sample Exam II – October 26, 2001

I. Give a formula for a rational function r(x) = \frac{ax^{2}+b}{x^{2}+dx+e} which has vertical asymptotes at x = 1, 3, x-axis intercepts at x = −2 and x = 2, and which has y = 6 as a horizontal asymptote.

II. The following graph shows y = f(x). Using the information here, construct a “qualitative” plot of y = f′(x).

III. Using the limit definition, find f′(x) for f(x) = x^{2} − x + 4.

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

A) (10) f(x) = 3^{x} + x^{6} − \frac{3}{\sqrt[3]{x}}

B) (10) g(x) = x^{e^{2}} + (e^{x})^{2}

C) (10) h(x) = \frac{4x^{2}−3}{x^{5}}

V. The price of an antique rocking chair at time t years after 1975 is approximated by P (t) = 100(1.25)^{t}. What was the price in 1990, and how fast was the price changing then? Was it increasing or decreasing?

VI. It costs C(q) = 50q^{1/4}+1000 dollars for an independent record company to manufacture q copies of its star alternative band’s latest CD. Compute C′′(q). Is the graph of the cost function concave up or down? What does your answer mean in everyday terms?