Mathematics 133 – Intensive Calculus for Science 1 Makeup Exam I – October 4, 2005

Directions: Do all work in the blue exam booklet. There are 100 possible points, distributed as indicated before the questions – (10) means the question is worth 10 points for instance. No calculators on this exam.

I. The following table contains values for three different functions: f(x), g(x), h(x).

\begin{matrix}x & 0 & 1 & 2 & 3 & 4 \\ f(x) & 0 & 1.5 & 3.0 & 4.5 & 6.0 \\ g(x) & 3 & 12 & 24 & 48 & 96 \\ h(x) & 3.7 & 3.4 & 3.1 & 2.8 & 2.5 \end{matrix}

A) (10) Which (if any) of these are linear functions? For those that are linear give

formula(s). B) (10) Which (if any) of these are exponential functions? For those that are exponential,

give a formula.

II. (20) This year’s Nobel Prize in Medicine went to two Australian researchers who showed that bacteria called Helicobacter pylori actually cause most stomach ulcers in humans (not excess stomach acid as most people thought before). A colony of Helicobacter pylori in a certain math prof’s stomach is growing at a continuous rate of 0.1% per month. How long will it take for the colony to double in size? (When it doubles, he gets an ulcer.)

III. The back of this sheet shows the graph of a polynomial function f(x). A) (15) Find a possible formula for f(x). B) (10) Does f(x) have an inverse function? Why or why not?

IV. (20) The average depth of the water in a bay is 20ft. Because of tides, the depth varies sinusoidally with amplitude 5ft. The period is 12.5 hours. Give a possible formula for the depth as a function of t (in hours). Say what the height is at t = 0 using your formula.

V. All parts of this question refer to the rational function

r(x) = \frac{x^{2} − 6x + 8}{3x^{2} − 3}.

A) (5) Where does the graph of r(x) have vertical asymptotes? B) (5) Where does the graph of r(x) cross the x-axis? C) (5) Where does the graph have a horizontal asymptote?