MATH 122, Exam #1 October 4, 2006
Please show all of your work on each of these problems. This is extremely important both for receiving partial credit on a question and for receiving full credit on problems you answer correctly. You may not use a calculator on this exam.
1. Simplify the following quantities or state why they do not exist.
a) b) c)
2. Assume you have a calculator that has buttons for computing and but does not have an inverse secant button. Think hard about what we mean by inverse functions in general, and then give a careful set of directions for how to use this calculator to compute something like .
3. Compute the following limits:
a) b) c)
4. Find the indicated particular solution for each of the differential equations:
a) where .
b) where
5. Below is a sketch of the graph of .
a) Find the area of the shaded region.
b) Set up an integral, but don’t evaluate it, that represents the length of the indicated path (the bold part of the graph) in the diagram.
6. A portion of the line corresponding
to the interval is rotated around the y-axis
to produce the cone. (See my rough sketch.)
a) Find the volume of the cone.
b) Now assume that this cone is actually a
tank that is full of water and we
want to pump the water up to a height of y=10 meters.
Set upan integral that represents the work required. You are
NOT required to evaluate this integral.) Recall that
Work = (force)(distance)=(volume)(distance)
7. Three epidemiologists are consulted about the way a rare disease ("calculitis") is spreading through a population. Use P(t) to denote the number of people infected with the illness at time t.
opinion A: The rate the disease is spreading is constant..
opinion B: The rate the disease is spreading is proportional to the number of people infected.
opinion C: The rate the disease is spreading is proportional to the square root of the number of
people infected.
a) Write down three differential equations that model each of these three opinions and find the general solution in each case.
b) Assuming that the initial infected population is 25 (hmmmm, the number of people in our class), find the particular solution in each case. (Without another data point, you will not be able to find a value for the proportionality constant , so just leave this as an unknown.) On one set of axes, sketch a qualitative picture of what each of the functions looks like for
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