College of the Holy Cross, Fall Semester, 2004 Math 131, Midterm 1 (All Sections) Wednesday, September 22, 6 PM

Name:

Instructions Please write your answers in the spaces provided, and show work on the test itself. Use the back of the preceding page if you need more space for scratch work.

Please do not write in the space below

Problem Points/Poss
1  / 15
2  / 20
3  / 20
4  / 10
5  / 20
6  / 15
Total  /100

1. The graph y = f(x) and four graphs obtained by transforming it are shown. Match the given formulas with the corresponding graph. Note that there is an extra graph.

−1  ^{0}  ^{1}  ^{2}  ^{3}_{−1}^{1}  y = f(x)
(a) y = f(\frac{1}{2}x):  (b) y = \frac{1}{2}f(x):  (c) y = \frac{1}{2} − f(x):

−1  ^{0}  ^{1}  ^{2}  ^{3}_{−1}^{1}  (i)  ^{−1}  ^{0}  ^{1}  ^{2}  ^{3}_{−1}^{1}  (ii)

−1  ^{0}  ^{1}  ^{2}  ^{3}_{−1}^{1}  (iii)  ^{−2}  ^{0}  ^{2}  ^{4}  ^{6}_{−1}^{1}  (iv)

2. [20 points] The desert temperature H varies sinusoidally from a high of 80^{∘}F at 5 PM to a low of 40^{∘}F at 5 AM. Find a formula for H as a function of t, with t measured in hours from 5 PM. You may use the graph below for reference; it’s a good idea to start by labeling the vertical and horizontal scales.

5 PM  5 AM

Math 131, Midterm 1 4

3. [10 points each] In each part, fill in the table as indicated. Give exact answers: Use fractions, square roots, logarithms, etc. as needed.

(a) Assuming f is a linear function.

x −1 2 5 10
f(x) 24 12

(b) Assuming f is an exponential function.

x −1 2 5 10
f(x) 24 12

4. [10 points] Find the polynomial whose graph is shown; expand (multiply out) your answer.

−2  −1  0  1  2  3  4 6 12 18  y = p(x)

5. [10 points each] Let y = f(x) = e^{x} − 1.

(a) Find the inverse function; express your answer in the form y = f^{−1}(x).

(b) Find the domain of f^{−1}.

6. Alice buys an automobile for $25,000 that depreciates in value by 20% per year. Bob buys an automobile for $18,000 that depreciates in value by 10% per year.

(a) [5 points] Find formulas for A(t) and B(t), the values of the cars after t years.

(b) [10 points] After how many years are their cars worth the same amount? Give an exact answer (using logarithms) and round your answer to two decimal places.