College of the Holy Cross, Fall 2007 Math 131, Midterm 1 (All Sections) Wednesday, September 19, 6 PM

Your Name:

Your Section:

Little (8:00am) Ballantine (9:00am) DeStefano (10:00am) DeStefano (noon)

Instructions: Please write your answers in the boxes provided, and show work on the test pages. Use the back of the preceding page if you need more space for scratch work. The numbers next to each part of the questions are their point values.

Please do not write in the space below

Problem Points/Poss
I  / 12
II  / 15
III  / 18
IV  / 20
V  / 20
VI  / 15
Total  /100

I. Match the plots below with the following formulas. Note that there is an extra graph.

(4) A) y = 3 − (x − 1)^{2} is Plot:

(4) B) y = sin(x/2) is Plot:

(4) C) y = 1 − e^{−x} is Plot:

Plot 1: Plot 2:

Plot 3: Plot 4:

II. The manager of a furniture factory has collected the following data for the cost of manufacturing chairs.

# Chairs (per day) Cost (in dollars)
100  2400
150  3100
250  4500
300  5200

(6) A) Express the cost (y) as a function of the number of chairs produced per day (x), assuming that it is linear.

Cost function:

(3) B) What is the slope, and what does it represent in real-world terms?

(3) C) What is the y-intercept, and what does it represent in terms of cost?

(3) D) Using your model, determine how much it will cost to produce 350 chairs per day.

Cost:

III. Given f(x) = 4 − x^{2} and g(x) = \sqrt{3x − 2}, answer the following questions.

(6) A) Find the domain of f(x) and the domain of g(x). Domain of f: Domain of g:

(6) B) Find the function f/g and its domain. The function f/g: Domain of f/g:

(6) C) Find the function f ∘ g and its domain. f ∘ g: Domain of f ∘ g:

IV. Answer the following questions.

(5) A) Find all values of x in (−\frac{π}{2} , \frac{π}{2} ) for which |tan x| > 1. Values of x:

(5) B) If sin θ = \frac{2}{3} and \frac{π}{2} < θ < π, give the exact value of cos θ.

cos θ:

(5) C) Express as a single logarithm: \frac{1}{2} ln 3 − 3 ln 2 + ln 6.

Single logarithm:

(5) E) Complete the following table with the values for the functions f, g and h given that f

is even, g is odd, and h = f ∘ g.

x f(x) g(x) h(x)
−3 −1 0  1
−2 3  1 −2
−1 −2 −3 −1
0 1 2 3

1 2

3  1

V. Consider the function f(x) = 2e^{x+3} + 1.

(5) A) What are the domain and range of f?

Domain of f: Range of f:

(5) B) Given that f is one-to-one find a formula for the inverse function of f.

f^{−1}(x) =

(5) C) What are the domain and range of f^{−1}?

Domain of f^{−1}: Range of f^{−1}:

(5) D) On the next page, plot the graph of the functions f and f^{−1} on the same set of axes. Label one point on each graph with its coordinates.

Place graph for V D on this page. 7 VI. The half-life of radioactive polonium-210 is about 140 days.

(5) A) If a sample contains 200g of polonium-210 originally, find the amount remaining after 560 days.

(5) B) Find the amount remaining after t days.

(5) C) How long will it take the amount in the sample to decay to 10g? (Give your answer in exact form.)