College of the Holy Cross, Fall 2013

Math 135, Section 1, Midterm 1 Friday, September 20

Your Name:

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  / 20
II  / 25
III  / 20
IV  / 15
V  / 20
Total  /100

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

(5) A) y = 3 − (x − 1)^{2} is Plot:

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

(5) C) y = 1 − e^{−x} is Plot:

(5) D) y = sin(2x) is Plot:

Plot 1: Plot 2:

Plot 3: Plot 4:

Plot 5:

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

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

(10) A) Given that y is a linear function of x, determine a formula for it. Cost function:

(5) B) What does the slope represent in real-world terms?

(5) C) What does the y-intercept represent in terms of cost?

(5) 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.

(10) A) Find the domain of f(x) and the domain of g(x).

Domain of f: Domain of g:

(5) B) What is the domain of the function g(x)/f(x)?

Domain of g(x)/f(x):

(5) C) Find the function g ∘ f.

g ∘ f:

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:

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

(15) A) Given that f is one-to-one, find a formula for the inverse function of f.

f^{−1}(x) =

(10) B) In the space below, plot the graphs of the functions f and f^{−1} on the same set of axes. Label one point on each graph with its coordinates.