Math 180 – Make up Exam – Chapters 1, 2, 3 – Spring 2007Name______
SHOW STEPS IN ALL PROBLEMS. NO STEPS = NO CREDIT –
TIME YOURSELF!!!!! NO MORE THAN 8 MINUTES PER PAGE!
1) Given the function, find the difference quotient
Write answer in simplest form.
2) Find the domain for each of the functions given below? Show how you find it, and very clearly specify the answer.
Domain of f(x) ______Domain of g(x) ______
3) Given the function
a) Find the average rate of change of the function from x = 2 to x = 4.
b) Geometrically, what does this number represent? Be very specific in your explanation.
4) Sam has 6000 ft of fencing available to enclose a rectangular field. One side of the field lies along a river, so it does not need a fence along that side. He also plans to put ONE interior fence perpendicular to the river in order to divide the rectangular field into three plots.
a) Express the area A of the rectangle as a function of x, where x is the length of the side parallel to the river.
b) Graph A = A(x) using the calculator. Show graph. Label axes with variables, words and units. Indicate the window values used.
c) For what value of x is the area largest? What is the largest area? What is the width of the rectangular field? Include units in your answers.
Optimal x =
Largest area =
Optimal width =
5) Factor the given polynomial as a product of linear factors
a) In order to do this, first find all the zeros. Make sure you show all your reasoning/work here.
(You will only receive credit IF YOU USE the following method:
List all potential zeros, do long division, and show all steps to find the remaining zeros)
List here all the zeros: ______
(b) Write the factorization of the polynomial as a product of linear factors
6) United Parcel Service has contracted you to design a closed box with a square base that has a volume of 5,000 cubic inches.
a) Find a function for the surface area of the box, in terms of x, the side of the base of the box.
(Remember, there are two equations: the constraint equation, which is the one that relates x, y and the volume, and the objective equation which is the one for the surface area that you want to optimize.)
b) Use the calculator to graph. Show graph, label with variables, words and units, indicate window values used.
c) Use the graph to find the minimum amount of cardboard that can be used to construct the box. Give the dimensions of the optimal box.
Minimum surface area is =
Optimal side for the base of the box=
Optimal height of the box =
d) Why might UPS be interested in designing a box that minimizes the surface area?
7) Use the following graph to answer the questions given below.
a)Find f(1)
b)Solve f(x) = -1
c)For what values of x is f(x) > 1?
d)Give intervals for which the function is decreasing
8) Construct a polynomial with the given properties:
All zeros are real. The zeros are -1 of multiplicity 3, and -3 of multiplicity 2. The y-intercept is 20. Leave answer in factored form.
The polynomial is ______
9) The graphs of two functions f and g are given below. The coordinates of the points labeled in the graph are given on the table.
/ x / yA / -3.5 / -10.9
B / -2 / 0
C / 0 / 8
D / 3.5 / 2.9
E / 4 / 0
F / 2 / 0
G / 0 / -4
a)Label the functions in the graph. (Do not use your calculator!!)
b)USE THE GRAPH to solve the inequality
State the solution(s) to the inequality using interval notation.
The solution is......
1