MAT 111 – Section 007
MAT 115 – Section 001 Name: ____KEY________________
Exam 1 Grade: ___110_______
Directions: Answer each of the following questions. Be sure to show all work in order to receive partial credit. If you need additional space, you may use the back of the test paper.
1-10pts) Solve the following equation for x algebraically (decimal approximations are not acceptable):
[pic]
We first bring all terms to the right-hand side and then use the quadratic formula.
[pic]
2-10pts) Solve the following system of equations algebraically (i.e. find all points where the two graphs intersect). Decimal approximations are not acceptable!
[pic]
We plug in the bottom expression for y into the top expression. This gives an equation for x that we can factor. Once we find all possible x-values, we plug these into either of the two equation (I used the bottom one) to find the corresponding y-values.
[pic]So solutions are (0,0) ; (-4,3) ; (4,3)
3-10pts) Solve the following inequality algebraically. Express your answer either in interval notation or with inequalities.
[pic]
[pic]
4-10pts) Solve the following inequality algebraically. Express your answer either in interval notation or with inequalities.
[pic]
[pic]
5-10pts) Solve the following inequality algebraically. Express your answer either in interval notation or with inequalities. (Hint: Use a sign chart.)
[pic]
[pic]So the solution is [pic]
6-10pts) Give the equation of the line that is parallel to 2y – 3x = 10 and goes through the point (-4,1).
We first find the slope of the given line. The line parallel will have the same slope. We then use the point-slope form to find the equation of the line.
[pic]
7-10pts) Give the domain of the following two functions. In addition, determine whether the functions are even, odd, or have no symmetry.
a) [pic] b) [pic]
Domain: Domain:
[pic] [pic]
Symmetry: Symmetry:
[pic] [pic]
8-10pts) Consider the following quadratic function:
[pic]
a) Find the x and y-intercepts for the graph y = f(x).
y-int: (0,6)
x-int: (2,0) , (-3,0)
b) Find the coordinates of the vertex for the graph y = f(x)
x= -b/2a = -1/2
y=f(-1/2) = 25/4
VERTEX = (-1/2, 25/4)
c) Does the graph of this parabola open up or down?
Opens Down as a (which is -1) is less than zero.
d) Does the vertex represent a maximum or minimum?
Since this parabola opens down, the vertex is a maximum.
9-10pts) The cost to manufacture x number of cell phones is given by the function
[pic].
How many cell phones should be manufactured to minimize this cost?
The minimum will occur at the vertex. The x-coordinate of the vertes is x = -b/2a.
Since b = -200 and a = 5 we have that the vertex has an x-coordinate of 20.
Hence, 20 cell phones should be made.
10-10pts) Given that he polynomial function
[pic]
has a zero at [pic], find the remaining zeros.
We begin by factoring out the term (2x – 1) from the polynomial and then factoring the remaining quadratic. This gives us that
p(x) = (2x – 1)(x + 2)(x – 1).
Hence, the zeros of this polynomial are ½, 1, and -2.
BONUS – 10 pts ) The expected profit (in hundreds of dollars) from producing x amount (in hundreds of kilograms) of a particular chemical is given by the function
[pic].
Due to manufacturing constraints, x must be in the range [pic]. Using a graphing utility, find the value of x that maximizes the profit in the allowable range of x. What is the maximum expected profit? (Round answers to 2 decimal places.)
Using a graphing calculator, we find that the function given has a maximum on the interval [0, 10] at the point (7.48, 1073.49).
Hence, we should produce 748 kilos of the chemical. The expected profit will be
$107,349.
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