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Math 1630 – Online Spring 2009 Midterm Name: (86 possible points)
You must show all work to receive full credit. Work in pencil only.
(^2 ) 2 x b^ b^ ac a
=^ −^ ±^ −
1. (4 points) Solve 3 x − 5 y = 25 for y in terms of x.
3 5 25 3 3 5 3 25 5 3 25 5 3 25 5 5 (^3 ) 5
x y x x y x y x y x
y x
2. (5 points) Solve the following equation for x: 2 1 2 3 2
x (^) − = x −
x x
x (^) x
x x x x x x x x x
3. Let f ( ) x = x^2 − 3 x + 1 and g x ( ) = 3 x − 5. Determine the following (simplify as much as possible):
3a. (3 points) ( f − g )( x ) 3b. (3 points) g ( f ( x ))
2 2
f g x x x x
x x x
x x
2 2 2 2
g f x g x x
x x
x x
x x
3c. (3 points) f ( 2)− f ( 2)− = ( − 2 ) 2 − 3 ( − 2 ) + 1 = 4 + 6 + 1 = 11
4. (3 points) Determine the domain of the following function.
( ) 2 3 h x x
The denominator of a fraction cannot equal 0. Therefore, 3 0 3 3 0 3 3
x x x
The domain is the set of all Real Numbers except -3.
5. (3 points) Determine whether the relation defined by the table below is a function. Explain specifically why or why not.
x -1 1 2 2 8 y 1 1 1 8 1
A function is a relation in which each member of input is associated with exactly one member of output. The relation in this table satisfies this definition. Notice that the input of 2 has two outputs, 8 and 1. Therefore, this relation does not satisfy the definition of a function. This relation is not a function.
6. (6 points) Solve the following system of equations by the substitution method. 4 5 14 3 16
x y x y
Solve the second equation for y. 3 16 3 16 3 16 3 16 16 16 3 16
x y x y y y x y x y y x
Substitute for y in the second equation and solve for x
( )
( )
The solution is 6, 2.
x y
x x
x x
x
x
x
x
x
y x
y
y
x y
9. (4 points) The profit function for a certain commodity is P x ( )^^ = −.4^ x^^2 +^80 x −^200 where x represents the
number of units of the commodity. Find the level of production that yields maximum profit, and find the maximum profit.
The profit function is quadratic. The lead coefficient (value of “a”) is negative. Thus the parabola points down. The vertex is at the top. The vertex gives the maximum value of “y” (profit). So we need to determine the coordinates of the vertex.
( ) ( ) ( ) ( )
2
x b a P
= −^ = −^ = − =
Profit is maximized when the production level is 100 units. The maximum profit is $3800.
10. The supply for a tool is linear and determined to be p = 15 q + 10 and a linear demand of p = − 5 q + 330. 10a. (4 points) Showing all work, determine the coordinates of the equilibrium point.
We need to solve the system of equations. I’ll solve by substitution
( )
q q q q q q q q q q
q p q p p
Market equilibrium is achieved when the price is set at $250. The amount demanded and supplied will each be
10b. (3 points) If the company sets the price at $230.00, will there be a shortage of goods on the market or a surplus? Show all work. (Round to nearest whole numbers)
Amount Supplied Amount Demanded
5 330 230 5 330 230 330 5 330 330 5 100 5 100 5 5 20
p q q q q q
q
Since the amount demanded is greater than the amount supplied, there will be a shortage of goods.
10c. (2 points) Based on your answers to part b. above and the principles of supply and demand, explain whether the price will more likely remain at $230, fall below $230, or climb above $230.
Since there is a shortage of goods, then the market needs to either increase supply or decrease demand (or both). The way to accomplish this is to increase prices. Thus we would expect prices to climb above $230.
11. (6 points) The total costs for a company are given by C ( x ) = x^2 + 40 x + 2000 and the total revenues given
by R ( x ) = 130 x. Calculate the break-even point(s).
I’ll set Revenue = Cost and solve for x.
( )( )
2 2 2
I'll solve by factoring. 40 50 0 40 0 40 40 0 40 40 50 0 50 50 0 50 50
x x x x x x x x x x
x x x x x x x x
There are two break-even points; make and sell 40 units, or make and sell 50 units.
p q q q q q
q
13. ( 3 points) Managers rate employees according to job performance and attitude. The results for several employees randomly selected are given below. Using your calculator, develop the linear model that best fits the data.
Performance (x)
Attitude (y) 72 67 78 82 75 87 92 83 87 78
The problem asks us to develop a linear model (i.e. y = mx + b) that uses Performance as the x-variable and Attitude as the y-variable.
This is a linear regression problem. We’ll let our calculator determine the equation of the line. First, we enter our data in the calculator.
We then use the Linear Regression Command. Press the STAT button and right arrow to the CALC menu. Select command 4: LinReg (ax + b)
This returns us to our home screen. The x’s are in column L1 and the y’s are in column L2.
Press ENTER
Your screen probably will not show the r and r-squared values. Don’t worry, you do not need these numbers for this.
The equation of our line is y = 1.021505376x + 11.65913978 or
Predicted Attitude Score = 1.021505376 (Performance Score) + 11.
13b. (2 points) Use your model to predict an employee’s Attitude score if his/her Performance Score is 71.
We enter 71 for Performance Score in the above equation.
Predicted Attitude Score = 1.021505376 (71) + 11.65913978 = 84.
A
B
C
14. Use the above matrices to determine the following quantities. You must show your work. If an operation cannot be performed, demonstrate why.
14a. (2 points) Give the dimensions of matrix C. 2 rows, 3 columns (or 2x3)
14b(4 points) Determine 3A - 5B
7 3 21 9 3 3 2 5 6 15 3 2 15 10 5 5 0 2 0 10 21 9 15 10 36 19 3 5 6 15 0 10 6 5
A
B
A B
⎡^ − ⎤ ⎡ − ⎤
⎡ −^ ⎤ ⎡− ⎤
⎡ −^ ⎤ ⎡ −^ ⎤ ⎡ − ⎤
