Math 141, Fall 2015, c Dr. Craig Gin
Sections 1.3-1.4
Relevant Equations:
Slope:
Point-Slope Form: m = y
2
− y
1 y − y
1 x
2
− x
1
= m ( x − x
1
)
Slope-Intercept Form: y = mx + b
Cost Function:
Revenue Function:
Profit Function:
C
R
P
(
(
( x x x
) =
) =
) = cx sx
R (
+ x )
F
− C ( x )
Key Terms:
• Function: A rule that assigns to each value of x one and only one value of y
• Domain: The set of all possible values that x can assume
• Range: The set of all possible values that y can assume
• Market Equilibrium: The intersection of the supply and demand curves
• Break-even Point: The intersection of the revenue and cost functions
1.
Consider the line that goes throught the points (-6,3) and (2,1).
y
5
( − 6 , 3)
(2 , 1)
− 5 5 x
− 5 a) What is the slope of the line?
b) Write the equation of the line in point-slope form.
c) Find the y-intercept and write the equation in slope-intercept form.
d) Find the x-intercept of the line.
2.
Find the equations of the following lines.
(a) Find the equation of the horizontal line that passes through the point (2,3).
(b) Find the equation of the vertical line that passes through the point (-1,4).
(c) A line passes through the point (2,-1). As x increases by 2, y increases by 5. Find the equation of the line.
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Math 141, Fall 2015, c Dr. Craig Gin
3.
It costs a manufacturer $ 6200 to produce 40 units and $ 6536 to produce 64 units. The product sells for $ 24 per unit.
(a) Given that the cost, C, is a linear function of the number of units, x, find the cost function. What are the fixed costs?
(b) What is the revenue function?
(c) What is the profit function?
(d) If the production level is 500 units, does the company make a profit or does it lose money?
4.
A new car costs $ 18,000 and is depreciated linearly over ten years with a scrap value of $ 1000. What is the value of the car after 4 years?
5.
At a unit price of $ 35, the quantity demanded of a certain commodity is 1200 units.
At a unit price of $ 75, the demand drops to 600 units.
(a) Given that it is linear, find the demand equation.
(b) Sketch the demand curve.
(c) What is the highest price (theoretically) that anyone would pay for this commodity?
(d) What quantity would be demanded if the commodity were free?
6.
Producers of a certain brand of hammock will make 2500 hammocks available if the unit price is $ 45. At a unit price of $ 60, 3100 hammocks will be made available. Assuming that the relationship between the unit price and the quantity supplied is linear, find the supply equation. How many hammocks will be made available when the unit price is
$ 70?
7.
Consider the following two linear equations:
2 y − 4 x + 28 = 0
3 y + 9 x + 12 = 0
(a) Find the intersection of the lines algebraically.
(b) Find the intersection using your calculator.
8.
A watch manufacturer has a monthly fixed cost of $ 27,000 and a production cost of
$ 30 for each watch manufactured. The watches sell for $ 75.
(a) Find the cost and revenue functions. Sketch the graph of each.
(b) Find the break-even point.
9.
The quantity demanded of a certain tablet is 3100 per week when the unit price is
$ 360. The demand is 3400 per week when the unit price is $ 315. The suppliers will not market the tablets if the unit price is $ 250 or lower. At a unit price of $ 400, they are willing to make 3000 tablets available.
(a) Find the demand equation.
(b) Find the supply equation.
(c) Find the equilibrium quantity and price.
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