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Applications with Linear Functions
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Cost, revenue, profit
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Marginals for linear functions

Break Even points

Supply and Demand Equilibrium
Cost, Revenue, Profit, Marginals

Cost: C(x) = variable costs + fixed costs
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Revenue: R(x) = (price)(# sold)
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Profit: P(x) = C(x) – R(x)
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Marginals: what would happen if one more
item were produced (for marginal cost) and sold
(for marginal revenue or marginal profit)
Example 1
C ( x)  22 x  60
R( x)  30 x
Find C(50), R(50), P(50) and interpret.
Find all marginals when x = 50 and interpret.
Example 1 – continued
C ( x)  22 x  60
R( x)  30 x
Find all marginals when x = 50 and interpret.
For linear functions, the marginals are the slopes of the lines.
Break Even Points
Companies break even when costs = revenues
or when profit = 0.
Example 2
C ( x)  75 x  1400
R( x)  89 x
Example 3
If P(10) = -150 and P(50) = 450, how many units are needed
to break even if the profit function is linear?
y  y1  m( x  x1 )
The company breaks even by producing and selling

Law of Demand: quantity demanded goes up as
price goes down. Likewise, as price goes up,
quantity demanded goes down.

Law of Supply: quantity supplied goes up as
price goes up. Likewise, as price goes down,
quantity supplied goes down.

Market Equilibrium: where quantity demanded
equals quantity supplied
Example 4
p
Demand:
p  480  3q
Supply:
p  17q  80
q