Chapter 1 Linear Functions

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Chapter 1Linear Functions

• Section 1.2
• Linear Functions and Applications

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Linear Functions

• Many situations involve two variables related by
  a linear equation.
• When we express the variable y in terms of x, we
  say that y is a linear function of x.
• Independent variable x
• Dependent variable y
• f(x) is used to sometimes denote y.

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Linear Function
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Example 1

• Given a linear function f(x) 2x 5, find the
  following.
• a.) f(-2)
• b.) f(0)
• c.) f(4)

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Supply and Demand

• Linear functions are often good choices for
  supply and demand curves.
• Typically, there is an inverse relationship
  between supply and demand in that as one
  increases, the other usually decreases.

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Supply and Demand Graphs

• While economists consider price to be the
  independent variable, they will plot price, p, on
  the vertical axis. (Usually the independent
  variable is graphed on the horizontal axis.)
• We will write p, the price, as a function of q,
  the quantity produced, and plot p on the vertical
  axis.
• Remember, though price determines how much
  consumers demand and producers supply.

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General Set-Up for Supply and Demand Graph
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Example 2

• Suppose that the demand and price for a certain
  model of electric can opener are related by
• p D(q) 16 5/4 q Demand
• where p is the price (in dollars) and q is the
  demand (in hundreds).
• a.) Find the price when there is a demand for
  
• 500 can openers.

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Example 2

• p D(q) 16 5/4 q Demand
• b.) Graph the function.

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Example 2 continued

• Suppose the price and supply of the electric can
  opener are related by
• p S(q) 3/4q Supply
• where p is the price (in dollars) and q is the
  demand (in hundreds).
• c.) Find the demand for electric can openers
  with a
• price of 9 each.

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Example 2 continued

• p S(q) 3/4q Supply
• d.) Graph this function on the same axes used
  for the
• demand function.

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Supply and Demand Graph
D(q)
S(q)
Equilibrium point
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Equilibrium Point
The equilibrium price of a commodity is the price
found at the point where the supply and demand
graphs for that commodity intersect.
The equilibrium quantity is the demand and supply
at that same point.
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Example 2 continued

• p D(q) 16 5/4 q Demand
• p S(q) 3/4q Supply
• Use the functions above to find the
  equilibrium quantity and the equilibrium price
  for the can openers.

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Cost Analysis

• The cost of manufacturing an item commonly
  consists of two parts the fixed cost and the
  cost per item.
• The fixed cost is constant (for the most part)
  and doesnt change as more items are made.
• The total value of the second cost does depend on
  the number of items made.

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Marginal Cost

• In economics, marginal cost is the rate of change
  of cost C(x) at a level of production x and is
  equal to the slope of the cost function at x.
• The marginal cost is considered to be constant
  with linear functions.

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Cost Function
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Example 3

• Write a linear cost function for each situation
  below. Identify all variables used.
• a.) A car rental agency charges 35 a day plus
  25
• cents a mile.
• b.) A copy center charges 4.75 to create a
  flier and
• 10 cents for every copy made of the
  flier.

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Example 4

• Assume that each situation can be expressed as a
  linear cost function. Find the cost function in
  each case.
• a.) Fixed cost is 2000 36 units cost 8480

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Example 4

• Assume that each situation can be expressed as a
  linear cost function. Find the cost function in
  each case.
• b.) Marginal cost is 75 25 units cost 3770

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Break-Even Analysis

• The revenue R(x) from selling x units of an item
  is the product of the price per unit p and the
  number of units sold (demand) x, so that
• R(x) p(x).
• The corresponding profit P(x) is the difference
  between revenue R(x) and cost C(x).
• P(x) R(x) - C(x)

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Break-Even Analysis

• A profit can be made only if the revenue received
  from its customers exceeds the cost of producing
  and selling its goods and services.
• The number of units x at which revenue just
  equals cost is the break-even quantity the
  corresponding ordered pair gives the break-even
  point.

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Break-Even Point

• As long as revenue just equals cost, the company,
  etc. will break even (no profit and no loss).
• R(x) C(x)

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Example 5

• The cost function for flavored coffee at an
  upscale coffeehouse is given in dollars by C(x)
  3x 160, where x is in pounds. The coffee sells
  for 7 per pound.
• a.) Find the break-even quantity.

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Example 5

• The cost function for flavored coffee at an
  upscale coffeehouse is given in dollars by C(x)
  3x 160, where x is in pounds. The coffee sells
  for 7 per pound.
• b.) What will the revenue be at the break-even
  point?

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Example 5

• The cost function for flavored coffee at an
  upscale coffeehouse is given in dollars by C(x)
  3x 160, where x is in pounds. The coffee sells
  for 7 per pound.
• c.) What is the profit from 100 pounds?

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Example 5

• The cost function for flavored coffee at an
  upscale coffeehouse is given in dollars by C(x)
  3x 160, where x is in pounds. The coffee sells
  for 7 per pound.
• d.) How many pounds of coffee will produce a
  
• profit of 500?

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Example 6

• In deciding whether or not to set up a new
  manufacturing plant, analysts for a popcorn
  company have decided that a linear function is a
  reasonable estimation for the total cost C(x) in
  dollars to produce x bags of microwave popcorn.
• They estimate the cost to produce 10,000 bags as
  5480 and the cost to produce 15,000 bags as
  7780.
• Find the marginal cost and fixed cost of the
  bags of microwave popcorn to be produced in this
  plant, then write the cost function.

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Example 6

• The cost to produce 10,000 bags is 5480 and the
  cost to produce 15,000 bags is 7780.
• Find the marginal cost and fixed cost of the
  bags of microwave popcorn to be produced in this
  plant, then write the cost function.