Chapter 1 Functions and Linear Models Sections 1.3 and 1.4

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Chapter 1 Functions and Linear Models Sections 1.3 and 1.4

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Chapter 1 Functions and Linear Models Sections 1.3 and 1.4

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Linear Function

A linear function can be expressed in the form

Function notation

Equation notation

where m and b are fixed numbers.

Graph of a Linear Function

The graph of a linear function is a straight line.

This means that we need only two points to completely determine its graph.

m is called the slope of the line and

b is the y-intercept of the line.

Example: Sketch the graph of f (x) = 3x – 1

y-axis

Arbitrary point

(1,2)

(0,-1)

x-axis

y-intercept

Role of m and b in f (x) = mx + b

The Role of m (slope)

f changes m units for each one-unit change in x.

The Role of b (y-intercept)

When x = 0, f (0) = b

Role of m and b in f (x) = mx + b

To see how f changes, consider a unit change in x.

Then, the change in f is given by

Role of m and b in f (x) = mx + b

Role of m and b in f (x) = mx + b

Role of m and b in f (x) = mx + b

Slope = 3/1

y-intercept

Example:Sketch the graph of f (x) = 3x – 1

y-axis

(1,2)

x-axis

Example:Sketch 3x + 2y = 6

y-axis

x-intercept (y = 0)

x-axis

y-intercept (x = 0)

If aquantity q changes from q1 to q2 , the change in q is denoted by q and it is computed as

Example: If x is changed from 2 to 5, we write

Delta Notation

Example:the slope of a non-vertical line that passes through the points (x1,y1) and (x2,y2) is given by:

Example: Find the slope of the line that passes through the points (4,0) and (6, -3)

Delta Notation

Zero Slope and Undefined Slope

Example: Find the slope of the line that passes through the points (4,5) and (2, 5).

This is a horizontal line

Example:Find the slope of the line that passes through the points (4,1) and (4, 3).

This is a vertical line

Undefined

Examples

Estimate the slope of all line segments in the figure

An equation of a line that passes through the point (x1,y1) with slope m is given by:

Example: Find an equation of the line that passes through (3,1) and has slope m = 4

Can be expressed in the form y = b

y = 2

Can be expressed in the form x = a

x = 3

- A cost function specifies the cost C as a function of the number of items x produced. Thus, C(x) is the cost of x items.
- The cost functions is made up of two parts:
C(x)= “variable costs” + “fixed costs”

- If the graph of a cost function is a straight line, then we have a Linear Cost Function.
- If the graph is not a straight line, then we have a Nonlinear Cost Function.

Dollars

Dollars

Cost

Cost

Units

Units

Dollars

Dollars

Cost

Cost

Units

Units

- The revenue function specifies the total payment received R from selling x items. Thus, R(x) is the revenue from selling x items.
- A revenue function may be Linear or Nonlinear depending on the expression that defines it.

Dollars

Revenue

Units

Dollars

Dollars

Revenue

Revenue

Units

Units

- The profit function specifies the net proceeds P. P represents what remains of the revenue when costs are subtracted. Thus, P(x) is the profit from selling x items.
- A profit function may be linear or nonlinear depending on the expression that defines it.

Profit = Revenue – Cost

Dollars

Profit

Units

Dollars

Dollars

Profit

Profit

Units

Units

The Linear Models are

Cost Function:

** m is the marginal cost (cost per item), b is fixed cost.

Revenue Function:

** m is the marginal revenue.

Profit Function:

where x = number of items (produced and sold)

The break-even point is the level of production that results in no profit and no loss.

To find the break-even point we set the profit function equal to zero and solve for x.

The break-even point is the level of production that results in no profit and no loss.

Profit = 0 means Revenue = Cost

Dollars

Revenue

profit

loss

Break-even Revenue

Cost

Units

Break-even point

Example: A shirt producer has a fixed monthly cost of $3600. If each shirt has a cost of $3 and sells for $12 find:

a.The cost function

C (x) = 3x + 3600 where x is the number of shirts produced.

b.The revenue function

R (x) = 12x where x is the number of shirts sold.

c.The profit from 900 shirts

P (x) = R(x) – C(x)

P (x) = 12x – (3x + 3600) = 9x – 3600

P(900) = 9(900) – 3600 = $4500

Example: A shirt producer has a fixed monthly cost of $3600. If each shirt has a cost of $3 and sells for $12 find the break-even point.

The break even point is the solution of the equation

C (x) = R (x)

Therefore, at 400 units the break-even revenue is $4800

- A demand function or demand equation expresses the number q of items demanded as a function of the unit price p (the price per item).
- Thus, q(p) is the number of items demanded when the price of each item is p.
- As in the previous cases we have linear and nonlinear demand functions.

q = items demanded

Price p

q = items demanded

q = items demanded

Price p

Price p

- A supply function or supply equation expresses the number q of items, a supplier is willing to make available, as a function of the unit price p (the price per item).
- Thus, q(p) is the number of items supplied when the price of each item is p.
- As in the previous cases we have linear and nonlinear supply functions.

q = items supplied

Price p

q = items supplied

q = items supplied

Price p

Price p

Market Equilibrium occurs when the quantity produced is equal to the quantity demanded.

supply curve

q

surplus

shortage

demand curve

p

Equilibrium Point

Market Equilibrium occurs when the quantity produced is equal to the quantity demanded.

q

supply curve

surplus

shortage

demand curve

Equilibrium demand

p

Equilibrium price

- To find theEquilibrium price set the demand equation equal to the supply equation and solve for the price p.
- To find theEquilibrium demand evaluate the demand (or supply) function at the equilibrium price found in the previous step.

Example of Linear Demand

The quantity demanded of a particular computer game is 5000 games when the unit price is $6. At $10 per unit the quantity demanded drops to 3400 games.

Find a linear demand equation relating the price p, and the quantity demanded, q (in units of 100).

Example: The maker of a plastic container has determined that the demand for its product is 400 units if the unit price is $3 and 900 units if the unit price is $2.50.

The manufacturer will not supply any containers for less than $1 but for each $0.30 increase in unit price above the $1, the manufacturer will market an additional 200 units. Assume that the supply and demand functions are linear. Let p be the price in dollars, q be in units of 100 and find:

a. The demand function

b. The supply function

c. The equilibrium price and equilibrium demand

a. The demand function

b. The supply function

c. The equilibrium price and equilibrium demand

The equilibrium demand is 960 units at a price of $2.44 per unit.

Rate of change of q

Quantity at time t = 0

Linear Change over Time

A quantity q, as a linear function of time t:

If q represents the position of a moving object, then the rate of change is velocity.

We have seen how to find a linear model given two data points. We find the equation of the line passing through them.

However, we usually have more than two data points, and they will rarely all lie on a single straight line, but may often come close to doing so.

The problem is to find the line coming closest to passing through all of the points.

We use the method of least squares to determine a straight line that bestfits a set of data points when the points are scattered about a straight line.

least squares line

Given the following n data points:

The least-squares (regression) line for the data is given by y = mx + b, where m and b satisfy:

and

Example: Find the equation of least-squares for the data (1,2), (2,3), (3,7).

The scatter plot of the points is

Solution: We complete the following table

Sum: 6 12 29 14

Example: Find the equation of least-squares for the data (1,2), (2,3), (3,7).

The scatter plot of the points and the least squares line is

Coefficient of Correlation

A measurement of the closeness of fit of the least squares line. Denoted r, it is between –1 and 1, the better the fit, the closer it is to 1 or –1.

Example: Find the correlation coefficient for the least-squares line from the last example.

Points: (1 , 2), (2,3), (3,7)

= 0.9449