Linear vs. Exponential:

Linear: a fixed absolute amount of change per unit of time but the percentage change varies across time.

Exponential relationship: there is a fixed percent change unit of time but the absolute amount of change varies.

Two methods for solving exponential growth problems:

1. Simulation (aka “table method”)

2. Direct solution using exponential growth formula

Compute the value of a variable xA B

1 Type X variable name

2 Type initial value of X

3 1 =B2*(1+r)

4 =A3+1 Copy B3

5 Copy A4 Copy B4

Simulation of exponential growth

Copy the formula in Row 5 for all necessary time periods.

Demonstration in lab

Compute the value of a variable xwhose initial value is 100

after growth for 50 time periodsat a 4% growth rate.

Direct solution using exponential growth formula:

Deriving the formula

Exponential growth is a successive percentage change

phenomenon.

The percentage change just stays the same.

Successive percents formula:

Y = X*(1+p1)*(1+ p2)*(1+ p3)

Change in notation

Y = X*(1+r)*(1+ r)*(1+ r)

Where r is the percentage rate of change expressed as a proportion.

Two reminders:

1. To convert a percentage into a decimal (proportion), move the decimal point over two places to the left.

2. The plus sign is converted to a negative for percentage decreases.

Deriving the general equation for exponential functions:

Y = X*(1+r)

Deriving the general equation for exponential functions:

Y = X*(1+r)

Y = X*(1+r)(1+r)

Deriving the general equation for exponential functions:

Y = X*(1+r)

Y = X*(1+r)(1+r)

or

Y = X*(1+r)2

Deriving the general equation for exponential functions:

Y = X*(1+r)

Y = X*(1+r)(1+r)

Y = X*(1+r)(1+r)(1+r)

Deriving the general equation for exponential functions:

Y = X*(1+r)

Y = X*(1+r)(1+r)

Y = X*(1+r)(1+r)(1+r)

or

Y = X*(1+r)3

General equation for exponential functions:

Y = X*(1+r)t

Where:

Y is the amount of the variable at any point in time

X is the initial amount of X

r is the rate of change

t is the number of time periods of change in Y

Direct solution using exponential growth formula:

Steps in solving for Y:

1. Y=X*(1+r)t

2. Substitute known information into equation

3. Compute– Exponentiation in Excel: ^t– Y=X*(1+r)^t

Between 1970 and 2000, DePaul tuition increased from $1,620 per

year to $15,390. The average annual tuition increase was 7.9%.

Assuming the same rate of change, what will tuition be in 2010?

• Y=$15,390*(1+.079)10

• Y=$15,390*1.07910

• Y=$15,390*2.139018

• Y=$32,919.49

Computing the rate of change (solving for r)

1. Start with the formula: Y=X*(1+r)t

Computing the rate of change (solving for r)

1. Start with the formula: Y=X*(1+r)t

2. Enter the values for Y, X and t

Computing the rate of change (solving for r)

1. Start with the formula: Y=X*(1+r)t

2. Enter the values for Y, X and t

3. Divide Y by X: Y/X = (1+r)t

Computing the rate of change (solving for r)

1. Start with the formula: Y=X*(1+r)t

2. Enter the values for Y, X and t

3. Divide Y by X: Y/X = (1+r)t

4. Extract the tth root of both sides. • In Excel, roots are extracted by ^(1/t)

Computing the rate of change (solving for r)

1. Start with the formula: Y=X*(1+r)t

2. Enter the values for Y, X and t

3. Divide Y by X: Y/X = (1+r)t

4. Extract the tth root of both sides. • In Excel, roots are extracted by ^(1/t)• (Y/X)^(1/t)=(1+r)^t^(1/t)

Computing the rate of change (solving for r)

1. Start with the formula: Y=X*(1+r)t

2. Enter the values for Y, X and t

3. Divide Y by X: Y/X = (1+r)t

4. Extract the tth root of both sides. • In Excel, roots are extracted by ^(1/t)• (Y/X)^(1/t)=(1+r)^t^(1/t)• Extracting the root cancels out the

exponentiation of (1+r)t

Computing the rate of change (solving for r)

1. Start with the formula: Y=X*(1+r)t

2. Enter the values for Y, X and t

3. Divide Y by X: Y/X = (1+r)t

4. Extract the tth root of both sides. • In Excel, roots are extracted by ^(1/t)• (Y/X)^(1/t)=(1+r)^t^(1/t)• Extracting the root cancels out the

exponentiation of (1+r)t

• (Y/X)^(1/t) = 1+r

Computing the rate of change (solving for r)

1. Start with the formula: Y=X*(1+r)t

2. Enter the values for Y, X and t

3. Divide Y by X: Y/X = (1+r)t

4. Extract the tth root of Y/X both sides: (Y/X)^(1/t)=1+r

5. Subtract 1 from the result of step 4

(Y/X)^(1/t)-1 = r

Solving for t

• How long will it take for a variable increasing or decreasing exponentially to reach a known or specified value of Y?

• To solve requires use of logarithms. . . Jump to logarithms presentation

Linear or exponential?

Over a 15 year period,

the number of workplace injuries at International Widget Inc. fell by

3 per year.

Linear or exponential?

From 1997 through 2005, revenues at Bogus Pictures rose by 7.5% per year.

Linear or exponential?Computational method

1. For two sets of time periods, compute the absolute change from one time period to the next. If it is the same, the change is linear.

2. For two sets of time periods, compute the percentage change from one time period to the next. If it is the same, the change is exponential.

Linear or exponential?Graphing method

1. Make an XY scatter diagram of the data.

2. If it appears to fall on a straight line, the function is linear.

3. If it has an accelerating curve upward or a decelerating curve downward it is exponential.

Linear or exponential?Graphing method

Graphing method can be risky: Linear functions can closely approximate exponentially growing data over the short run.

Linear model, exponential growth

y = 157.67x + 738.61

R2 = 0.9827

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Exponential model, exponential growth

y = 909.09e0.0953x

R2 = 1

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