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
x
A
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 x
whose initial value is 100
after growth for 50 time periods
at 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.
2.
Start with the formula: Y=X*(1+r)t
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.
2.
3.
Start with the formula: Y=X*(1+r)t
Enter the values for Y, X and t
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.
2.
3.
Start with the formula: Y=X*(1+r)t
Enter the values for Y, X and t
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.
2.
3.
Start with the formula: Y=X*(1+r)t
Enter the values for Y, X and t
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.
2.
3.
Start with the formula: Y=X*(1+r)t
Enter the values for Y, X and t
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.
2.
3.
4.
Start with the formula: Y=X*(1+r)t
Enter the values for Y, X and t
Divide Y by X: Y/X = (1+r)t
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
3000.0
2500.0
2000.0
1500.0
1000.0
y = 157.67x + 738.61
R2 = 0.9827
500.0
0.0
0
2
4
6
8
10
12
Exponential model,
exponential growth
3000.0
2500.0
2000.0
1500.0
y = 909.09e0.0953x
R2 = 1
1000.0
500.0
0.0
0
2
4
6
8
10
12