Nuffield Free-Standing
Mathematics Activity
Exponential
rates of change
© Nuffield Foundation 2012
Exponential rates of change
There are many examples of exponential growth and decay in
everyday life, such as bacteria growth and radioactive decay.
Population growth depends on
the size of the current population.
The rate of population growth
can be used to predict the size
of the population at any given time.
The number of people in the
population, N, at any time t,
can be written as an exponential function:
N = Aert where A and r are constants
© Nuffield Foundation 2010
Graphs of exponential growth and decay have distinctive shapes
Exponential growth curve
Exponential decay curve
y
y
0
x
0
x
Think about …
Can you suggest everyday examples of exponential growth or decay?
What can you say about the gradient of the exponential growth graph?
What can you say about the gradient of the exponential decay graph?
© Nuffield Foundation 2010
Estimating gradients by calculation
Q
The sketch shows points P and Q on a curve.
Their coordinates can be used to find the gradient
of the chord PQ.
P
The nearer that Q is to P, the nearer the gradient of
PQ is to the gradient of the tangent at P.
In general, the gradient at a point P where x = a, on
the curve y = f(x) is estimated using:
gradient 
© Nuffield Foundation 2010
f a  h   f a 
where h is a small increment.
h
Estimating gradients by calculation
The spreadsheet can be used to investigate gradients at
points on the curve y = ex.
Think about …
What do the formulae in each column do?
© Nuffield Foundation 2010
Exponential
rates of
change
Graph of y = e2x and its gradient function
y
110
100
90
80
70
60
y = EXP(2x)
Gradient
50
40
30
20
10
0
© Nuffield Foundation 2010
-2
-1
0
1
2
x
Exponential
rates of
change
Graph of y = e0.5x and its gradient function
y
3
2
y = EXP(0.5x)
Gradient
1
0
-2
© Nuffield Foundation 2010
-1
0
1
2 x
Exponential
rates of
change
Graph of y = e –x and its gradient function
y
8
6
4
f(x) = EXP(-x)
2
Gradient
0
-2
-1
0
-2
-4
-6
© Nuffield Foundation 2010
-8
1
2 x
Reflect on your work
• Compare the two methods you have used for finding
the gradients of exponential functions. What are their
advantages and disadvantages?
• What can you say in general about the gradient function
of y = ekx?
© Nuffield Foundation 2012