This lesson aims to introduce the concept of exponential growth, and to demonstrate its importance in modelling. By mastering the content in this session you will:
To understand what exponential growth is and its role in mathematical models
To identify situations in which you’d expect exponential growth
To understand that exponential growth leads to some quite unintuitive outcomes
Exponentials are a family of functions which can be modelled by the following equation.
In this equation is known as the base and is known as the exponent. This function is most is most often used to represent growth and decay as it is able to model repeated multiplication or division.
Growth vs Decay
Exponential growth is a phenomena that describes a rapid increase in a functions output as the input increases. This is in contrast to exponential decay, which describes the rapid decrease in a functions output as the input increases. The conditions which create exponential growth and decay are shown below:
Base Value
Effect on Exponential
Exponential / rapid growth
Exponential / rapid decay
Constant function
The Number e
The number also known as Eulers number is a special mathematical constant that is approximately equal to . This number is very important when working with exponentials, as an exponential with a base of is the only function that differentiates into itself ( this concept is explained further in the exponential derivates lesson).
This implies that the height of the function is always the same as its gradient for all inputs.
The number also has many other useful applications in maths.
The number is much like , in the sense that its transcendental ( goes on forever ) and can be found in many mathematical fields including:
Compound interest
Radioactive decay
Spring Oscillation systems
and many many more
Converting between exponential bases
Converting between exponential bases can be done using the following formula.
To understand how this works, we require a basic understanding of logarithms ( the undo function of exponentials ). Namely, the fact that exponentials and logarithms are inverse functions of each other.
Because of this property, we are able to apply any exponential function followed by its log counterpart and have no net change to the original equation. Hence we can modify the function in the following way:
We can then use our log power down law ( taught in the later parts of this topic ) to get:
We now have two ways to express , which means:
This is particularly useful when trying to differentiate exponentials which are not expressed in base
Lesson Loading
Difficulty
00
Time
SOLUTION
Difficulty
3 Mins
Time
Lets examine a simple exponential model.
If a = 3, draw up a table of values for to study the effect of changing x. Do you notice anything interesting?
If x = 3 draw up a table of values for a = -3, -2, -1, 0, 1, 2, 3 to study the effect of changing a. Repeat for x = 2.
SOLUTION
Part A
x
-3
-2
-1
0
1
2
3
f(x)
0.04
0.11
0.33
1
3
9
27
We can see that as x gets more positive, the function gets bigger. On the other hand as x gets more negative, the function approaches 0. We can also see that the function is never negative.
Part B
a
-3
-2
-1
0
1
2
3
f(x)
-27
-8
-1
0
1
8
27
a
-3
-2
-1
0
1
2
3
f(x)
9
4
1
0
1
4
9
We can see that when a is negative, the function can be negative or positive. Hence we do not normally choose a < 0 since it demonstrates strange behaviour.
Difficulty
4 Mins
Time
Newton’s law of cooling states that if an object has a higher temperature () than its surrounding temperature (), it cools down according to the following exponential model.
A cup of water just below boiling water at is placed in a fridge that is . What is the temperature of the water after 5 hours if the coefficient of cooling (k) is 0.5?
What happens as we leave the cup of water in the fridge for longer? With this in mind, what does the model approach as t approaches infinity?
SOLUTION
Part A
Substituting gives:
Part B
We can see that as t grows, T approaches 4. This is because:
So in general, as t approaches infinity, T approaches the surrounding temperature.
Difficulty
5 Mins
Time
The money earnt from compound interest can be represented by the following exponential function:
where P is the initial amount of money, r is the annual interest, t is the number of years and n is the number of times the money is compounded.
If I placed $30000 into an account, how much money would I have after 3 years of 3% annual interest if it were compounded quarterly?
What is the new formula if we instead compounded an infinite amount of times per year?
Using this new model, how much money would we have in our account?
SOLUTION
Part A
All we need to do is substitute our values into the model provided:
Part B
We know that the maximum amount of money that could be made per dollar we borrowed after one year with an infinite amount of compounding periods is given by:
So if we borrowed dollars, we would have:
However, for the next year, we would simply treat the amount at the end of the first year as the the amount we started with for the second year:
This trend continues onto three years, so after three years we would have:
So after years, we would just have:
Part C
We just want to substitute our values into our new model:
Difficulty
8 Mins
Time
Population growth can be represented by a simple exponential function, for example a bacterium culture can be modelled as doubling in population every hour.
If the bacteria population starts at 50 cells, what is the equation that would model the population after t hours?
What would the population be after 16 hours?
What would the equation be if we instead represented it in t days?
What would the population be after 3 days?
SOLUTION
Part A
Since the population starts at 50 cells and doubles every hour we know that:
We can see from this, we can establish the exponential model:
Part B
We just need to substitute into the model we derived:
Part C
We know that there are 24 hours in a day, so the number of times the population should double is 24 times the number of days passed, hence our new model is:
Part D
We just need to substitute in our new model:
Difficulty
6 Mins
Time
Carbon-14 is a radioactive material that can be found in organic materials. The amount of Carbon-14 found in an organism halves every 5,700 years - which makes it a really useful tool to determine the age of fossils.
If the initial amount of Carbon-14 in an organism is , what is the model for the amount of Carbon-14 after t years? Can we represent this model without a fraction?
If a fossil is found to be 20,000 years old, what is the percentage of Carbon-14 left in the organism?
SOLUTION
Part A
We know that:
So we can deduce that:
We can also use index laws to remove the fraction from our model:
Part B
We want to substitute the number of years into our model, and find the percentage of the initial amount left over.
So we can see that 8.79% of the initial amount of Carbon-14 is left over.
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Presets
Indices
Cambridge 2 Unit Year 11 | Exercise 7A
Question 1 (all)
Question 12 (all)
Question 22 (all)
Fractional Indices
Cambridge 2 Unit Year 11 | Exercise 7B
Question 15 (all)
Application of These Functions
Cambridge 2 Unit Year 11 | Exercise 7G
Question 3 (a, b)
Question 4 (a, b)
Question 6 (a, b, c)
Question 7 (all)
Lesson Loading
×
×
Exponential Growth
Review Questions
What causes exponential growth
Repeated multiplication
Repeated addition
Repeated subtraction
Repeated division
If we started with a square meter of land, and doubled it every day for a month, we would have at least:
One thousand square meters
One million square meters
Ten million square meters
One billion square meters
Ten thousand square meters
Assuming is a variable and is a constant, an exponential function is any function of the form:
Why can we not allow the base of an exponential ( ) to be a negative number?
Negative numbers can’t be raised to negative powers
Negative exponentials would lead to complex valued inputs
Negative numbers can’t be raised to fractional powers
Negative numbers can’t be easily graphed
Negative values would cause to no longer by smooth
Negative exponentials would lead to negative y-values
×
Exponential Growth
Review Questions
What causes exponential growth
Repeated multiplication
Repeated subtraction
Repeated division
Repeated addition
If we started with a square meter of land, and doubled it every day for a month, we would have at least:
One million square meters
One thousand square meters
One billion square meters
Ten million square meters
Ten thousand square meters
Assuming is a variable and is a constant, an exponential function is any function of the form:
Why can we not allow the base of an exponential ( ) to be a negative number?
Negative numbers can’t be easily graphed
Negative exponentials would lead to negative y-values
Negative numbers can’t be raised to fractional powers
Negative exponentials would lead to complex valued inputs
Negative numbers can’t be raised to negative powers
Negative values would cause to no longer by smooth
×
Exponential Growth
Review Questions
Which of the following is the best approximation to Euler’s Number
3.141592654
1.618033988
2.718281828
1.41421356
What is the definition of Euler’s Number
The total amount of interest that could be charged after continuously compounding a loan of one dollar with an interest rate of 100% for a year
The total amount of money that could be made from continuously compounding a loan of one dollar with an interest rate of 100% for a year
The amount of money that was loaned by a bank when the interest rate was 100% over an entire year
The limit of the interest that could be charged by a bank as the number of compounding periods approaches infinity
What happens to the amount of extra interest a bank collects as the number of compounding periods increase?
The total amount of interest charged always increases, but it does so at a decreasing rate
The total amount of interest charged always decreases as the number of compounding periods grows
The total amount of interest eventually stops increasing after millions of compounding periods
How do we calculate the total amount of money charged at the end of a year if we have an interest rate of instead of 100%?
We just raise to the power of , so
We take the th root of , so
We multiply by , so we get
We divide by , so we get
×
Exponential Growth
Review Questions
Which of the following is the best approximation to Euler’s Number
2.718281828
3.141592654
1.41421356
1.618033988
What is the definition of Euler’s Number
The limit of the interest that could be charged by a bank as the number of compounding periods approaches infinity
The total amount of money that could be made from continuously compounding a loan of one dollar with an interest rate of 100% for a year
The total amount of interest that could be charged after continuously compounding a loan of one dollar with an interest rate of 100% for a year
The amount of money that was loaned by a bank when the interest rate was 100% over an entire year
What happens to the amount of extra interest a bank collects as the number of compounding periods increase?
The total amount of interest eventually stops increasing after millions of compounding periods
The total amount of interest charged always decreases as the number of compounding periods grows
The total amount of interest charged always increases, but it does so at a decreasing rate
How do we calculate the total amount of money charged at the end of a year if we have an interest rate of instead of 100%?
