C3 Chapter 3: Exponential and Log Functions

Dr J Frost (jfrost@tiffin.kingston.sch.uk)

Last modified: 1st August 2014

-2 -1 1 2 3 4 5 6 7

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6

4

2

-2

-4

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𝑦=2𝑥x -2 -1 0 1 2 3

y 0.25 0.5 1 2 4 8? ? ? ? ? ?

Click to Brosketch

This is known as an exponential function.It is useful for modelling things like: population growth/savings with compound interest.

The key property of exponential growth is that: the output gets multiplied by some constant each time the input increases (by a unit). e.g. A rabbit population might get 40% larger each year. This is in contrast to linear growth where we add some constant each time.

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You won’t yet be able to differentiate exponential functions till C4.

But I’ve calculated some gradients for you – click the black arrow to reveal the graph and gradient function.

Gradients of Exponential Functions

Function Gradient

𝑦=1𝑥

𝑦=1.5𝑥

𝑦=2𝑥

𝑦=2.5𝑥

𝑦=3𝑥

𝑦=3.5𝑥

𝑑𝑦𝑑𝑥

=0

𝑑𝑦𝑑𝑥

=0.41×1.5𝑥

>

>𝑑𝑦𝑑𝑥

=0.69×2𝑥 >𝑑𝑦𝑑𝑥

=0.92×2.5𝑥 >𝑑𝑦𝑑𝑥

=1.10×3𝑥 >𝑑𝑦𝑑𝑥

=1.25×3.5𝑥 >

Can you estimate the base of the exponential function where the gradient function is the same as the function itself?

“The” Exponential Function

Function Gradient

𝑦=1𝑥

𝑦=1.5𝑥

𝑦=2𝑥

𝑦=2.5𝑥

𝑦=𝑒𝑥𝑑𝑦𝑑𝑥

=𝑒𝑥

𝑑𝑦𝑑𝑥

=0

𝑑𝑦𝑑𝑥

=0.41×1.5𝑥

>

>𝑑𝑦𝑑𝑥

=0.69×2𝑥 >𝑑𝑦𝑑𝑥

=0.92×2.5𝑥 >

>𝑦=3𝑥𝑑𝑦𝑑𝑥

=1.10×3𝑥 >

𝑦=3.5𝑥𝑑𝑦𝑑𝑥

=1.25×3.5𝑥 >

is known as Euler’s Constant.

It is one of the five most fundamental constants in mathematics (0, 1, , , ) and we will explore it in the coming slides.

But for the purposes of this chapter, the most important thing to appreciate is that:

𝑦=𝑒𝑥→𝑑𝑦𝑑𝑥

=𝑒𝑥

Although any function of the form is known as an exponential function, is known as “the” exponential function.Yeah, it’s that big a deal…

You can find the exponential function on your calculator, to the right (above the “ln” key)

Bernoulli’s Compound Interest Problem(This won’t be examined)

You have £1. If you put it in a bank account with 100% interest, how much do you have a year later? What if the interest is split into 2 instalments of 50% interest, how much will I have? What about 3 instalments of 33.3%? And so on…

No. Instalments Money at Maturity

As becomes large, the money at maturity approaches

𝑒= lim𝑛→∞ (1+ 1𝑛 )

𝑛

Bernoulli is credited as first finding the value of ‘’, as the solution to this problem. Euler introduced the letter to represent the value for the base of a logarithm (which we’ll see).

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ExamplesSketch graphs of:

𝑦=𝑒2𝑥 𝑦=𝑒−𝑥 𝑦=12+4 𝑒

12𝑥

1

𝑥

𝑦

1

, so we’re just squaring the values. Note that squaring values less than 1 make them smaller, so the line is below on the left.

𝑦=𝑒𝑥

𝑥

𝑦

1𝑥

𝑦

𝑦=12

. So start with graph of and reciprocate values.This is known as exponential decay. We have decay whenever the in .

The in the power doesn’t do anything terribly exciting except to make the line grow slower (as )

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4.5

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The price of a used car can be represented by the formula:

Where is the price in £s and is the age in years. Calculate:a) The new price: b) The value after 5 years: c) The long term eventual value of the car? As , will tend to 0. Thus

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d) Sketch the function.17000

1000?

Test Your Understanding

The population of Tiffin Boys since 1970 is modelled using the equation

Where is the number of years since 1970 and is the number of boys.

How boys were there in:a) 1970b) 2014c) Far into the futured) Sketch a graph to represent the number of Tiffin Boys over time.

1200

800

𝑥

𝑃

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Inverse of Exponentials

In C2, we learnt what the inverse is of an exponential function.

4 𝑥5 5√x 41024

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4 3𝑥 log3 x 481

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4 𝑒𝑥 loge x 454.59

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! is the natural log of and is written (or in ‘proper maths’, just simply . Although confusingly on a calculator, means base 10, not base )

Solving EquationsSolve the following:

E1

E2

E3

E4

E5

E6

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Test Your Understanding

Edexcel C3 June 2012 Q6

3

3

𝑦= 𝑓 (𝑥 )

𝑦=𝑓−1 (𝑥

)

e)????

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Exercises

Solve (giving exact solutions)

Sketch the following:

The price of a new car varies according to

where P is price and t is age in years.State its value when new:

Calculate its age after 5 years.£9098Find its age when the price falls below £5000

Exercise 3B

1

3

f

i

4

a

b

c

5State the value of .

Find the value of for which

State the range of the function.

Find the exact coordinate of the point where the graph intercepts the -axis, in terms of .

Find stating its domain.

Domain is (i.e. range of )

Solve

Solve

Solve

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