Cambridge Additional Mathematics

Surds, indices, and exponentials (Chapter 4) 123

We have seen that the simplest exponential functions are of

the form f(x)=bx where b> 0 , b 6 =1.

Graphs of some of these functions are shown alongside.

We can see that for all positive values of the baseb, the

graph is always positive.

Hence bx> 0 for all b> 0.

There are an infinite number of possible choices for the base

number.

However, where exponential data is examined in science, engineering, and finance, the base e¼ 2 : 7183

is commonly used.

eis a special number in mathematics. It is irrational like¼, and just as¼is the ratio of a circle’s circumference

to its diameter,ealso has a physical meaning. We explore this meaning in the followingDiscovery.

Discovery 2 Continuous compound interest

A formula for calculating the amount to which an investment grows is un=u 0 (1 +i)n where:

un is the final amount, u 0 is the initial amount,

i is the interest rate per compounding period,

n is the number of periods or number of times the interest is compounded.

We will investigate the final value of an investment for various values ofn, and allownto get extremely

large.

What to do:

1 Suppose $ 1000 is invested for one year at a fixed rate of6%per annum. Use your calculator to

find the final amount ormaturing valueif the interest is paid:

a annually (n=1, i=6%=0:06) b quarterly (n=4, i=6% 4 =0:015)

c monthly d daily

e by the second f by the millisecond.

2 Comment on your answers from 1.

3 If r is the percentage rate per year, t is the number of years, and N is the number of interest

payments per year, then i=

r

N

and n=Nt.

The growth formula becomes un=u 0

³

1+

r

N

́Nt

.

If we let a=

N

r

, show that un=u 0

h³

1+

1

a

́airt

.

H The natural exponential

x

H

1

O x

y

y=1 2.x

y=2x

y = (0.2)x y=5x

y = (0.5)x

4037 Cambridge

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Y:\HAESE\CAM4037\CamAdd_04\123CamAdd_04.cdr Tuesday, 14 January 2014 10:28:28 AM BRIAN