Understanding Engineering Mathematics

curve is

f(a+h)−f(a)

a+h−a

=

f(a+h)−f(a)

h

Ash→0, the chord approaches the tangent to the curve atx=a, so the slope of the
curve atx=ais

lim

h→ 0

f(a+h)−f(a)

h

An equivalent and possibly more useful form is

xlim→a

f(x)−f(a)

x−a

Note that whenh=0, orx=a, this is of the indeterminate form 0/0. It is called the
derivative off.x/atx=a(231

➤

). We use the notation

lim

x→a

(

f(x)−f(a)

x−a

)

=

df(x)

dx

atx=a

Problem 14.7

Determine the slope of the curvef.x/=xnat the pointx=a.

From above, the slope off(x)=xnatx=ais given by

lim

x→a

f(x)−f(a)

x−a

=lim

h→ 0

(

xn−an

x−a

)

=nan−^1

from Section 14.3.

Exercise on 14.5

Evaluate the slope of the functionf(x)= 3 x^3 using the limit definition.

Answer

9 x^2

14.6 Introduction to infinite series

The purpose of this section is to ease you gently into the topic of infinite series, before we
embark on a more formal treatment. How would you evaluate cosxforx= 0 .1radians?
Probably tap it out on your calculator. But suppose you wanted greater accuracy than your
calculator can give? Writing a short computer programme might be the answer. But how do
you instruct the computer? You can’t tell it to evaluate cosxfrom its geometrical definition
‘adjacent over hypotenuse’. The logic circuits of computers can only do very simple