Understanding Engineering Mathematics

21.Expand( 1 + 2 x)

3

(^2) as far as the term inx^2. How many terms of the series would be
required to give( 1. 02 )
3
(^2) correct to three decimal places? For what range of values of
xis the series valid?
22.Obtain terms up tox^4 in the Maclaurin’s series for the functions
(i) sin 2x (ii) ex
2
(iii)
1
1 − 3 x
(iv) ln( 1 + 2 x) (v) tanx (vi)
1
( 1 +x)^2
State the values ofxfor which the series are convergent.

14.13 Applications

1.Since the derivative is defined in terms of a limit, the whole of the theory of differ-
entiation can be based on limits and their properties. While an engineer may not need
to know much about the details of this theory, it is useful to have an idea of the basic
principles, since limits tell us when we can, for example, approximate a derivative by
a simple algebraic difference, for the purposes of numerical methods. This is important
in such things as numerical solution of differential equations, which is a common task
faced by the engineer.
Using the limit definition of the derivative, prove the rules of differentiation: sum,
product, quotient, and chain rule (see Chapter 8).

2.Show that ifhis small then near to a pointx=aany differentiable function,f(x),
can be approximated by

f(x)=f(a+h) f(a)+hf′(a)

This is alinear (inh)approximationtof(x)near tox=a. It is often used as a

basis for numerical differentiation. It essentially replaces the curvef(x)nearx=aby

a portion of the tangent – Figure 14.10.

y

0 a x

y = f(x)

f(a) + hf′(a)

f(a+h)

f(a)

a+h

Figure 14.10Linear approximation to a function.