Understanding Engineering Mathematics

The functiong(x)is continuousand indeed is equivalent to

g(x)=

1

x− 1

forx> 2

Exercise on 14.4

Consider the following functions for 0<x<∞, and discuss whether or not they are
continuous for these values ofx.

(i) x+ 1 (ii)

1

x

(iii)

1

x− 1

(iv) sinx (v) lnx (vi)

x^2 − 1

x+ 1

(vii)

x+ 1

x^2 − 1

(viii)

x^2 − 4

x+ 2

(ix)

f(x)=

x^2 − 4

x− 2

x

= 2

= 4 x= 2

Answer

(i) C (ii) C (iii) D (iv) C (v) C

(vi) C (vii) D (viii) C (ix) C

14.5 The slope of a curve

The slope of a curve at a point is defined as the slope of the tangent at that point. It can be
evaluated by a limiting process illustrated in Figure 14.8 (We adopted a similar approach
in Chapter 8, using a different but equivalent notation (230

➤

).)

y = f(x)

y

0 a a+h x

Figure 14.8Definition of the derivative.

The value of the function atx=aisf(a), and the value atx=a+hisf(a+h),so
the slope of the extended chord (also called thesecant)joining these two points on the