Chapter 34

Introduction to

differentiation

34.1 Introduction to calculus

Calculusis a branch of mathematics involving or lead-
ing to calculations dealing with continuously varying
functions such as velocity and acceleration, rates of
change and maximum and minimum values of curves.
Calculus has widespread applications in science and
engineering and is used to solve complicated problems
for which algebra alone is insufficient.
Calculus is a subject that falls into two parts:

(a) differential calculus(ordifferentiation),

(b) integral calculus(orintegration).

This chapter provides an introduction to differentiation
andappliesdifferentiationtoratesofchange.Chapter35
introduces integration and applies it to determine areas
under curves.
Further applications of differentiation and integration
are explored inEngineering Mathematics(Bird, 2010).

34.2 Functional notation

In an equation such asy= 3 x^2 + 2 x− 5 ,yis said to be
a function ofxand may be written asy=f(x).
An equation written in the form f(x)= 3 x^2 + 2 x− 5
is termed functional notation.Thevalueoff(x)
whenx=0 is denoted byf( 0 ), and the value off(x)
whenx=2 is denoted byf( 2 ), and so on. Thus, when
f(x)= 3 x^2 + 2 x−5,

f( 0 )= 3 ( 0 )^2 + 2 ( 0 )− 5 =− 5

and f( 2 )= 3 ( 2 )^2 + 2 ( 2 )− 5 = 11 ,and so on.

Problem 1. Iff(x)= 4 x^2 − 3 x+2, find

f( 0 ),f( 3 ),f(− 1 )andf( 3 )−f(− 1 )

f(x)= 4 x^2 − 3 x+ 2

f( 0 )= 4 ( 0 )^2 − 3 ( 0 )+ 2 = 2

f( 3 )= 4 ( 3 )^2 − 3 ( 3 )+ 2 = 36 − 9 + 2 = 29

f(− 1 )= 4 (− 1 )^2 − 3 (− 1 )+ 2 = 4 + 3 + 2 = 9

f( 3 )−f(− 1 )= 29 − 9 = 20

Problem 2. Given thatf(x)= 5 x^2 +x−7,

determine (a)f(− 2 ) (b)f( 2 )÷f( 1 )

(a) f(− 2 )= 5 (− 2 )^2 +(− 2 )− 7 = 20 − 2 − 7 = 11

(b) f( 2 )= 5 ( 2 )^2 + 2 − 7 = 15

f( 1 )= 5 ( 1 )^2 + 1 − 7 =− 1

f( 2 )÷f( 1 )=

15

− 1

=− 15

Now try the following Practice Exercise

PracticeExercise 131 Functional notation

(answers on page 354)

1. If f(x)= 6 x^2 − 2 x+1, find f( 0 ),f( 1 ),
   f( 2 ),f(− 1 )and f(− 3 ).


2. If f(x)= 2 x^2 + 5 x−7, find f( 1 ),f( 2 ),
   f(− 1 ),f( 2 )−f(− 1 ).

DOI: 10.1016/B978-1-85617-697-2.00034-X