4.2 The process of differentiation 95
More generally, if the variable changes by ∆x, from p 1 = 1 xto q 1 = 1 x 1 + 1 ∆x, the
corresponding change in the function is
∆y 1 = 1 f(x 1 + 1 ∆x) 1 − 1 f(x) (4.3)
and the corresponding average rate of change is
(4.4)
EXAMPLE 4.1The general quadratic function,y 1 = 1 ax
21 + 1 bx 1 + 1 c:
At point P y
P1 = 1 f(x) 1 = 1 ax
21 + 1 bx 1 + 1 c
At point Q y
Q1 = 1 f(x 1 + 1 ∆x) 1 = 1 a(x 1 + 1 ∆x)
21 + 1 b(x 1 + 1 ∆x) 1 + 1 c
The change in the function on going from P to Q is therefore
= 1 (2ax 1 + 1 b)∆x 1 + 1 a(∆x)
2and the gradient of the line PQ is
(4.5)
0 Exercise 1
Figure 4.4 shows how the quantity∆y 2 ∆xchanges as the point Q is moved along the
curve towards P (as ∆xis decreased in magnitude).
∆
∆
∆
y
x
=++()2ax b a x
∆= − = +∆ + +∆ +
−++
y y y a x x b x x c ax bx c
QP()()
22
∆
∆
∆
∆
y
x
fx x fx
x
=
()()+−
P
Q
′
Q
tangentatp
.
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Figure 4.4