Example: Let 𝑓(𝑥) = 𝑥 3 − 𝑥. Calculate the difference quotient
Solution:
Way 1: find
𝑓(5+ℎ)−𝑓(5)
ℎ
𝑓(5+ℎ)−𝑓(5)
ℎ
for ℎ = 0.1.
and then plug in ℎ.
𝑓(5 + ℎ) − 𝑓(5) [(5 + ℎ)3 − (5 + ℎ)] − [(5)3 − 5]
=
ℎ
ℎ
=
=
=
=
[(53 + 3 ∙ 52 ℎ + 3 ∙ 5ℎ2 + ℎ3 ) − 5 − ℎ] − [125 − 5]
ℎ
[125 + 3 ∙ 25ℎ + 15ℎ2 + ℎ3 − 5 − ℎ] − [120]
ℎ
[120 + 74ℎ + 15ℎ2 + ℎ3 ] − 120
ℎ
74ℎ + 15ℎ2 + ℎ3
= 74 + 15ℎ + ℎ2
ℎ
Plug in ℎ = 0.1:
74 + 15(0.1) + (0.1)2 = 74 + 1.5 + .01 = 75.51
Way 2: plug ℎ into
𝑓(2+ℎ)−𝑓(2)
ℎ
and then compute that fraction
𝑓(5 + 0.1) − 𝑓(5) 𝑓(5.1) − 𝑓(5) [(5.1)3 − 5.1] − [(5)3 − 5]
=
=
0.1
0.1
0.1
=
[127.551] − [120] 7.551
=
= 75.51
0.1
0.1