MA107 Precalculus Algebra
17 January 2008
Difference Quotient Examples
1. Compute f(x+h)−f(x)
hwhere f(x) = √x
f(x+h)−f(x)
h=√x+h−√x
h
=√x+h−√x
h
√x+h+√x
hMultiply by conjugate
=x+h−x
h(√x+h+√x)
=h
h(√x+h+√x)
=1
(√x+h+√x)
Done, because I could let h=0and not divide by 0.
2. Compute f(x+h)−f(x)
hwhere f(x) = 1
x
f(x+h)−f(x)
h=
1
x+h−1
x
hEvaluated f(x+h)and f(x)
= ( 1
h)( 1
x+h−1
x)Factored out 1
h
= ( 1
h)( 1
x+h(x
x) − 1
x(x+h
x+h)) Multiply by “1” to force denominators to match
= ( 1
h)( x
x(x+h)−x+h
x(x+h))
= ( 1
h)x−(x+h)
x(x+h)Subtracted the fractions
= ( 1
h)x−x−h
x(x+h)Distributed the negative
= ( 1
h)−h
x(x+h)
=−h
hx(x+h)Multiplied the 1
h
=−1
x(x+h)Cancelled out the ‘h’
1