Using the difference quotient

• Jan 28th 2010, 12:49 PM
jtowery
Using the difference quotient

Let $f(x)=(2/x)-4$, then the expression

$[f(x+h)-f(x)]/h$ can be written in the form

$(A)/(Bx^2+Cxh)$, where A, B, and C are constants.

Find A, B, and C.

Any help would be greatly appreciated. Thanks!
• Jan 28th 2010, 12:59 PM
1005
Quote:

Originally Posted by jtowery

Let $f(x)=(2/x)-4$, then the expression

$[f(x+h)-f(x)]/h$ can be written in the form

$(A)/(Bx^2+Cxh)$, where A, B, and C are constants.

Find A, B, and C.

Any help would be greatly appreciated. Thanks!

$\frac{\frac{2}{x+h}-4-(\frac{2}{x}-4)}{h}$
the 4s cancel
$\frac{\frac{2}{x+h}-\frac{2}{x}}{h}$
divide the h into each part
$\frac{2}{h(x+h)}-\frac{2}{hx}$
make each denominator the same
$\frac{2hx}{h^2x(x+h)}-\frac{2h(x+h)}{h^2x(x+h)}$
combine
$\frac{2hx-2h(x+h)}{h^2x(x+h)}$
1 h on top cancels with 1 h on bottom + factor out 2
$\frac{2(x-x-h)}{hx(x+h)}$
x's on top cancel leaving another h to cancel with the h below
$\frac{-2}{x(x+h)}$
expand denominator
$\frac{-2}{x^2+xh}$
$\therefore a = -2$
$b=1$
$c = 1$
• Jan 28th 2010, 01:11 PM
jtowery
nvm?
• Jan 28th 2010, 01:13 PM
pickslides
Quote:

Originally Posted by 1005
nvm

??

$
\frac{f(x+h)-f(x)}{h}=\frac{\left(\frac{2}{x+h}-4\right)-\left(\frac{2}{x}-4\right)}{h}=\dots
$

Now simplify.
• Jan 28th 2010, 01:16 PM
jtowery
Awesome. Thank you both so much