Question

the difference quotient of f(x)=4x^2-x+2

Answer 1

Difference quotient: \(\dfrac{f(x+h)-f(x)}{h}\)

If \(f(x)=4x^2-x+2\), then
\[\begin{align*}f(x+h)&=4(x+h)^2-(x+h)+2\\
&=4\left(x^2+2xh+h^2\right)-x-h+2\\
&=4x^2+8xh+4h^2-x-h+2
\end{align*}\]
So you have
\[\begin{align*}\frac{f(x+h)-f(x)}{h}&=\frac{\left(4x^2+8xh+4h^2-x-h+2\right)-\left(4x^2-x+2\right)}{h}\\
&=\frac{8xh+4h^2-h}{h}\\
&=8x+4h-1

\end{align*}\]

Answer 2

thanks