Question

find the difference quotient of f, that is, find $\frac{f(x + h)-f(x)}{h}$, h $
eq$ 0, for the following function. be sure to

$f(x)=x^{2}-5x + 4$

Explanation:

Step1: Find f(x + h)

Substitute \(x+h\) into \(f(x)\):
\[

$$\begin{align*} f(x + h)&=(x + h)^2-5(x + h)+4\\ &=x^{2}+2xh+h^{2}-5x-5h + 4 \end{align*}$$

\]

Step2: Calculate f(x + h) - f(x)

\[

$$\begin{align*} f(x + h)-f(x)&=(x^{2}+2xh+h^{2}-5x-5h + 4)-(x^{2}-5x + 4)\\ &=x^{2}+2xh+h^{2}-5x-5h + 4-x^{2}+5x - 4\\ &=2xh+h^{2}-5h \end{align*}$$

\]

Step3: Find the difference quotient

\[

$$\begin{align*} \frac{f(x + h)-f(x)}{h}&=\frac{2xh+h^{2}-5h}{h}\\ &=\frac{h(2x + h-5)}{h}\\ &=2x+h - 5 \end{align*}$$

\]

Answer:

\(2x+h - 5\)