Calculus and Its Applications, Chapter 1, 1.3, Section 1.3, Problem 16

For the function $\displaystyle f(x) = x^2 - 3x + 5$
(a) Determine the simplified form of the difference quotient
(b) Complete the table.

a.) For $\displaystyle f(x) = x^2 - 3x + 5$

$
\begin{equation}
\begin{aligned}
f(x+h) &= (x +h)^2 - 3(x + h) + 5 \\
\\
&= x^2 + 2xh + h^2 - 3x - 3h + 5
\end{aligned}
\end{equation}
$


Then,

$
\begin{equation}
\begin{aligned}
f(x + h) - f(x) &= x^2 + 2xh + h^2 - 3x - 3h + 5 - (x^2 - 3x + 5)\\
\\
&= x^2 + 2xh + h^2 - 3x - 3h + 5 - (x^2 - 3x + 5) - x^2 + 3x - 5\\
\\
&= 2xh + h^2 - 3h
\end{aligned}
\end{equation}
$


Thus,

$
\begin{equation}
\begin{aligned}
\frac{f(x +h)- f(x)}{h} &= \frac{2xh + h^2 - 3h}{h}\\
\\
&= \frac{h(2x + h - 3)}{h}\\
\\
&= 2x + h - 3
\end{aligned}
\end{equation}
$


b.)


$
\begin{array}{|c|c|c|}
\hline
x & h & \displaystyle \frac{f(x+h)-f(x)}{h} \\
\hline
5 & 2 & 9\\
\hline
5 & 1 & 8 \\
\hline
5 & 0.1 & 7.10 \\
\hline
5 & 0.01 & 7.01 \\
\hline
\end{array}
$