Calculus and Its Applications, Chapter 1, 1.3, Section 1.3, Problem 6

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

a.) For $f(x) = x^2 - x$
$f(x + h) = (x + h)^2 - (x + h) = x^2 + 2xh + h^2 - x - h$
Then,

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


Thus,
$\displaystyle \frac{f(x + h) - f(x)}{h} = \frac{2xh + h^2 - h}{h} = \frac{h(2x + h - 1)}{h} = 2x + h - 1$

b.)


$
\begin{array}{|c|c|c|}
\hline
x & h & \displaystyle \frac{f(x+h)-f(x)}{h} \\
\hline
5 & 2 & 11 \\
\hline
5 & 1 & 10 \\
\hline
5 & 0.1 & 9.1 \\
\hline
5 & 0.01 & 9.01 \\
\hline
\end{array}
$

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