Calculus and Its Applications, Chapter 1, 1.3, Section 1.3, Problem 10

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

a.) For $\displaystyle f(x) = 2x + 3$
$\displaystyle f(x + h) = 2(x + h) + 3 = 2x + 2h + 3$
Then,

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


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

b.)


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