Calculus and Its Applications, Chapter 1, 1.3, Section 1.3, Problem 12

For the function $\displaystyle f(x) = 12x^3$
(a) Determine the simplified form of the difference quotient
(b) Complete the table.

a.) For $\displaystyle f(x) = 12x^3$

$
\begin{equation}
\begin{aligned}
f(x+h) = 12 (x + h)^3 &= 12 \left[ x^3 + 3x^2 h + 3xh^2 + h^3 \right]\\
\\
&= 12x^3 + 36x^2 h + 36 xh^2 + 12h^3
\end{aligned}
\end{equation}
$


Then,

$
\begin{equation}
\begin{aligned}
f(x + h) - f(x) &= 12x^3 + 36x^2 h + 36xh^2 + 12h^3 - 12x^3\\
\\
&= 36x^2 h + 36xh^2 + 12h^3
\end{aligned}
\end{equation}
$


Thus,

$
\begin{equation}
\begin{aligned}
\frac{f(x +h)- f(x)}{h} &= \frac{36x^2h + 36xh^2 + 12h^3}{h}\\
\\
&= \frac{h \left( 36x^2 + 36xh + 12h^2 \right)}{h}\\
\\
&= 36x^2 + 36xh + 12h^2
\end{aligned}
\end{equation}
$


b.)


$
\begin{array}{|c|c|c|}
\hline
x & h & \displaystyle \frac{f(x+h)-f(x)}{h} \\
\hline
5 & 2 & 1308 \\
\hline
5 & 1 & 1092 \\
\hline
5 & 0.1 & 918.12 \\
\hline
5 & 0.01 & 901.8012 \\
\hline
\end{array}
$