Calculus and Its Applications, Chapter 1, 1.3, Section 1.3, Problem 2

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

a.) For $f(x) = 5x^2$
$f(x + h) = 5(x + h)^2 = 5x^2 + 10xh + 5h^2$
Then,
$f(x + h) - f(x) = 5x^2 + 10xh + 5h^2 - 5x^2 = 10xh + 5h^2$
Thus,
$\displaystyle \frac{f(x+h)-f(x)}{h} = \frac{10xh+5h^2}{h} = \frac{h(10x+5h)}{h} = 10x + 5h$

b.)


$
\begin{array}{|c|c|c|}
\hline
x & h & \displaystyle \frac{f(x+h)-f(x)}{h} \\
\hline
5 & 2 & 60 \\
\hline
5 & 1 & 55 \\
\hline
5 & 0.1 & 50.5 \\
\hline
5 & 0.01 & 50.05 \\
\hline
\end{array}
$

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