Calculus and Its Applications, Chapter 1, 1.3, Section 1.3, Problem 4

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,

$
\begin{equation}
\begin{aligned}
f(x + h) - f(x) &= -5x^2 - 10xh - 5h^2 - (-5x^2)\\
\\
&= -5x^2 - 10xh - 5h^2 + 5x^2\\
\\
&= -10xh - 5h^2
\end{aligned}
\end{equation}
$


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}
$