Calculating the Difference Quotient

Problem 254: Consider $f(x) = 3x^3 + 3x - 38$ . Calculate

$\frac{f(a+h) - f(a)}{h}.$

Solution: One can compute the pieces of the quotient first and then put it all together.

$f(a) = 3a^3 + 3(a) - 38=3a^2 + 3a - 38$
$f(x) = 3x^3 + 3x - 28$

$\begin{align*} f(a +h) & = 3(a+h)^3 + 3(a + h) - 38 \\ & = 3(a + h)^2(a+h) + 3a + 3h - 38 \\ & = 3(a+h)(a^2 + 2ah + h^2) + 3a + 3h - 38 \\ & = (3a + 3h)(a^2 + 2ah + h^2) + 3a + 3h - 38 \\ & = 3a^3 + 9a^2h + 9ah^2 + 3a + 3h - 38 \end{align*}$

Putting it all together we have.

$\begin{align*} \frac{f(a+h) - f(a)}{h} & = \frac{3a^3 + 9a^2h + 9ah^2 + 3a + 3h - 38 - (3a^2 + 3a - 38)}{h} \\ & = \frac{3a^3 + 9a^2h + 9ah^2 + 3a + 3h - 38 - 3a^2 - 3a + 38}{h} \\ & = \frac{9a^2h + 9ah + 3h}{h} \\ & = 9a^2 + 9a + 3.\end{align*}$

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