Calculating the Difference Quotient II

Problem 255: Consider $f(x) = \sqrt{3x - 8}$ . Compute

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

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

$\begin{align*} f(a + h) & = \sqrt{3(a+h) - 8} \\ & = \sqrt{3a + 3h - 8} \end{align*}$

Putting all together.

$\begin{align*}\frac{f(a+h) - f(a)}{h} & = \frac{ \sqrt{3a + 3h - 8} - \sqrt{3a - 8}}{h} \end{align*}$

Since we can’t simplify further, we leave our answer like that.

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