Originally Posted by

**Kakariki** I am unsure as to which forum to put this in. I took a shot and put it here, apologies if wrong.

Okay, to the question I am having trouble with.

"Find the average rate of change of $\displaystyle f(x) = \sqrt{x + 11}$ with respect to x from $\displaystyle x = 5$ to $\displaystyle x = 5 +h$. [4 marks]"

My attempt to answer:

$\displaystyle f(5) = \sqrt{5 + 11}$

$\displaystyle f(5) = \sqrt{16}$

$\displaystyle f(5) = 4 $

$\displaystyle f(5 + h) = \sqrt{5 + h + 11}$

$\displaystyle f(5 + h) = \sqrt{16 + h}$

$\displaystyle f(5 + h) = 4 + \sqrt{h}$

*I am unsure if I am correct in square rooting the 16, so having 4 + the square root of h.*

Then I use the $\displaystyle \frac{y2 - y1}{x2 - x1}$ formula to get the slope of the secant *rate of change*:

$\displaystyle \frac{4 + \sqrt{h} - 4}{5 - 5 + h}$

$\displaystyle = \frac{\sqrt{h}}{h}$

That's my final solution. My problem with this solution is when I sub a number in for h, I get a different answer from running the numbers through, and simply going *square root of the number divided by the number*.

Example:

Let's pretend h is 20.

So I have $\displaystyle f(5) = 4$.

I can sub in 20 for h:

$\displaystyle f(5 + 20) = \sqrt{5 + 20 + 11} $

$\displaystyle f(25) = \sqrt{36}$

$\displaystyle f(25) = 6 $

Run that through the slope formula:

$\displaystyle \frac{6 - 4}{20 - 5}$

$\displaystyle = \frac{2}{5} $

$\displaystyle = 0.4$

Then, using my average rate of change thing: $\displaystyle \frac{\sqrt[h]][h]]$ I get:

$\displaystyle \frac{\sqrt{20}}{20}$

$\displaystyle = 0.223606797$

which is a completely different answer. So I must be doing something wrong here.

Okay, I tried using 25 for h in the $\displaystyle \frac{\sqrt{h}}{h}$ and it gives the same ansewr as running the math through. So it is not technically h, but it is h + 5, so is this actually right???