Question

I have the this function: y=x^(1/2) and I need to calculate the average rate of change.
can someon help me?

Answer 1

I got \[\frac{ 1 }{ \Delta x }\]
but the answer in book is
\[\frac{ \Delta y }{ \Delta x }=\frac{ 1 }{ \sqrt{x0} + \sqrt{x0+ \Delta x}}\]

Answer 2

by average rate of change they mean the slope \( \frac{\Delta y}{\Delta x}\)

if you pick a point \(x_0 \) its \(y_0\) value will be \(\sqrt{x_0}\)
if you go a small distance \(\Delta x \) so that \(x_1=x_0 +\Delta x\)
and \(y_1= \sqrt{x_1}= \sqrt{x_0+\Delta x} \)
you can write the average rate of changes as
\[ \frac{y_1 -y_0}{x_1- x_0} = \frac{\sqrt{x_0+\Delta x} - \sqrt{x_0}}{\Delta x}\]

so far, this looks ugly, because we do not want a \(\Delta x \) in the denominator (we want to let \(\Delta x \) approach 0)
however, if you multiply top and bottom by the conjugate (note the plus sign)
\[ \sqrt{x_0+\Delta x} + \sqrt{x_0} \]
you get a nice result.

Answer 3

Thank you Phi. I have to study too much both, english and mathematics. 8)
thank you very much.

Answer 4

I understood the solution but how did you get \[\sqrt{x0+\Delta x} + \sqrt{x0}\] ?

Answer 5

\[ \frac{y_1 -y_0}{x_1- x_0} = \frac{\sqrt{x_0+\Delta x} - \sqrt{x_0}}{\Delta x} \cdot \frac{\sqrt{x_0+\Delta x} + \sqrt{x_0}}{\sqrt{x_0+\Delta x} + \sqrt{x_0}}\]
\[= \frac{x_0+\Delta x - x_0}{\Delta x \left(\sqrt{x_0+\Delta x} + \sqrt{x_0}\right)} \]
\[ = \frac{1}{ \sqrt{x_0+\Delta x} + \sqrt{x_0}} \]

I don't understand your question
**but how did you get
\( \sqrt{x0+\Delta x} + \sqrt{x0}\) ?

If you are asking why do we multiply top and bottom by that expression, it's because multiplying by a conjugate is a "known trick" (to those who know it) to get rid of the radicals

Answer 6

that was my question. You need to get some experience ( and many exercices) to find this solutions. At this point, this site is (almost) perfect, because here we can get help when we need it. Where else I would get help in mathematics on sunday? 8)

Phi, Thank you very much for your help. I hope you can help me many more times.

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