Monica's question at Yahoo Answers regarding Linear Approximation

[MarkFL]

Here is the question:

Linear Approximation?

Find the linear approximation of the function
g(x) = fifth sqrt(1 + x) at a = 0.

Use it to approximate the numbers
fifth sqrt(0.95) and fifth sqrt(1.1)

I have posted a link there to this thread so the OP can view my work.

 

[MarkFL]

Hello Monica,

Consider the approximation:

\(\displaystyle \frac{\Delta g}{\Delta x}\approx\frac{dg}{dx}\)

Now this implies:

\(\displaystyle \Delta g\approx\frac{dg}{dx}\Delta x\)

We may rewrite $\Delta g$ as follows:

\(\displaystyle g\left(x+\Delta x \right)-g(x)\approx\frac{dg}{dx}\Delta x\)

And so we have:

\(\displaystyle g\left(x+\Delta x \right)\approx\frac{dg}{dx}\Delta x+g(x)\)

Now, with $g$ defined as:

\(\displaystyle g(x)\equiv x^{\frac{1}{5}}\implies \frac{dg}{dx}=\frac{1}{5}x^{-\frac{4}{5}}\)

And with $x=1$, our formula becomes:

\(\displaystyle g\left(1+\Delta x \right)\approx\frac{1}{5}\Delta x+1\)

And so for:

i) \(\displaystyle \Delta x=-0.05\)

We have:

\(\displaystyle g\left(1-0.05 \right)\approx\frac{-0.05}{5}+1\)

\(\displaystyle \sqrt[5]{0.95}\approx0.99\)

ii) \(\displaystyle \Delta x=0.1\)

We have:

\(\displaystyle g\left(1+0.1 \right)\approx\frac{0.1}{5}+1\)

\(\displaystyle \sqrt[5]{1.1}\approx1.02\)