Another Linear approximation question

[bcahmel]

Homework Statement

stimate Δf using the Linear Approximation and use a calculator to compute both the error and the percentage error.
f(x) =1/(1+x^2) , a = 3, Δx = 0.5

Homework Equations

f'(a)(x)
percentage error= abs(error) divided by actual value

The Attempt at a Solution

So first I got the derivative which is -2x/(1+x)^2.
Then I plugged the a value, 3 into it which came out to be -3/50 and then multiplied it by x, 0.5, to get my linear approximation of -0.3. I understand this part.
Now to find the error, I first have to find the actual value on the calculator. So I plugged 3.5 into the original function, 1/(1+3.5^2)= 1/13.25. Is this right so far?
Plugging just 3 into the equation I get 1/10.

Now 1/13.25- 1/10 is about -.0245. This is the error, I think...

and percentage error would be .0055/-.0245=22.44% Seems high..

Basically I'm confused on the error part. I would be really grateful if anyone could straighten out how to do this!

 

[lanedance]

shouldn't the derivative be -2x/(1+x^2)^2

 

[vela]

The linear approximation gives you an estimate of f(3.5), so the error you want to calculate is

$$\frac{f_\mathrm{est}(3.5)-f(3.5)}{f(3.5)}$$

 

[bcahmel]

thanks lanedance, I typed it wrong on the computer- but you're right that is the derivative.

And vela, that's makes sense. So I think I did the error part right!

 

[vela]

The error was something like 7% or so, not 22%, so you should recheck what you did.