How can I find the error and percentage error in a Linear Approximation?

Question

f(x) = sqrt(3+x)
a = 6
delta(x) = 1

I get delta(f) = 1/2 * 9^(-1/2), but I don't understand how you can get the error since the formula says:
Error: abs( delta(f) - f '(a)*delta(x) )

I keep getting 0 as the error, but I know it isn't correct.

Answer 1

Linear approximation is L = f(a) + f'(a)(x - a)
f(6) = 3, f'(x) = 1/(2sqrt(3+x)), f'(6) = 1/6, so finally
L(x) = 3 + (x - 6)/6 = 2 + x/6
At a - 1 = 5 we have L(5) - f(5) = 17/6 - sqrt(8) = 0.0049
At a + 1 = 7 we have L(7) - f(7) = 19/6 - sqrt(10) = 0.0043

Answer 2

It looks as if you may be taking the delta of f' when you should be taking the delta of f (possibly among other errors).

When x changes from 6 to 7, f changes from 3 to sqrt(10).

Also -- SOMETIMES the error happens to be zero. There generally would have to be an inflection point in the curve nearby for that to happen, but it's possible.

Answer 3

wrong way