Accuracy of Tangent-Line Approximation for f(x) = x^2

[misterau]

Homework Statement

Let the function f be given by f (x) = x^2
(a) Determine the tangent line to the graph of f at x = 1. Denote this by y = g (x) .
(b) Let \epsilon be a positive number. Solve the inequality|f (x) - g (x)| <\epsilon
(c) What does part b) tell us about the accuracy of the tangent-line approximation to f ?
Illustrate your answer by using the values \epsilon = 0.01 and \epsilon = 0.0001.

Homework Equations

The Attempt at a Solution

(a) For y = mx + b
f'(x) = 2*x
f'(1) = 2*1 = 2
b =1 -2(1)
= -1
g(x) = 2x -1
(b) This where I am having problems
|f (x) - g (x)| = | x^2 -2*x +1| = |(x-1)(x-1)|
so
|(x-1)(x-1)|<\epsilon
|(x-1)(x-1)|<K|(x-1)|<\epsilon
assuming |x-1|<K
Then I realized I can not make this assumption..
I am not sure what I can assume to get: |x-1| < (\epsilon/k)
Now do I get rid of the other |x-1|?
Basically confused, I find these Definition of limits Q's difficult.

 

[Dick]

You've got |x-1|^2<epsilon. How about taking the square root of each side? In the solution |x-1|<K, K should be a function of epsilon.

 

[misterau]

You've got |x-1|^2<epsilon. How about taking the square root of each side? In the solution |x-1|<K, K should be a function of epsilon.

Oh yeah didn't think of that! Thanks!