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Definition of limits Question

  1. Apr 6, 2009 #1
    1. The problem statement, all variables and given/known data
    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 [tex]\epsilon[/tex] be a positive number. Solve the inequality|f (x) - g (x)| <[tex]\epsilon[/tex]
    (c) What does part b) tell us about the accuracy of the tangent-line approximation to f ?
    Illustrate your answer by using the values [tex]\epsilon[/tex] = 0.01 and [tex]\epsilon[/tex] = 0.0001.
    2. Relevant equations



    3. 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)|<[tex]\epsilon[/tex]
    |(x-1)(x-1)|<K|(x-1)|<[tex]\epsilon[/tex]
    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| < ([tex]\epsilon[/tex]/k)
    Now do I get rid of the other |x-1|?
    Basically confused, I find these Definition of limits Q's difficult.
     
  2. jcsd
  3. Apr 6, 2009 #2

    Dick

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    Science Advisor
    Homework Helper

    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.
     
  4. Apr 6, 2009 #3
    Oh yeah didn't think of that!:smile: Thanks!
     
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