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Use the Given Graph to find δ

  1. Jan 21, 2017 #1
    1. The problem statement, all variables and given/known data

    Use the given graph (see attachment) of [itex]f(x) = x^2[/itex] to find a number δ (delta) such that

    if: [itex]|x-1|<δ[/itex] then: [itex]|x^2-1|<\dfrac{1}{2}[/itex].

    (Round your answer down to three decimal places.)

    2. Relevant equations

    No equations used.

    3. The attempt at a solution

    I need to find the smallest value of δ

    [itex]x^2=0.5[/itex]

    [itex]\sqrt{x^2}=\sqrt{0.5}[/itex]

    [itex]x=0.707106781[/itex]

    [itex]x^2=1.5[/itex]

    [itex]\sqrt{x^2}=\sqrt{1.5}[/itex]

    [itex]|x-1| ---> |8-0.0701067|=7.292893[/itex]

    [itex]|x-1| ---> |8-1.224744871|=6.775255[/itex]

    Now I would pick the smaller value and round:

    [itex]6.775[/itex]

    I use WebAssign, and it says I got it wrong. I don't know what I should try.
     

    Attached Files:

  2. jcsd
  3. Jan 21, 2017 #2

    Mark44

    Staff: Mentor

    Your value for δ is way too large. You want an interval around x = 1 on the x-axis so that the y values on the graph are between .5 and 1.5.
     
  4. Jan 21, 2017 #3
    Uh oh, I have no idea why i was subtracting 8 when it says | x - 1 | in the question. Thanks for saying that Mark, I got 0.225 and got it correct. Thank you!
     
  5. Jan 21, 2017 #4

    Mark44

    Staff: Mentor

    That's why Greg pays us the big bucks!

    Oh, wait, we don't get paid at all!
     
  6. Jan 21, 2017 #5

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    Why bother finding the smallest possible value of ##\delta > 0##? The question does not tell you to do that; it just tells you to find a ##\delta## that "works".
     
  7. Jan 21, 2017 #6
    Well this is what happens:

    [itex]|x-1|--->|1-0.0701067|=0.2928932[/itex]

    [itex]|x-1|--->|1-1.2247448|=0.2247448[/itex]

    Now I rounded the first one 0.293 and it says:

    "Please try again. For finding δ such that |x − a| < δ implies |x2 − a2| < b, start by finding the solutions to |x2 − a2| = b. Choose δ so that neither of these solutions are at a distance smaller than δ of a."

    So that's why I had to choose the second line instead.
     
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