What is the smallest value of δ that satisfies the given graph and equation?

[FritoTaco]

Homework Statement

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

if: $|x-1|<δ$ then: $|x^2-1|<\dfrac{1}{2}$.

(Round your answer down to three decimal places.)

Homework Equations

No equations used.

The Attempt at a Solution

I need to find the smallest value of δ

$x^2=0.5$

$\sqrt{x^2}=\sqrt{0.5}$

$x=0.707106781$

$x^2=1.5$

$\sqrt{x^2}=\sqrt{1.5}$

$|x-1| ---> |8-0.0701067|=7.292893$

$|x-1| ---> |8-1.224744871|=6.775255$

Now I would pick the smaller value and round:

$6.775$

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

 

[Mark44]

Homework Statement

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

if: $|x-1|<δ$ then: $|x^2-1|<\dfrac{1}{2}$.

(Round your answer down to three decimal places.)

Homework Equations

No equations used.

The Attempt at a Solution

I need to find the smallest value of δ

$x^2=0.5$

$\sqrt{x^2}=\sqrt{0.5}$

$x=0.707106781$

$x^2=1.5$

$\sqrt{x^2}=\sqrt{1.5}$

$|x-1| ---> |8-0.0701067|=7.292893$

$|x-1| ---> |8-1.224744871|=6.775255$

Now I would pick the smaller value and round:

$6.775$

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

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.

 

[FritoTaco]

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.

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!

 

[Mark44]

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!

That's why Greg pays us the big bucks!

Oh, wait, we don't get paid at all!

 

[Ray Vickson]

Homework Statement

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

if: $|x-1|<δ$ then: $|x^2-1|<\dfrac{1}{2}$.

(Round your answer down to three decimal places.)

Homework Equations

No equations used.

The Attempt at a Solution

I need to find the smallest value of δ

$x^2=0.5$

$\sqrt{x^2}=\sqrt{0.5}$

$x=0.707106781$

$x^2=1.5$

$\sqrt{x^2}=\sqrt{1.5}$

$|x-1| ---> |8-0.0701067|=7.292893$

$|x-1| ---> |8-1.224744871|=6.775255$

Now I would pick the smaller value and round:

$6.775$

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

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".

 

[FritoTaco]

Well this is what happens:

$|x-1|--->|1-0.0701067|=0.2928932$

$|x-1|--->|1-1.2247448|=0.2247448$

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.