How to Find the Radius of a Disk for Delta-Epsilon Proof?

[dtl42]

Homework Statement

Let f(x,y)=2x+3y.
Let $$\epsilon$$ be any positive number. Show that there is a disk with center (1,1) such that whenever P is in that disk, $$|f(P)-5|< \epsilon$$. (Give $$\delta$$ as a function of $$\epsilon$$.)

Homework Equations

None.

The Attempt at a Solution

Um, I tried to rewrite stuff in a form that's needed, but I can't really get anything. My trouble with these problems is setting everything up and then rearranging it cleverly to get what we need.

 

[statdad]

$$|f(x,y) - 5| = |(2x-2)+(3y-3)| \le 2|x-1| + 3|y-3|$$

so now ...

 

[dtl42]

So that is epsilon? How would we find the radius of the disk then?

 

[HallsofIvy]

$$|f(x,y) - 5| = |(2x-2)+(3y-3)| \le 2|x-1| + 3|y-3|$$

so now ...

So that is epsilon? How would we find the radius of the disk then?

The form statdad gave gives you almost immediately the dimensions of a rectangle that will work. Can you find the radius of a disk that will be contained in that rectangle?

By the way, you say "the" disk. You are only asked to find the radius of "a" disk. There are an infinite number that will work.