Question

Calculus help please?

Answer 1

Find a relationship between a and b so that f is continuous at x = 2.

Answer 2

Just a sec, I'm getting the rest of the info.

Answer 3

http://imgur.com/gSCUUWZ

Answer 4

So I guess, find a relationship for a and b, so that lim as x -> 5x-10 = lim x->ax^2 + bx?

Answer 5

Ah, yes. Good.
For continuity, we need the pieces to connect together nicely.
So the left and right limits must agree at x=2,\[\Large\rm \lim_{x\to2^-}f(x)=\lim_{x\to2^+}f(x)\]

Answer 6

Is the equation the one I just typed? It doesn't show correctly

Answer 7

Notice that we'll also need continuity at x=1,\[\Large\rm \lim_{x\to1^-}f(x)=\lim_{x\to1^+}f(x)\]

Answer 8

\[\Large\rm \lim_{x\to2^-}f(x)=\lim_{x\to2^+}f(x)\]Yes, your use of the arrow notation was a little strange, but you have the "pieces" correct.\[\Large\rm \lim_{x\to2^-}ax^2+bx=\lim_{x\to2^+}5x-10\]

Answer 9

I see, will it be a pretty linear process from there, or is it not that simple?

Answer 10

Yes, you're correct.
We don't have any funny business going on.
So you simply plug x=2 into both sides, that takes care of the limiting process.

Answer 11

You'll have `one equation` involving `two unknowns`.
So you'll have to make use of the x->1 limits before you can make further progress :)

Answer 12

So, a(2)^2 + b(2) = 5(2) - 10

Answer 13

\[\large\rm 4a+2b=0\]Mmm k looks good so far.

Answer 14

so, b = (-2)a
or
a = -(1/2)b

Answer 15

So like I mentioned before,
Notice that you have `one equation` involving `two unknowns`.
So you can't actually do anything with it just yet.

You need to set up the limits for continuity at x=1.

Answer 16

To get the other equation

Answer 17

OHHH I get it.

Answer 18

So we need to do the same thing basically, but with the first two pieces?

Answer 19

Yes :) Good.

Answer 20

Alrighty. One sec.

Answer 21

Then we do substitution?

If a = 2 - b (from the lim x->1 equation)
and
a = (-1/2)b

then (-1/2)b = 2 - b

right?

Then the same for a?

Answer 22

a = -2, b = 4

Answer 23

These were your two equations?
4a + 2b = 0
a + b = 2

I think elimination was easier than substitution :)
But seems to have worked out, cool.

Answer 24

There is another version of this problem that you might run into later.

It will be worded like this:
Find a relationship between a and b so that f is `differentiable` at x = 2.

Answer 25

It will be largely the same process,
but you'll also have to make the slopes connect,
so you'll have the limit of the derivatives equal on each side as well.

Answer 26

Yes, they were. Sweet though.

Same deal, then?

Answer 27

Oh okay. Sweet thanks!

Answer 28

np