Question

Find the values of m and b that make f differentiable everywhere. f(x)=x^2, if x2. I know how to do these types of problems with only one variable, but am lost with two.

Answer 1

ok so we do need it to be continuous to be differentiable

Answer 2

so we need the left limit to equal the right limit as we approach x=2

Answer 3

\[\lim_{x \rightarrow 2^-}f(x)=\lim_{x \rightarrow 2^+}f(x) \\ \lim_{x \rightarrow 2^-}x^2=\lim_{x \rightarrow 2^+}(mx+b) \\ \]

Answer 4

evaluate the limit on both sides

Answer 5

and since you want it to also need the function to be smooth in order to be differentiable then you need the left derivative to equal the right derivative

Answer 6

\[\lim_{x \rightarrow 2^-}f'(x)=\lim_{x \rightarrow 2^+}f'(x) \\ \lim_{x \rightarrow 2^-}2x=\lim_{x \rightarrow 2^+}m\]

Answer 7

aha. so we will have a system of equations

Answer 8

you therefore have a system of equations to solve

Answer 9

lol yeah

Answer 10

thanks a lot

Answer 11

do you think you need further assistance with this one

Answer 12

nah i got it thanks man

Answer 13

I probably need assistance in slowing down my typing so my sentences will actually make sense

Answer 14

or woman lol

Answer 15

nah i prefer not waiting forever

Answer 16

hmmm are you sure?

Answer 17

i understand them at least haha

Answer 18

oh yeah, you are sure

Answer 19

I could be an alien without a sexual orientation

Answer 20

am i sure?

Answer 21

a unicorn with a rainbow is most likely female

Answer 22

like a truck with mudflaps

Answer 23

Maybe it is to deceive you

Answer 24

yeah thats why i added that haha. But you never know nowadays