Question

Find the values of m and b that make f differentiable everywhere

Answer 1

You need to first make it continuous

Answer 2

Then make it smooth

Answer 3

that is
left limit=right limit as we approach 12

and

left slope limit=right slope limit as we approach 12

Answer 4

how do I set this up?

Answer 5

you will have a system of equations
\[\lim_{x \rightarrow 12^-}f(x)=\lim_{x \rightarrow 12^+}f(x) \\ \lim_{x \rightarrow 12^-}f'(x)=\lim_{x \rightarrow 12^+}f'(x) \\ \]

Answer 6

so am I taking the derivative of both pieces?

Answer 7

you will need to do that yep

Answer 8

\[\lim_{x \rightarrow 12-} = 2x\]

Answer 9

I do not know how to do the second part

Answer 10

so it looks like we are trying to do the second line right?
but we will go back to the first line of:
\[\lim_{x \rightarrow 12^-}f(x)=\lim_{x \rightarrow 12^+}f(x) \\ \lim_{x \rightarrow 12^-}f'(x)=\lim_{x \rightarrow 12^+}f'(x) \\
\]

\[\lim_{x \rightarrow 12^-}(x^2)'=\lim_{x \rightarrow 12^-}2x=? \\ \lim_{x \rightarrow 12^+}(mx+b)'=?\]

What if I asked you to find the derivative of y=2x+3?

Answer 11

would it be \[2 + 0\]

Answer 12

yes so the derivative of (mx+b) is just the same

Answer 13

except not 2 but the thing that is in front of the x

Answer 14

(mx+b)'=m is what I'm saying

Answer 15

m and b were just just constants

Answer 16

just like 2 and 3 are in the example I chose

Answer 17

\[\lim_{x \rightarrow 12^-}(x^2)'=\lim_{x \rightarrow 12^-}2x=? \\ \lim_{x \rightarrow 12^+}(mx+b)'=\lim_{x \rightarrow 12^+}m=?\]

Answer 18

can you take care of that limit part?

Answer 19

let me see, \[\lim_{x \rightarrow 12-} 2x = 24\] and I do see a place to put 12 when x is gone, is m = 0?

Answer 20

it doesn't matter what x approaches y will always remain m

Answer 21

that is \[\lim_{x \rightarrow a}m=m\]

Answer 22

okay, but when I am filling in the answer m = 24

Answer 23

yes m is 24 great stuff you recalled the equality above that we must have in order for the function to be smooth at x=12

Answer 24

we also need the function to be continuous at x=12

Answer 25

we needed \[\lim_{x \rightarrow 12^-}f(x)=\lim_{x \rightarrow 12^+}f(x)\]

Answer 26

so work out both limits on both sides and solve for b

Answer 27

and you can replace the m here with 24 since we already found that constant number

Answer 28

left side is 144 and right hand side is 288 so \[144 = 288 +b\]

Answer 29

am I correct so far

Answer 30

well I didn't do 12*24
I did 12*12*2=144*2
144=144*2+b

Answer 31

so b is - 144?

Answer 32

yes because 144-2(144)=-1(144)=-144

Answer 33

Awesome, thanks for you patience.

Answer 34

I didn't feel like using a calculator :P

Answer 35

and np