Question

Is g(x) differentiable at 0? Is g(x) differentiable at 2?...

Answer 1

2x if x is less than or equal to 0
2-x^(2) if 0<x<2
2-x if x is greater than or equal to 2

Answer 2

Piecewise function :) Thanks!

Answer 3

Hint: if g ' (x) is continuous at x = c, then g(x) is differentiable at x = c

Answer 4

Is it continuous at x=2?

Answer 5

if y = 2x, then y ' = ???

Answer 6

2

Answer 7

if y = 2 - x^2, then y ' = ???

Answer 8

2-2x

Answer 9

oh also, I should have stated that g(x) needs to be continuous at x = c to be differentiable at x = c (along with the hint I provided)

Answer 10

no just -2x

Answer 11

if y = 2 - x, then y ' = ???

Answer 12

but wouldn't it be 2x-x^(2)= 2(1)-2x^(2-1)

Answer 13

and the last one is -1?

Answer 14

wait, the original is 2x - x^2 ? or 2 - x^2 ?

Answer 15

Oh, I forgot the x.

Answer 16

Sorry about that.

Answer 17

ok so the second piece is 2x - x^2 correct?

Answer 18

just want to confirm

Answer 19

Yes.

Answer 20

ok one moment

Answer 21

I'm going to let p(x) be the piecewise function

f(x) will be the first piece (I've graphed in red)
g(x) is the second piece (graphed in blue)
h(x) is the third piece (in green)

see attached

Notice how the different pieces connect. So this function is definitely continuous at x = 0 and x = 2. It is continuous everywhere else as well.

Answer 22

now if we derive p(x), we have to derive everything that makes up p(x)

so we derive f(x) to get f ' (x), g to get g ', etc etc

this is shown in the next attachment

notice how p ' (x) is continuous at x = 0 (the red and blue pieces connect); however, p ' (x) is NOT continuous at x = 2. There's a jump from (2,-2) on the blue piece to (2,-1) on the green piece.

Because of this discontinuity on p ' (x) at x = 2, this means p(x) is NOT differentiable at x = 2

Answer 23

If you look back at the first attachment

http://assets.openstudy.com/updates/attachments/54484a4be4b0f84976800a4f-jim_thompson5910-1414025861146-1.png

you'll notice that at x = 2, there's a sharp point. This is usually a good indication the function isn't differentiable here. However, it's always a good idea to derive and find out.

Answer 24

Thank you!

Answer 25

you're welcome