## clara1223 one year ago At x=3, is the function given by f(x)={x^2 when x<3 and 6x-9 when x>=3} continuous and/or differentiable? I know that the graph looks like a parabola but just continues at x=3, so I know it is continuous but would it be considered a sharp turn (would it be differentiable)?

1. clara1223

|dw:1442957985635:dw| That's what it looks like.

2. zepdrix

Oh these are kinda neat :o So ummmm

3. zepdrix

Ya so you have a line segment, and a curve. For continuous you need, what I like to call, connectedness. The pieces have to connect together. For differentiable you need, uh I guess we'll call it, smoothness. The slope from the left side needs to be equal to the slope from the right side.

4. zepdrix

So for continuity, you need:$\large\rm \lim_{x\to3^-}f(x)=\lim_{x\to3^+}f(x)$

5. zepdrix

The 3^- shows that we're approaching from the left side. When we're on the left side of 3, are we dealing with the line segment, or the parabola?

6. zepdrix

Oh you already figured out continuous :p Woops I didn't read the whole post lol

7. zepdrix

For differentiable, you need the slopes to be the same from the left and right,$\large\rm \lim_{x\to3^-}f'(x)=\lim_{x\to3^+}f'(x)$

8. zepdrix

Take some derivatives :o

9. clara1223

@zepdrix what do you mean? Like take the derivative and plug in x=2 and 4? or what?

10. zepdrix

The only value you'll end up plugging in is x=3. That's the only one that matters. When we're on the left side of 3, the function is $$\large\rm f(x)=x^2$$ So on the left side of 3, $$\large\rm f'(x)=2x$$. $\large\rm \lim_{x\to3^-}f'(x)=\lim_{x\to3^+}f'(x)$$\large\rm \lim_{x\to3^-}2x=\lim_{x\to3^+}f'(x)$ How bout from the right side of 3? the function $$\large\rm f(x)=?$$

11. clara1223

well if f'(x)=2x, wouldn't it just be 2x?

12. clara1223

Wait I don't think that was what you were asking, I'm confused.

13. zepdrix

We have a piece-wise function. When we're on the RIGHT of 3, our function is defined by a different piece.

14. clara1223

yes, I get that

15. clara1223

So would we take f'(x) of 6x-9?

16. clara1223

which would be 6

17. zepdrix

|dw:1442961574141:dw|Yes, good. We're using that other piece to define our function on the right side o f3.

18. zepdrix

Ugh, I got a new mouse. It messed up my tablet settings :c Grr I'll have to fix that later..

19. clara1223

Haha still completely legible, continue

20. clara1223

if f'(x)=2(3)=6 then they're equal, does that make it differentiable?

21. zepdrix

Maybe I should have written it like this. We are trying to see if these are equal, not just say that they are.$\large\rm \lim_{x\to3^-}2x\stackrel{?}{=}\lim_{x\to3^+}f'(x)$$\large\rm \lim_{x\to3^-}2x\stackrel{?}{=}\lim_{x\to3^+}6$

22. zepdrix

So you plug in your 3, and you determined that the slope values are the same from the left and right. So yayyy it's differentiable! \c:/

23. clara1223

Thanks!

24. zepdrix

Looks kinda cool when you graph it :o https://www.desmos.com/calculator/klqptvzagg