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clara1223
 one year ago
At x=3, is the function given by f(x)={x^2 when x<3 and 6x9 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)?
clara1223
 one year ago
At x=3, is the function given by f(x)={x^2 when x<3 and 6x9 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)?

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clara1223
 one year ago
Best ResponseYou've already chosen the best response.0dw:1442957985635:dw That's what it looks like.

zepdrix
 one year ago
Best ResponseYou've already chosen the best response.1Oh these are kinda neat :o So ummmm

zepdrix
 one year ago
Best ResponseYou've already chosen the best response.1Ya 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.

zepdrix
 one year ago
Best ResponseYou've already chosen the best response.1So for continuity, you need:\[\large\rm \lim_{x\to3^}f(x)=\lim_{x\to3^+}f(x)\]

zepdrix
 one year ago
Best ResponseYou've already chosen the best response.1The 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?

zepdrix
 one year ago
Best ResponseYou've already chosen the best response.1Oh you already figured out continuous :p Woops I didn't read the whole post lol

zepdrix
 one year ago
Best ResponseYou've already chosen the best response.1For differentiable, you need the slopes to be the same from the left and right,\[\large\rm \lim_{x\to3^}f'(x)=\lim_{x\to3^+}f'(x)\]

zepdrix
 one year ago
Best ResponseYou've already chosen the best response.1Take some derivatives :o

clara1223
 one year ago
Best ResponseYou've already chosen the best response.0@zepdrix what do you mean? Like take the derivative and plug in x=2 and 4? or what?

zepdrix
 one year ago
Best ResponseYou've already chosen the best response.1The 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 \(\large\rm f(x)=?\)

clara1223
 one year ago
Best ResponseYou've already chosen the best response.0well if f'(x)=2x, wouldn't it just be 2x?

clara1223
 one year ago
Best ResponseYou've already chosen the best response.0Wait I don't think that was what you were asking, I'm confused.

zepdrix
 one year ago
Best ResponseYou've already chosen the best response.1We have a `piecewise function`. When we're on the RIGHT of 3, our function is defined by a different `piece`.

clara1223
 one year ago
Best ResponseYou've already chosen the best response.0So would we take f'(x) of 6x9?

zepdrix
 one year ago
Best ResponseYou've already chosen the best response.1dw:1442961574141:dwYes, good. We're using that other piece to define our function on the right side o f3.

zepdrix
 one year ago
Best ResponseYou've already chosen the best response.1Ugh, I got a new mouse. It messed up my tablet settings :c Grr I'll have to fix that later..

clara1223
 one year ago
Best ResponseYou've already chosen the best response.0Haha still completely legible, continue

clara1223
 one year ago
Best ResponseYou've already chosen the best response.0if f'(x)=2(3)=6 then they're equal, does that make it differentiable?

zepdrix
 one year ago
Best ResponseYou've already chosen the best response.1Maybe I should have written it like this. We are trying to see if these are equal, not just say that they are.\[\large\rm \lim_{x\to3^}2x\stackrel{?}{=}\lim_{x\to3^+}f'(x)\]\[\large\rm \lim_{x\to3^}2x\stackrel{?}{=}\lim_{x\to3^+}6\]

zepdrix
 one year ago
Best ResponseYou've already chosen the best response.1So 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:/

zepdrix
 one year ago
Best ResponseYou've already chosen the best response.1Looks kinda cool when you graph it :o https://www.desmos.com/calculator/klqptvzagg
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