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Problem 82 Hard Difficulty

Let $ f(x) = \left\{
\begin{array}{ll}
0 & \mbox{if $ x < 0 $}\\
x & \mbox{if $ 0 \le x \le 1 $}\\
2 - x & \mbox{if $ 1 < x \le 2 $}\\
0 & \mbox{if $ x > 2 $}
\end{array} \right.$
and $$ g(x) = \int^x_0 f(t) \, dt $$

(a) Find an expression for $ g(x) $ similar to the one for $ f(x) $.
(b) Sketch the graphs of $ f $ and $ g $.
(c) Where is $ f $ differentiable? Where is $ g $ differentiable?

Answer

a.$\frac{-x^{2}}{2}+2 x-1$
b$\frac{x^{2}}{2}$
c.$\mathbb{R}-\{0,1,2\}$

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Video Transcript

okay we're getting a function X. That is the key slice function. zero. If X is less than zero this is the chance X. If X is between zero and 1 inclusive. What two minus X? Yeah If X is greater than one. Less than or equal to two. Yes And zero is actually greater than two. It's kind of like spiraling out the art of war shit like and G. Of X. Is defined to be The integral from 0 to X. of Epic TV. T. I think that's the first rule every morning have a prison in part a gratifying expression Fergie similar to the 14 F. So I want to write G. As a piecewise function. They still have sex. Well we can integrate piecewise. So if X is less than zero area then it follows that G. Fx is the integral from zero X. Uh F. F. T. Which in this case is just zero E. T. Can't sleep. Dude. Fuck. Which is zero. If X is greater than or equal to zero but less than 4-1 way out. Then G. Fx is the integral from zero to ex uh In this case that the T. S. X. And say sorry just T. V. Two which is one half X squared. Now if X is greater than one and less than or equal to two 10 G. of X. is the integral from 0 to X. Of mine effects. Yes. Which is two minus T. Here Z. T. Taking anti derivatives. This is uh two X minus one half X squared Thailand, wow every level of Thailand going back to Thailand, we're not exactly because X has to lie between 12 So this is actually equal to six. Some integral from 0 to 1. Uh F. F. T. Plus the integral from +12 X. Of fft so low and this is equal to yeah. Mhm. Service. He could have Well plugging in integral from 0 to 1 uh X. Which is actually T between the T. B. T. Today plus the integral from 12 x. And because exit between one and two this is a tu minus T. E. T. Now taking what to do if it is this is uh one half A. T squared from 0 to 1 plus two T minus one half A. T squared from +12 X. Alright folks, we gotta get some funny moms. This simplifies to yeah. FDA one half, three of the ribs, stopping by bees plus two X minus one half X. Squared minus two plus one half. Which is negative one half X squared plus two X. It's like I only saw three days only like three dead trying this one. But that's all. And finally if X is greater than two then G. of X. This is the integral from zero, the XFFT. Which in this case is going to be GM two plus the integral from +22 X F F T. Which is equal to Well using the previous party of two. She put this in, this is uh negative two squared over two plus two times two minus one plus the integral from two X. Of zero G. T. And this simplifies here, which is just one. I've never done that summarizing It follows that GFX to the key slice function zero X is last 30 X squared over two. If X is greater than or equal to zero and less than or equal to one and negative X squared over two plus two. X. Good minus one. If X is greater than one and less than or equal to two. And finally one, if X. Is greater than two, I would love that. You're getting head from strippers. Um That's the answer to part A and part B. Were asked to accept the graphs of both G and F. I mean, I probably would do that would be really got home problem. What is it like a level based games like Mario world, Mario, like supermarket? It's what Louise just mentioned. Well, the graph of that, something like this, I played donkey kong country, Tony hawk's pro skater. Thank God is not a sport that is sports. You try to be neocortex by the way. Everyone, if you didn't know car claim you were never gay. Yeah, you're over here trying to prove you're not getting that. You don't think. I feel like if I was five years younger I would have sucked. So I draw the graph of actually read. So we see that F lies around the X axis for all X less zero. And then from zero up to one it's just straight line Men from 1 to it's another straight line but it goes down like this. And then from two onwards it lies on the texas. Now as Fergie will use the result we found in part A. So once again for all points to the left of zero G lies along the X axis. But then between zero and one we have a quadratic form X where there was two, you know something. And so the increase up to one half like this. And then from 1-2 we have another quadratic form which is going to increase scale up to do you have to which we saw was one. Talk to one, you feel it looks something like this and finally straw. That's greater than one. We just follow the horizontal line wise was one. And so this is the graph of G. Finally in part C. Whereas to determine where at this differential and their G. Is different to do this. The easiest way is to use the grass. Looking at the graph of F. You see that F is differentiable everywhere. Except at the points these corners, X equals zero, X equals one and X equals two. On the other hand, look at the graph of G still contrived but less stupid than that. Reason if you tangle 1990 G is differential everywhere and in particular, G is differentiable. Also when x equals 01 and two as those points are no longer corners special. Mhm. So he is just differentiable everywhere. In this way we see that the integral can change a function which is not differential at some point into a function which is differentiable everywhere. Wow! You know. And I love my mom, hmm.