1) Sketch one graph for a continuous function g that satisfy ALL the following:a) 9(0) 9(2) 9(4) = 0 b) g' (1) = 9' (3) = 0 9'(4) = 1,d) 9' (2) ...

1-2 Use the given graph to estimate the value of each derivative. Then sketch the graph of $f !$
(a) $f^{\prime}(-3)$
(b) $f^{\prime}(-2)$
(c) $f^{\prime}(-1)$
(d) $f^{\prime}(0)$
(e) $f^{\prime}(1)$
(f) $f^{\prime}(2)$
(g) $f^{\prime}(3)$

Consider this function DX. Okay, let's analyze this function at X equals two one. Okay, so, uh, that's it. This is executes 21 This point is that it goes to one. Okay. What is the value of the function? It x equal to one now, See if you approach the graph from left hand side, that is we're pressing this path from left hand side to this. So what is the video off? G X at X equals to one the value of G excess Negative too. So limit off the function GXE when extends to one from negative side. He's given us negative. So we're approaching the function from negative side. Right. So it is negative to This is also called. This limit is also called us left hand side limit. This is limited also far left hand side women because we're approaching from left hand side. Okay, Now let's assume that we're approaching the same function from right hand side. Okay, this to exit construction. So we're taking this path approaching from right inside. So let's see what will be the value off x. Sorry. We're for the value of the function, g X, but at execute toe one. It will again be negative too. That is limit off GXE extends to one for from positive side is also we could tow negative too right? So it this is called right hand side limit The first one Waas left inside limit when we were approaching the function from left inside. Now this is right inside limit Now notice that that these limits are equal Whether we approach X equals to one but they approach executes 21 from negative or positive side The value of the function the same grade Yeah, Whether we're approaching from any side the value of the function is same That is minus two negative too. This is sorry. This is negative Two So whether we're approaching from left side or right said that is the limit of the function From left hand side, It calls to the limit of the function from right hand side at X equals to one. Therefore the limit exists at X equals to one. So if we will ask, we will be asked that limit off gxe when extents toe one is what So that is negative two Because look, the limit exist because left inside limit is all it calls to write inside limit to limit exist. So, um, April, somebody will ask What is the value off given what is even when we substitute X equal to one in GX What is the value? Jeevan has value negative too equals to negative two. So the function G X is continuous at X equals to one because both limits are same The limit at off gxf xto one is negative toe the value off Jeevan is negative. Two again. Now let's analyze the same function at X equal to negative toe. This value at X equals too negative too. Now let's see if we will oppose the living from Sorry, We will approach the image from depth inside. Get it This site We're tracing this part of it This part. So we're approaching the limit from left hand side. What will be the value off GX will be approaching as we approach the American etc The limit off G X X here and I'm side okay limited GX, when we're approaching extends to negative two from negative side. The value of GX will be this this for it will be positive four or just four right so the value of GX when we will approach the remit from negative side is positive. Full. Now, let's see. We will approve the function from right. Answered the site. Let's choose a different color. Okay, so we're dressing this path, right? So we're approaching the value. Negative two negative to the next from right inside. So limit off the function DX when we extends to negative two from right and said that this positive side is so we're tracing this part and at X equals two minus two. What is the value of GX? It is negative three. So this value is equals to negative three. So what do you think? The limits said? Theme or different. See the limit from left hand side. The limit from, uh Okay. Okay. The limit from left inside is different from limit from right inside. They are not equal. These limits are not equal. Because when we approach the same function from left inside, it is four. When we approach the function from right inside, it is negative three. So the limits are different. Therefore, limit at X equal to minus two does not exist. Does not exist so limited off the function G X, It X equals two negative too. It does not exist. So what will be the value off g two If somebody will ask What is the value of the function it executes or negative toe? What will be the function off? The value of the function at X equals two negative to what will be the value? Here are two values. One is four and one is minus three. Let's see the value it food has a whole like it is an open circle. The value off open circle is not included in the function. Our function can never have too values that same value off X. So this is a full circle. This is actual value of G X at X equals two minus two. That means here G minus two will be equal to negative three. This will be value off G minus two. Although the limit does not exist at negative two, you can see from the graph also. Okay, you can see from graph also that this there is a discontinuity. There is a discontinuity. Oh, okay. They form a different color. This is a discontinued in the graph at X equals two minus two. So the graph is this continuous at X equals two minus two

Consider the shown function. Okay, this is the function. What? Ethics. So let's see what will be done. Limit off X when extends to one from negative side. That means access Taking the value negative. One story, uh, access taking the value one from negative side this side. So if you will just follow the trace the function when it the X becomes one the value the function is approaching value off two so x when extends to one from negative side value of the LTD's too. Okay, Now let's see what will be the limit of the function. Limit off the function ethics. Ben X is approaching one from positive side. Now, if you will raise the function from positive side Okay, days, the functional positive side, the value of the function it x equals to one is negative one, right. The value of the function here will be negative. One. You can see that Negative one, right. So you can see the values are different. Trump in X take the value from positive and negative side. The values are different. This one is called this one when thesis gold. Sorry. Great camp. This is corn left hand side limit and this is called right inside limit. Now this these limits are different. These gimmicks are not equal. The point is same. X equals to one. But when we we approached this value from left inside, the function approaches the value off to and when we approach X equals to one from right inside, the value of the function is minus negative one. So these limits are not equal. That means limit off ethics particular extends to one does not exist. The limit does not does not exist here because left inside limit is different from right inside limit. Yeah, So the left hand side limit is not equal to write inside limit. Therefore, the limit of the function at X equals toe one does not exist. So if someone will ask what is the value off f one? Okay, so we have to will use here. One is negative one and one is bold steps to. But now look, this is a whole actually hole. So whenever there is a hole, that value is not included in the function. So that means this value FX equals to do is excluded. So if someone will ask what is the value of F one when executed one nobody off the fixes Everyone. Everyone. So so we live. Affects is my negative ones. Value F one is negative one. Okay, because this is a full circle and this is a open circle. So this well is not included and this well is included. So if someone will ask the value off FX at one, so it will be negative one. One more thing if a function ah function cannot have two will lose that they believe affects Like we cannot have two full circles here because then it will not be called off election. It will be a relation. So for same value of X r function always has a unique value. So this here f one is negative one. Now let's see. Now let's see whether the things continuous and X equals to one. Since the limits are different. The right hand side and sorry, This left hand side limit the left hand side and the right hand side limits are different. Therefore, F X is not a continuous function. It has a discontinuity at X equals toe one. Look, this is a discontinuity. This is a discontinuity. So the function has a discontinuity at X equals toe one. The function is not continuous. Set X equals to one. So now let's analyze the same function for X equals two minus two. Okay, analyze the same function for X equals two minus two. Since, uh, let's assume I'm cover addressing the function from this negative side and going towards X equals two minus two. What were the value of the function? The value of the function will be three. So limit off ethics when I come from negative side X towards negative too. And I'm coming from negative side. The limit of the function is three. The value of the function is three. Right? So Okay, now I'm pressing the function from positive side. Okay, I'm coming towards negative to again. So I'm pressing the function from positive side here. I'm pressing the function and then I've gone from here and then reached the pause Negative to the what is the value of the function here? Three. Again. Therefore, limit off F X when extends to negative two from pollster side is three again. So whenever I drew from or like whichever this direction, I traced the function from the limited. Same right. So I traced it from negative side. The limit was three and ideas pressed it from Pollster side The limited again three. Therefore, the left hand side limit this limit. The left hand side limit is funny. Okay. The left hand side limit is same as the right hand side limit. So this production is continuous at X equals two minus two. At this point, at this point, this function is continuous because left hand side limit is equals to write and said limit on what will be there limit off the function as extends to negative two. The limit of the function will be three because the value is same for both sides. Whether I approached the limit from left side or right side, the value of the function is same. So the function is continuous at X equals two minus two and the value of the function is three at X equals two minus two and the value of the limits from whether I approach it from negative side of positive side. The value of the function is same, that is three. The value of the function is three at exit. So what? It will be the venue off F off minus two. It will be clean and it is continuous that this continues

And welcome to problem. 63 years in a problem. Give us the plaque graph is the graph of function F in assets to be able to sketch the function F crime is derivative. Well, um, the graph in the book, actually, I said in my butt, Here we go. Now, first of all, you see that from the plaque? It actually has a critic point. You know, actually, it's a relative mean at X equals zero. That means you from your after derivative has to be zero. There's has to cross zero right there for the derivative. Now it also showing the graph that have two inflection points. Someone over here and over here, we used a different color. Oh, I wish you wouldn't see that. Uh, so where they're, like, the inflection point. And over here in flux. And, boy. So you are a point. What does that tell you about a firm where inflection point. Meaning f double from zero. Meaning Ephraim has, uh, critical. They're great. Go there. What else is answers? Tell you. Well, the function is decreasing if you see decreasing for everything and to zero, which means you're, um you're divert. Motive has to be negative for everything before zero before. So Ephraim has to be smaller than zero for all access lesson zero and all sort of function to keep increasing the black one. Keep increasing after Sarah. So you know there are real Begin zero for everything after, is there now with that? Enough to sketch it. What? What happened at Infinity for, um, the Ephraim function? Well, to say that we have to know that what happened in affinity for F Well, from the look of that for the plaque function, it's going to be kind of straight line, but it's not really touching the horizontal Simpson X equal tree right there. Actually, we can answer part B right now. When they asked what happened to limit excellent, infinity actually would be equal to tree for this part. Be right there and then because straight line. So what is a straight line for? If it's straight like, what is this derivative? What is the tension? Have to be zero. So what happened to Tangshan? His Ephraim. Therefore, it has to be zero then. No, I will get. Say why? Because, um, f at horizontal. All rights on two is, uh somewhat a straight line. Not really a straighter I write because it cannot really touching the light. What? Why Equal tree there, so derivative of a straight line, has to be zero so that what happened with the F Fram limits it's extra to infinity of Ephraim would be zero. So it will ever be able to trust the light, the X or the X axis execute zero. And so there we go. I got that dread line. This is my sketch of a program can everything is below zero and it has a critical point right there at court is wanting to this critical point and then it must have an inflection point here. So I must go up. And then it's have a critical another critical point correspondent that them going down and never touched her right. The only place a touch there actually is that there is there where it has where it is the relative mean of the original function. And that's it for this problem. I want to see you in the next one

Consider the given function. Let analyze dysfunction. Ethics at X equals to one first. Okay, where is except close to one? This is exit. Okay, so let's see whether this function is continuous or not at X equals to one. But first, let's see, what are the conditions? First condition. Is that limit off? HX. There should be X in here. Limit off. Attacks should exist at X contends toe one. Okay, this limits should exist. First condition is this. Second condition is the value off. Each one will be there. There is a value when X is substituted in, uh, objects as execute one is substituted in abject. Then there should be a value for each one. Right? And the third condition is these value should be equal. Okay, so let's see. So let's just put this aside for a minute. So how to check the limit? Exist or not. So, first we will check limit what? The function at X then extends to one from negative side. That means we're approaching the functions F X from paychecks from this side. Okay, so let's trace the function until we reach X equals to one. Were dressing from this side So when we read, X equals to one the value off Texas Positive too. So the value off limit here is positive Two or simply to Okay, the left hand side limit is no. Now let's approach the same function from right inside. So I'm just going on this function from right inside and approaching Mexico's toe one. Okay, so what is the video function at one X equals to one It is again to This is the same point. So the limit off at X when extends towards one positive side from positive side is to so we can see that the right inside and the left hand side limits on equal. The red one is Let me tell you, this is left inside limit and this is left hand side limit. And this is right inside limits. And these limits are equal. Favorable first condition is satisfied the limit at a Tex exist. So the limit off X when extends to one, is two left inside limited the called right inside, and both are equal to two. Therefore, the limit exists at extends to one and the value of the limiters to I believe the limited. Now let's see. What is the value off? Let's see. What is the value off H one? So what is the value of H one? H one? Okay, so this one did again. This even access equals to one. The position off this craft tells you the value of each one. What is such one? Is this again to? So you can see the limit. The value of the limit at X equals to one and the value as X approaches. One actually is two. And the value of function at Texaco Toe one is to both are same. Therefore, the function is continuous. The function is continuous at X equals to one. The function is continuous. The value of each one is this. The limit is this. The function is continuous at X equal to one. Now, let's analyze this function at that's equals too negative too. Okay, The first, the limit off the function at eggs. When X becomes negative two from negative side, that means we're approaching very is executing the negative to this is execute. So we're coming from that negative side. Okay? You come from the site and retracing this function and going towards X equal to negative to What is the value of why Here? I mean, sorry. What is the value of checks here? That's why the same thing. What is that? I live off H X here, the video of Texas zero. So the limit went, Uh, the X approaches the value minus two from negative side zero. Now we will approach the same limit from positive side. So let's go from positive side. Yeah, When we go from positive side, what is the value when X approaches negative to the objects value approaches zero again. So it is limit off H X when X approaches. Negative two from positive side is again zero. Now you can again see that the limits the same, right? I'm sorry. The limits are same. Keep to society. Okay. So left inside limit that that goes in of a blue color and the right inside limit our scene. So therefore the limit exists. At this point, the point X equals two negative to the limit exist here on what is the value of limit? The value of limit is the limit off at X when extends towards negative two is zero. You can see that both limits are same. So limit exist and the value is zero. So okay, the value zero. And what is the value? Off edge negative to that is Okay. What is the value of edge negative to that is at X equals two negative to what is the value of it. So if there was a hole or something here, the value would not be included in ethics. But there is a line graph passing through this point. That means HX has some value here on what is the value? The value is zero. So again, h negative to it. Zero what? I think Xnegative to it. Zero. Now you can see that at this point, at X equals two negative to limit exists well, you exist now and these values are same. Is values are equal. Therefore, the graph is continuous. This is again continuous at equals to negative two for therefore, the as you can also see from the graph that it is a passing through line. There's like a passing through graph. Does not It doesn't have any discontinue to here. So the graph is continuous at X equal to negative toe and the graphs continuous that in equal student one, is it? Yeah,