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Sketch the graph of a function $ g $ that is continuous on its domain $ (-5, 5) $ and where $ g(0) = 1 $, $ g'(0) = 1 $, $ g'(-2) = 0 $, $ \displaystyle \lim_{x \to -5^+} g(x) = \infty $, and $ \displaystyle \lim_{x \to 5^-} g(x) = 3 $.

We begin by drawing a curve through (0,1) with a slope of 1 to satisfy $g(0)=1$

and $g^{\prime}(0)=1 .$ We round off our figure at $x=-2$ to satisfy $g^{\prime}(-2)=0 .$ As

$x \rightarrow-5^{+}, y \rightarrow \infty,$ so we draw a vertical asymptote at $x=-5 .$ As $x \rightarrow 5^{-}$

$y \rightarrow 3,$ so we draw a dot at (5,3)$[\text { the dot could be open or closed }]$

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All right. We're gonna draw this function G. And here's what we know about it. It's continuous on its domain negative 5 to 5. That means when we draw it it's never gonna be past negative five on the left as positive five on the right. Do I have 10 or 24? So no. Okay. So here will be 12345 1234-. Right? Okay. So I can't go past those. It doesn't have any holes or breaks in it because it's continuous. Now, the only point that they gave us is that G. F. Zero equals one. So I'm gonna go to G. M. 20 on the X. And one on the Y. And I'm gonna put a point. Okay. However we do know that G. Prime of zero equals one. And what that means is this At X equals zero. The slope of the tangent line two G. Is one because that's what the derivative means. So I'm going to draw a dotted line and it's going to be the tangent line and the slope is gonna be one. So up one over lawn of one of the one. Oops, went crazy there. Okay. So when I draw the function, when it gets to that point it has to just touch the tangent line and then go back. Okay. How do I know it's underneath it? I don't could be over on the other side. It's okay. All we gotta do is make sure it goes to the point and that's tangent to that line cannot win. We get to negative two. It has to have a tangent of zero. Okay, well, we're already too negative to let me uh when he raced just a little bit there. Okay, when we get to negative two then we have to have a slope tangent line that has slope zero. So here it is. How do I know? How do I know I'm at negative two on the y axis. I don't Okay. In fact, I'm going to fix it. So you you're not confused by that? Oops. Okay, erase, erase, erase, erase, erase, no erase erase, erase. Trying to make the function go away. Well, almost there. All right. So when we get to negative two, we gotta have a slope of zero. So I'm gonna I'm gonna have that all the way down here. I can remember When it got to 0 1. It has to Uh tangent slope. The slope of the tangent has to be one. Okay, now, right here the slope has to be zero. So it has to turn back around and I go that way. All right then we know that when I get close to negative five from the right so when I'm drawing right now I have to go to positive infinity. Okay, so I'm gonna Put a dotted line at -5 because that's an aspirin tote what went crazy. Okay, so instead of putting the arrow there, I need to make it go like this of to positive infinity. Okay, then, when I go to five from the left I have to get close to three which is right here. three. Okay, but there is no point there because five isn't in the domain, so I have to put an empty circle and then I have to go and get close to it. Okay, so that's one drawing. Okay, there's no reason I had to go all the way down here to when X was negative to maybe it's just very smooth like that. Just up to there. Okay, so there's infinitely many Draw drawings of this. Okay, this is just how mine looked. So the important points out the important ideas are this is a .01. This is a slope of the tangent, slope of the tangent. This tells you what to do when you get close to -5, this tells you what to do when you Get close to positive five. Okay, I hope that helps

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