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Sketch the graph of a function $ f $ for which $ f(0) = 0 $, $ f'(0) = 3 $, $ f'(1) = 0 $, and $ f'(2) = -1 $.

Make sure that the curve has a slope of -1 as we pass over $x=2 .$ Two of the many possibilities are shown.

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Missouri State University

Harvey Mudd College

University of Michigan - Ann Arbor

University of Nottingham

Okay. We want to draw a graph of a function that has these qualities. So the only point that we know for sure that's on the graph is given by this one right here. What which says when x zero? Y is zero. So it goes through the 00.0. So there it is. Okay, so since this is all bunched around 01 and two, let's let's make this be one. This will be to this will be three. Run 23. Okay. The next thing we know is that the derivative at zero equals three. Okay. So what that says is this At X equals zero. The slope of the tangent line to f Is 3/1. So I have to drop, I'm gonna draw the tangent line and it's got to touch this red dot and a slope has to be 3/1. So I'm just gonna go ahead and do that up. 3/1. Okay, so here's the tangent line. Okay, that's not part of the function. That's just the tangent line. Okay, so the red part is going to be the function. So here comes the function. What's it doing down here? I don't know. Okay, we don't have any information so we just ignore that. Okay, so it's got to go through that point and it's got to be tangent. See that green dotted line? Yeah. Okay. Now the next one says when it gets to one, The slope of the Tangent Line is zero. Okay. So how where is it when it gets to one is it at Y equals one. Is that y equals two? We don't know. Just make it go there Can it? When it gets to one The Tangent Line has to have slope zero. So here's the new tangent line. This one has chemical zero. This one had chemicals three. So when that red function gets to one it has to stop and then turn around and go back down. Okay? Because the tangent line can only touch it once and it can't cross through it at that point it can cross through it later but it can't cross through it now. All right now when we get to to the slope of the tangent line has to be negative one. Okay? So let's say it's gonna get to right here then the slope has to be negative one right there. So well it's oops come back. Yeah let's make it get right on one right here. Okay so when it gets there it's tangent line has to have slope negative one. So down line over one down 1/1 has slope negative one. So when we get to that red point we have to come up to that tangent line, touch it and then move on. Oh my kind of touching more than one place. I'd better fix that. I needed to touch right on it and then go off and then it could cross over later. All right so What's it doing after this? I don't know what's it doing before this? I don't know all I know is that it goes through 00. Is tangent line there has sloped three. Its tangent line at one is has slope zero and its tangent line at X equals two has slope negative one. That's a graph. So if you look in the back of the answers in the back might not look exactly like that. Okay, But look for the important things. What's happening at zero? What's happening at one? What's happening at two? Okay. Hope that helps. It's hard when you first start but you'll get it.

Oklahoma State University