{Use of Tech} The Witch of Agnesi The graph of y = a³ / x²+a²,...

Question

{Use of Tech} The Witch of Agnesi The graph of y = a³ / x²+a², where a is a constant, is called the witch of Agnesi (named after the 18th-century Italian mathematician Maria Agnesi).

b. Plot the function and the tangent line found in part (a).

Accepted Answer

Welcome back, everyone. For the given equation, Y equals 125 divided by X2 + 25, plot the curve and its tangent line at the specified point on the same rectangular coordinate system. That is where X equals 4. Now This problem is basically made of two parts. First, we want to plot the curve, and then we want to plot its tangent line at the specified point. So first, let's try to draw the curve on our graph, and then we'll figure out how to get the tangent line at X equals 4. Now we can plot the curve if we have a few points. So let's set up a table, OK? So to plot the curve, let's set up a table. Of X and Y values, OK. And for our table here. Let's notice that by looking on our graph, it ranges from -15 to 15. So let's plot a table from -15 to 15 in increments of 5. So negative 50, -10, -5, 05, 10, and 150. I know, to find the Y values that correspond to these X values, we can simply plug the X values into our function. I'll allow you to do that, but let's just do one for reference. So, for example, at X equals -15, then Y, OK, Y. It's gonna be equal to 125 divided by negative 15 2 plus 25. -15 squad is 225. So that's 125 divided by 225 plus 25, which is gonna be 250. And that would be equal to 0.5. So in other words, when Y equals, sorry, when X equals -15, Y is 0.5. No, when you go ahead and plot the rest, you should find that when X is -10, Y is 1. When it's -5, Y is 2.5, 0, then it's 55, 2.5, 10, 1, and 15, 0.5. So now we have all the points that we need to plot the graph of uh or to plot our points on our graph. So, now, if we go over to 5, go over to our graph, X is -15, Y is 0.5, that would be here. OK. And it's -10, Y is 1, -5 and 2.5, OK, 0 and 5. 52.5, 101. And 15 0.5. So now we know our curve is probably gonna look something like this, OK? Let me just try one more time just to draw that as accurate as possible. OK. And now this is what our curve will look like. This is an idea of what it should look like. So we've completed the first part of our problem. Next, we need to find its tangent line tangent line at the point where X equals 4. In other words, we want a tangent at this point. So the question is, how are we going to find a tangent at X equals 4? What can we do? Well, first, let's think about the equation of a line. Recall that we can find the equation of a line by writing it in point slope form, that is. The form that says Y minus Y1 is equal to M multiplied by X minus X1. Where X1 Y1 are points on the line are the coordinates of a point on the line, sorry, and M is the slope of the line. Now. If we want to find the slope of the line since it's a tangent to the curve, then the slope will be equal to the derivative of the curve at that point, at that value of X. So if we can differentiate our curve Y, OK, and substitute. Our value where X equals 4, we'll be able to find the slope and then we can substitute X1 Y1 into the equation for our curve because the tangent line also runs through that point. So if we can figure out this point, then we should be able to get the point for the tangent line. So first let's start with our slope. Now we know that our slope M, let me put that in red actually. A slope Is going to be the derivative of Y at the point X equals 4, OK. So no, that means we're going to find the derivative of Y, OK. Let me, let me write that. I think I need to write this a little better. OK. So, wait, let me just fix that here. Or M equals the derivative of Y at the point where X equals 4. No, the derivative of Y with respect to X, OK, is going to be the derivative of 125 divided by X2 + 25. So now, let's go ahead and try to differentiate this expression. Now, we can use the quotient rule, OK? Recall that by the questionent rule. The derivative of Y with respect to X. is going to be equal to the multiplied by the derivative of the numerator U minus u multiplied by the derivative of the denominator V divided by V2. OK, where in this case, as we said, our numerator is U and our denominator is V. So in this case, applying the quotient rule. Then the derivative of Y with respect to X will be equal to the derivative of our numerator. Our numerators are constant, so we're gonna have V X2 + 25 multiplied by U 0, OK, minus U or numerator 125 multiplied by the derivative of V. The derivative of X2 + 25 is 2 X. OK. I know all of this is being divided by V2 as X2 + 252. Now, let me make some space over here. When we expand this, then we get the derivative of Y with respect to X to be equal to -250 X divided by X2 + 25 squad. So now this means then that at X equals 4, then our slope M. OK. Is going to be equal to -250 multiplied by 4 divided by 42 + 25 all squared. So now when we were at all negative 250 multiplied by 4, that's -1000. And our denominator, it will be 1,681. OK. So the slope is -1000 divided by 1,681. Now that we have the slope, we need to find the two point. We need to find the coordinates of a point on the line, OK? Now, we can do that by simply substituting X equals 4 into our equation, and at X E equals 4, then Y is going to be equal to 125 divided by 42 + 25. Which equals 125 divided by 41. OK? So in other words, 4, 125 divided by 41 is on the tangent line. So now that we have our values for M. X1 Y1. Then if we substitute that into our formula, point slope form of the line, OK, then this implies that Y minus 4. Oh sorry, Y minus Y1, 125 divided by 41 is going to be equal to M-1,000 divided by 1,681 multiplied by X minus 4. And now if we were to go ahead and solve for Y here, add, we expand our bracket and add 125 divided by 41 to both sides, then we should get Y to be equal to -1000 divided by 1,681 X plus 9,125 divided by 1,681. This is the equation of our tangent line. All we need to know to solve is to find some other point on our line because remember we need two points to plot. Now, we could find that point where X equals 0, OK? Because we already know that uh 4 125 divided by 41 is on our line. So we only need one more point. And at X equals 0, OK. Then on our line, Y will be equal to 9,125 divided by 1,681, which is equal to 5.428. OK? So 05.428 is also a point on our tangent line. So now if we go back to our line, OK, or graph, and we go to 0 5.428, which will probably be somewhere right here. Then now that we have two points, we can draw our line, draw a tangent line, OK? And of course, you know that my line isn't gonna be perfect, OK, but you will want it to be as close as possible here because we know that it will be tangential to X equals 4, OK? So now our line, our tangent line will probably look something like this. Alright, let me try to draw that just a little bit better just for accuracy here. So now you should, you should ensure that you draw your curve properly so that you don't end up in the same mess that I do, right? What we know that whatever line we draw, it should be tangential to or curve, OK? So that means I probably needed to draw my curve a little bit better. Thus, this is going to be the plot of the curve and the tangent line at the point where X equals 4. Thanks a lot for watching everyone. I hope this video helped.

{Use of Tech} The Witch of Agnesi The graph of y = a³ / x²+a², where a is a constant, is called the witch of Agnesi (named after the 18th-century Italian mathematician Maria Agnesi).
b. Plot the function and the tangent line found in part (a).

Key Concepts

Witch of Agnesi

Tangent Line

Graphing Functions

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