{Use of Tech} The Witch of Agnesi The graph of y = a³

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).

Let a = 3 and find an equation of the line tangent to y = 27 / (x² + 9) at x = 2.

Accepted Answer

Hello. In this video, we are going to be finding the equation of a tangent line. So, we are told that the Witch of Agnesi curve is given by the following function Y is equal to 125 divided by the quantity X2 + 25. We want to determine the equation of the tangent line to the curve at X is equal to 4. So What we are given is the Agnesi curve, which is Y is equal to 125 divided by X2 + 25. We are also told that we want to find the tangent line at the specific point when X is equal to 4. So how do we find the equation of the tangent line? Well, we are going to need a slope. We are also going to use the point slope form, Y minus Y1 is equal to M multiplied by X minus X1. This equation will give us the equation of a line for the tangent line. So, what we need is we need a specific point X, Y, we need to find the slope, then we are going to take that information and plug it into the point slope form, so that we can get the equation of the tangent line. Let's go ahead and first start by solving for our point X Y. Now, we want the tangent line specifically at X is equal to 4. Let's go ahead and take this value and plug it into our given equation. That will give us y equal to 125 divided by 42 + 25, and this will simplify to give us the value of 125 divided by 41. So, the corresponding Y value, when X is equal to 4, is going to be the point 4,125 divided by 41. Now that we have the point, we need to determine the slope, but how do we get the slope using the given equation? Well, remember that the derivative of an equation can give us the slope at any value of X for the original function. So what we are going to do is we are going to solve for the derivative of the function. Then we will plug in the value of X into that derivative to find the slope at that specific point. So, notice that the equation is given to us in the form of a rational function. In order to solve for the derivative, we will need to use the quotient rule. Now, the quotient rule states that if you want to take the derivative of a rational function, you assign the numerator and denominator as their own functions F of X and G of X. Then the quotient rule is defined as the following. It will be G of X multiplied by the derivative of F of X minus F of X multiplied by the derivative of G of X, all divided by G of X squared. So, let's go ahead and label the numerator and denominator as their respective F of X and G of X. Now, our numerator is just a constant value. So F of X is equal to 125. And by the constant rule, the derivative of the numerator is just going to be zero, because the derivative of a constant is 0. Next, we will label the denominator as G of X. The denominator is X2 + 25. By using the power rule, the derivative of the denominator G of X will be 2 X. Now that we have the 4 pieces of information we need for the quotient rule, let's go ahead and take the derivative. That's going to give us Y is equal to G of X, which is X2 + 25, multiplied by the derivative of the numerator, which is 0, minus the original numerator, which is 125, multiplied by the derivative of the denominator, which is 2 X, all divided by the original denominator squared. This will be X2 + 25 squad. Now what we need to do is we need to simplify. X2 + 25, multiplied by 0, will just be 0. And 125 multiplied by 2 will give us the value of 250. So, what we have here is 2, 250 X divided by the quantity X2 plus 25 squad. This is going to be the simplest form of the derivative. Now that we have the derivative, we will go ahead and take the value of X we were given, plug it into the derivative, so that we may find the slope of the tangent line at that specific point, and recall the value of X that we are interested in is 4. So let's go ahead and plug in 4 into the derivative. That is going to give us negative 250 multiplied by 4 divided. By 4 squad plus 25 this whole quantity squared. And simplifying this algebraically is going to give us a slope of -1000 divided by 1,681. This is going to be the slope respective to the point 4,125 divided by 41. Now that we have the point, and we have the slope at that point, we can use the point slope form to solve for the equation of the tangent line. So again, the point slope form is given to us as Y minus Y1 is equal to the slope M multiplied by X minus X1. The point that we solved for earlier is 4,125 divided by 41. This will be our perspective X1 and Y1. So if we take our M X1 and Y1 and plug it in, we get Y minus 125 divided by 41 is equal to negative 1000 divided by 1,681 multiplied by the quantity X minus 4. Now what we need to do is we need to algebraically simplify the equation of the tangent line. We will start by distributing our fraction 1000 over 1,681 to the quantity X minus 4. That will give us Y minus 125 divided by 41 equal to -1000 divided by 1,681 X plus the fraction 4000. Divided by 1,681. Next, we will go ahead and add -125 divided by 41 to both sides of the equation. And that will give us Y is equal to -1000 divided by 1,681 X plus 9,125 divided by 1,681. And this is going to be the equation to the tangent line at the specified value X is equal to 4. And with that being said, the solution to this problem is going to be C. So I hope this video helps you in understanding how to find the equation of a tangent line, and I will go ahead and see you all in the next video.

{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).
Let a = 3 and find an equation of the line tangent to y = 27 / (x² + 9) at x = 2.

Key Concepts

Tangent Line

Derivative

Function Evaluation

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