The eight curve Find the slopes of the curve ...

Question

The eight curve Find the slopes of the curve y⁴ = y² – x² at the two points shown here.

Accepted Answer

Hello there. Today we're going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Calculate the slope of the curve at the parentheses 2/3, the square rooted 2, for the equation of 4 is equal to 4 of 2. Minus 9 x to the power of 2. Awesome. So it appears for this particular problem we're ultimately trying to figure out how to calculate the slope of the curve at the specific point for our provided equation that's given to us by the prom itself. So now that we know that we're ultimately trying to figure out the slope of this curve, let us read off our multiple choice answers to see what our final answer might be. A is -1, B is 0, C is 1, and D is undefined. Awesome. So first off, in order to find the slope of the curve defined by the equation of 4 is equal to 4 to the power of 2 minus 9 X 2. At the parentheses 2/3 square rooted 2, we will need to recall and use implicit differentiation. So our first step to solve this problem is we need to differentiate both sides with respect to X. So once again, let us note that we are given the equation Y 4 is equal to 4 Y2 minus 9 X2. So we need to differentiate, once again, differentiate both sides with respect to X. So we need to take D by DX of Y to the power for, starting with the left side. And when, when we take the derivative of D by DX of Y to the power of 4, we will find that this will be equal to D by DX of 4Y. Squared minus, or actually we can, you can different, so there's two different ways. So I, I jumped ahead a little bit, so let me go back for a second so to avoid confusion. So you could split it into two separate problems to solve, so you could differentiate the left side first, then you could differentiate the right side, or we could do it all together at once, which is what I'm doing. So we're taking D by DX of Y to the power of 4 is equal to D by DX of 4 Y2 minus D by DX 9X2. Fantastic. So we can now doing it this method, so we can now at this stage we can apply and recall the chain rule to the terms involving Y, so we can go ahead and write 4 to the power of 3, multiplied by Dy by DX is equal to 8Y. Multiplied by Dy by DX minus 18 X. So now we need to solve. So our second step now is we need to solve for Dy by DX. So we need to collect all the terms containing Dy by DX and move it to one side of our equation so we can go ahead and write. 4 to the power of 3 multiplied by DY by DX minus 8Y, so we can say. Move our 8 Y over, so minus 8Y multiplied by DY by DX is equal to -18X. Fantastic So now our next step. we can go ahead and write 4 to 343 minus 8Y multiplied by Dy by DX is equal to -18X. So when we factor out Dy by DX, we will find that DY by DX is going to be equal to negative. 18 X divided by 43 minus 8Y. Fantastic. So now we need to simplify our denominator. So when we do that, we will find that DY by DX is going to be equal to negative 18X divided by 4Y multiplied by Y2 minus 2. Which will be equal to negative when we simplify again, 9 X divided by 2Y multiplied by Y2 minus 2. Fantastic. So moving right along here, our third step now is we need to evaluate at the point parentheses 2/3 square rooted 2. So now we're evaluating at the point. So, we need to calculate our numerator first. So in order to do that, we need to take 9 multiplied by 2/3, which will give us an answer of 6. And we also need to calculate the denominator. So there's our numinator, and we also need to calculate the denominator, which is 2 multiplied by the square root of 2 to the power of 3 minus 4 multiplied by the square root of 2. And we need to note that since the square rooted 2 to the power of 3 is equal to Once again, so let's make a note of this off to the side and blue here, let's make a note. So let us note that. The square rooted 2 to the power 3 is gonna be equal to 2 multiplied by the square rooted 2. So that will mean that we can go ahead and write that 2 multiplied by 2 multiplied by the square root of 2 minus 4 multiplied by the square root of 2 is going to be equal to 0, and we need to note that since the denominator is 0, anything divided by 0 will make it undefined. So Dy by DX will be undefined because once again the denominator is 0. So that will mean ultimately that this curve has a vertical tangent at the given point. So that means that our final answer without a doubt has to be undefined. Hooray, we did it. So looking up at our multiple choice answers, our final answer, without a doubt, will have to be multiple choice answer letter D, which once again is undefined. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

The eight curve Find the slopes of the curve y⁴ = y² – x² at the two points shown here.

Key Concepts

Implicit Differentiation

Slope of a Curve

Critical Points

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