90–93. {Use of Tech} Work carefully Proceed with caution when ...

Question

90–93. {Use of Tech} Work carefully Proceed with caution when using implicit differentiation to find points at which a curve has a specified slope. For the following curves, find the points on the curve (if they exist) at which the tangent line is horizontal or vertical. Once you have found possible points, make sure that they actually lie on the curve. Confirm your results with a graph.
x²(3y²−2y³) = 4

x²(3y²−2y³) = 4

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. Identify the points on the curve 3 X to the power of 3, all multiplied by 42 minus Y 3 equals 8. Where the tangent line is horizontal or vertical, so it appears for this particular problem we're asked to figure out, we're ultimately trying to figure out the horizontal and vertical tangent lines if. There exist. So we're ultimately trying to solve for two separate answers. We're trying to determine whether or not they are horizontal. That's our first answer, or vertical, that's our second answer, tangent lines. So we're trying to solve for vertical and horizontal tangent lines. So that's what we're ultimately trying to solve for. If they even exist. So, With that said, our first stop, in order to find points on the curve, and we're told that our curve once again is 3 X to the power of 3, multiplied by 4 multiplied or so it's going to be 4 squared. Minus y to the power of 3. Which is equal to 8. So we're trying to find points on this curve with horizontal or vertical tangent lines. We're trying to figure out the vertical and or Actually, not and or we're trying to figure out the vertical and horizontal tangent lines. So in order to do that, we are first way we could solve for this is we can use implicit differentiation. So we need to differentiate our equation, or rather our expression for the curve with respect to X. And when we do that, we will find that our derivative is 9 x 2, all multiplied by. For the 2 minus Y to the power of 3, plus 3 X to the power of 3 multiplied by 8 Y minus 3 Y to the power of 2. D by DX is all equal to 0. So now we need to solve for DY by DX. So DY by DX is equal to -9. X to the power of 2, all multiplied by. 42 minus Y to the power of 3. And then we need to divide everything by. 3 X to the power 3 multiplied by 8 Y minus 3Y to the power of 2, which is gonna all be equal to we do a little bit of simplifying, -3 multiplied by 4 to the power of 2 minus Y to the power of 3. Divided by X multiplied by 8 Y minus 3 Y to the power of 2. Fantastic. So, now we can figure out if there's a horizontal tangent line. So let us recall that for this condition to be met, we have to note that DY. By DX will have to be equal to 0. So, in order to do that, the see if the horizontal tangent is going to be equal to DY by DX is equal to 0. We need to set our numerator equal to 0. So let's do that, shall we? So we need to take. Or 2 minus Y 3 is equal to 0, and now we need to rearrange this expression to isolate and solve for Y. So when we do that, we will start to find that Y 2 multiplied by 4 minus Y is equal to 0, and we will find that Y is equal to 0 or Y is equal to 4. So now we need to substitute these values back into our original equation. So when Y is equal to 00 is equal to 8, so that will be invalid. So let's put a red X to mean invalid and right next to it. So now, let us focus our attention now for when Y is equal to 4. So when Y is equal to 4, that will mean that 0 is equal to 8, which is also invalid. So, therefore, as a result, there are no horizontal tangent lines, which I'll call as HTL. So there's no horizontal tangent lines. So that's one of our final answers. Hooray, we did it. We found one answer. So now we need to figure out if there's any vertical tangent lines. So, as you should recall, we need to note that DY by DX is going to be undefined, so. So undefined, so undefined. I'll just call it UD undefined. So in order to see if it's going to be undefined, we need to set the denominator equal to 0. So when we set the denominator value equal to 0, we will find that drum roll that X multiplied by 8 Y minus 32 is equal to 0. So that will mean that X is equal to 0 or Y is equal to 0, or Or or or Y is equal to 8 divided by 3. So now we need to substitute these values into our original expression. So when X is equal to 0, 0 is equal to 8, so this will be invalid. When Y is equal to 0, that means 0 is going to be equal to 8, which is also going to be invalid. And for the condition where Y, so now we're focusing that Y is equal to 8 divided by 3. So when Y is equal to 8 divided by 3, we have to do a little bit of work here to determine this. So we can write that 3. X 3 multiplied by 4 multiplied by 8, divided by 32 minus 8 divided by 3 to the power of 3 is equal to 8. So now what we're trying to do is we're trying to rearrange this equation to isolate and solve for X all by itself. So when we do that, I'm going to skip a few steps, but we're ultimately taking this big old long expression and we're rearranging it to isolate and solve for X and noting that because we have this X to the power 3, we're ultimately going to be having a cube root in our answer, so that's a hint. So when we simplified and we got X all by itself, we will find that X is equal to plus or minus the cube root of 9. Divided by 2 multiplied by the cube root of 4, and that's it. That is our final answer. For the vertical tangent lines. So that is for our vertical tangent lines. Our line, hooray, we did it. So therefore, without a doubt, Our final answers, as we should note, as a quick, like, let's put a little quick little recap here at the top here. So our final answers, without a doubt, As to be, once I go over it, so we have to note that our final two answers is that there's going to be no horizontal tangent lines. And let us also note that there will be vertical tangent lines at parenthesis, plus or minus the rooted 9 divided by 2 multiplied by the cuber of 4.8 divided by 3. And that's it. That is our final answer. He answers, I should say, thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Key Concepts

Implicit Differentiation

Tangent Lines and Slopes

Graphical Confirmation

Watch next