Vertical tangent lines
a. Determine the points where t...

Question

Vertical tangent lines

a. Determine the points where the curve x+y³−y=1 has a vertical tangent line (see Exercise 60).

Accepted Answer

Hello. In this video, we are going to be determining the slope of a vertical tangent line. So the problem asks us to determine the points where the curve, 2 X minus Y2 + 4 Y equal to 3, has a vertical tangent line. How can we start this process? Well, first, notice that we are asked to find the slope of a tangent line. Whenever we want to take the slope of a tangent line or find one, we will need to take the derivative of the given equation. So the equation given to us is 2 X minus Y2 + 4 Y is equal to 3. And in order to determine the slope of the tangent line, we are going to need to take the derivative of this equation with respect to X. So let's go ahead and start this process. On the left-hand side of the equation, we will need to take the derivative of each term with respect to X. The derivative of 2 X with respect to X is just going to be 2. Now, things get a little tricky once we take the derivative of a function of Y with respect to X. Now, the second term is negative Y2d. In order to take the derivative, we will still use the power rule. The derivative of negative Y2 will be negative to Y, but because we took the derivative of Y with respect to X, we will need to generate a DYDX term through implicit differentiation. For the next term, for Y, that derivative is going to be 4 by the power rule, but through the same reasoning as the second term, we will need to generate a DYDX term because we took the derivative of Y with respect to X. Now, on the right hand side, we have the constant 3, and the derivative of any constant is just going to be 0. So, now that we've taken the derivative of the entire equation with respect to X, we will need to isolate the DYDX term in order to find an equation for the derivative. So, let's go ahead and solve for DYDX. We will start by subtracting to to both sides of the equation. That will leave us with -2YDYDX plus 4 DYDX equal to -2. Notice that both sides of the equation have a DYDX term. So we can go ahead and factor out a DYDX from the left-hand side of the equation. That will give us DYDX multiplied by the quantity -2Y + 4 equal to -2. Let's go ahead and reorder the terms with respect to the positive term, and that will be DYDX multiplied by 4 minus 2 Y equal to -2. Next, what we would go ahead and do is divide both sides of the equation by 4 minus 2Y, and that will leave us with DYDX equal to negative 2 divided by 4 minus 2 Y. Notice that we can factor out a 2 from the denominator. If we factor out a 2 and then simplify the factors of 2, we can simplify the derivative to be -1 divided by 2 minus Y. This is going to be our derivative DY divided by DX. Now that we have the derivative, we need to determine the points where the equation has a vertical tangent line. Now recall, a vertical tangent line is a slope, whose slope is undefined since we have a vertical line. So, since we have a rational function, we need to determine the value of Y that causes our derivative to be undefined. In order to do this, we will take 2 minus Y and solve where it becomes zero, because if we have a denominator of zero, we get an undefined value. We subtract 2 to both sides of this equation, leaving us with negative Y equal to -2, and if we divide by -1, we get Y equal to 2. This is going to be the respective value of Y, where we have a vertical tangent line of the original function. Now that we have a value of Y that where the vertical tangent line is located at, we can take this value of Y, plug it back into the original function, or original equation, and solve for the value of X, and that will give us the missing piece we need to determine the point where a vertical tangent line exists. So again, Our original function is 2 X minus Y2 + 4 Y equal to 3. We will go ahead and take the value of Y we just solved for, which is 2, plug it into the original function, and solve for X. That is going to give us 2 X minus 22 + 4 multiplied by 2 equal to 3. Now we just have to simplify. The left hand side will simplify to be 2 X minus 4 + 8 equal to 3. -4 + 8 will give us 4, so we have 2 X + 4 equal to 3. Next, we will subtract 4 to both sides of the equation, and that will give us 2 X equal to -1, and dividing both sides of the equation by 2, will give us X is equal to -12. So we have X equal to -12 and Y equal to 2. That means the point where the curve has a vertical tangent line is going to be -2. And this is going to be the solution to the problem. And with that being said, the overall solution is going to be D. So, I hope this video helps you in understanding how to approach this type of problem, and I will go ahead and see you all in the next video.

Vertical​ ​tangent linesa. Determine the points where the curve x+y³−y=1 has a vertical tangent line (see Exercise 60).

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Vertical tangent lines
a. Determine the points where the curve x+y³−y=1 has a vertical tangent line (see Exercise 60).