yego.me
💡 Stop wasting time. Read Youtube instead of watch. Download Chrome Extension

Finding points with vertical tangents


3m read
·Nov 11, 2024

Consider the closed curve in the xy plane given by this expression. Here, find the coordinates of the two points on the curve where the line tangent to the curve is vertical. So, pause this video and see if you could have a go at it.

I don't know what the exact shape of this closed curve is, but if I were to draw some type of a closed curve, maybe it looks something like this. This isn't the one that's right over here. This one also has two points where my tangent line is vertical. At one point would be right over there; another point would be right over there.

Now, how do we figure this out? Well, what we could do is use implicit differentiation to find the derivative of y with respect to x and think about the x and y values that would give us a situation where that derivative is non-zero in the numerator and zero in the denominator. So let's do that.

Let me rewrite everything I have: (x^2 + 2x + y^4 + 4y = 5). I want to take the derivative with respect to x of both sides of this equation. I'm trying to find an expression for the derivative of y with respect to x. So what am I going to get? This is going to be equal to (2x + 2 +) the derivative of this with respect to y is (4y^3) and then times the derivative of y with respect to x; that's just straight out of the chain rule. Plus, the derivative of this with respect to y is (4) times the derivative of y with respect to x—once again straight out of the chain rule—is equal to, whoops, I want to take the derivative with respect to x here, is equal to (0).

Now we just have to solve for (\frac{dy}{dx}). A couple of things we could do: we could take the (2x + 2) and subtract it from both sides, and we could also factor out a (4\frac{dy}{dx}) out of this stuff right over here. So let's do that: let's subtract the (2x + 2) from both sides and factor out the (4\frac{dy}{dx}).

We will get (4 \cdot \frac{dy}{dx} \cdot (y^3 + 1) = -2(x + 1)). Now I just have to divide both sides by (4(y^3 + 1)), and I'm going to get the derivative of y with respect to x is equal to (\frac{-2(x + 1)}{4(y^3 + 1)}). Actually, this can be rewritten as being equal to (-\frac{x + 1}{2(y^3 + 1)}); I just divided the numerator and the denominator by (2).

Now, why is this useful? Well, we can think about what y-values—because y is the only variable we have in the denominator here—would make the denominator equal (0) and then find the corresponding x-values for those y-values by going to our original equation.

Well, this is going to be (0) when (y = -1). So when (y = -1), let's figure out what x is. To do that we just have to substitute (y = -1) back in our original equation and then solve for x.

Let's do that; let me clear this out since I need that real estate. If we go back and we substitute (y = -1) up here, we're going to get:

[ x^2 + 2x + 1 + 1 - 4 = 5. ]
This is going to be (-3). Subtract (5) from both sides, you get (x^2 + 2x - 8 = 0). This is just simple factoring, so it's going to be ((x + 4)(x - 2) = 0).

What two numbers, when I take the product, I get (-8)? Four and negative two. When I add four and negative two, I get a positive (2); there it is equal to (0). So (x) is equal to (-4) or (x) is equal to (2) when (y) is equal to (-1).

To answer their question, find the coordinates of the two points on the curve where the line tangent to the curve is vertical. Well, the answer here would be—get a little bit of a drum roll—it would be the points ((-4, -1)) and ((2, -1)), and we're done.

More Articles

View All
Differentiating rational functions | Derivative rules | AP Calculus AB | Khan Academy
Let’s say that Y is equal to five minus three X over x squared plus three X, and we want to figure out what is the derivative of Y with respect to X. Now, it might immediately jump out at you that look-look, Y is being defined as a rational expression her…
THIS VIDEO WILL CHANGE YOUR LIFE COMPLETELY | STOICISM BY MARCUS AURELIUS
Marcus Aurelius is a historical figure who is regarded as a symbol of ultimate authority. Apart from his position, he distinguished himself by his rigorous moral code, adherence to values, and deep philosophical views. He turned down ostentatious feasts b…
Discretionary and rulemaking authority of the federal bureaucracy | Khan Academy
In many videos, we have talked about how a bill can become a law. It first gets introduced into the legislative branch, which in the United States is the U.S. Congress at the federal level. If it passes both houses of Congress, then the bill will go to th…
10 Things I Wish I Knew Before Investing
Hey guys, welcome back to the channel. In this video, I’m going to be going through 10 things I wish I knew before I started investing, so hopefully we can get through these 10 in around about 10 minutes. So, time is on, let’s get stuck into it. The firs…
How price controls reallocate surplus | APⓇ Microeconomics | Khan Academy
What we’re going to talk about in this video is the effect of price controls on changing how the surplus, the total surplus, is reallocated between consumers and producers. We already touched on this in other videos, the video on rent control, the video o…
Ruby Jean's Juicery | Black Travel Across America
That same spirit is alive and healthy today all over this city. Black owned spaces have a knack for preserving our past while nurturing the future. So you brought her in? Case in point, Ruby Jean’s Juicery, which combines nutritious food with family root…