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

Second derivatives (vector-valued functions) | Advanced derivatives | AP Calculus BC | Khan Academy


3m read
·Nov 11, 2024

So I have a vector valued function H here. When I say vector valued, it means you give me a T; it's a function of T. So you give me a T, I'm not just going to give you a number; I'm going to give you a vector. As we'll see, you're going to get a two-dimensional vector.

You could view this as the X component of the vector and the Y component of the vector. You are probably familiar by now that there's multiple notations for even a two-dimensional vector. For example, you could use what's often viewed as engineering notation here, where the X component is being multiplied by the horizontal comp unit vector.

So you might see something like that, where that's the unit vector plus the Y component, 4T^4 + 2T + 1, is multiplied by the vertical unit vector. These are both representing the same thing; it just has a different notation. Sometimes you'll see vector valued functions with an arrow on top to make it explicit that this is a vector valued function.

Sometimes you'll just hear people say, "Well, let H be a vector valued function," and they might not write that arrow on top. So now that we have that out of the way, what we are interested in is, well, let's find the first and second derivatives of H with respect to T.

So let's first take the first derivative H prime of T. Well, as you'll see, that's actually quite straightforward. You're just going to take the respective components with respect to T. So the X component with respect to T, if you were to take the derivative, what are you going to get?

Well, we're going to use the power rule right over here: 5 * the negative 1, or time the negative, you're going to get -5 * T^(5 - 1) power, so T^4. The derivative with respect to T of -6, well that's just zero. So that's the rate of change of the X component with respect to T.

Now we go to the Y component, so we're going to do the same thing. The derivative with respect to T is going to be, and once again we just use the power rule. 4 * 4 is 16, T^3. The derivative of 2T is just 2, and then the derivative of a constant, well, that's zero; we've already seen that.

So there you have it. This is the rate of change of the X component with respect to T, this is the rate of change of the Y component with respect to T. One way to do it, and you know a vector can represent many, many, many different things, but the type of a two-dimensional vector like this, you could imagine this being H of T being a position vector in two dimensions.

If you're looking at the rate of change of position with respect to time, well then this would be the velocity vector. If we were to take the derivative of this with respect to time, well, we're going to get the acceleration vector.

So if we say H prime prime of T, what is that going to be equal to? H prime prime of T, well we just apply the power rule again. So 4 * -5 is equal to -20 T^(4 - 1), so T^3. Then we have 3 * 16 is 48 T^2, and then the derivative of 2 is just zero.

So there you have it. For any, if you view T as time, for any time, if you view this one as position, this one as velocity, and this is acceleration, you could, this would now give you the position, velocity, and acceleration. But it's important to realize that these vectors could represent anything of a two-dimensional nature.

More Articles

View All
My Lightbulb Moment: Using Solar Energy to Feed a Village | National Geographic
Energy is life. My light bulb moment came during a trip to a remote part of China in 1994. We delivered simple solar home systems to families that had never before experienced electricity. Witnessing these families flip a switch and have electric lights c…
Variance of a binomial variable | Random variables | AP Statistics | Khan Academy
What we’re going to do in this video is continue our journey trying to understand what the expected value and what the variance of a binomial variable is going to be, or what the expected value or the variance of a binomial distribution is going to be, wh…
WARNING: The Great Reset Of 2022 Explained
[Music] All right, fine, don’t ask me again. I will talk about the great reset. By the way, it feels weird starting a video off without the normal introductions, so let’s get that out of the way. What’s up, son? It’s Dad here, and yes, I will talk about t…
Safari Live - Day 276 | National Geographic
This program features live coverage of an African safari and may include animal kills and carcasses. Viewer discretion is advised. Good afternoon, everybody, and welcome to Open Skies in the Maasai Mara! You can see there’s a few little puffy clouds in t…
Why the Sky ISN'T Blue
Happy 500,000! Thank you guys so much for subscribing to my channel and for joining me on this scientific adventure. You know, if you got 500,000 people together and we all held hands in a line, it would stretch from Sydney to Melbourne or from San Franci…
Dashes | Punctuation | Khan Academy
Hey grammarians! Hey Paige! Hi David! Today we’re going to talk about dashes, which is a piece of punctuation that looks kind of like this—um, it’s just kind of a straight line. Later we’re going to talk about hyphens, which look like this. There is a dif…