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# Second derivatives (vector-valued functions)

AP.CALC:
CHA‑3 (EU)
,
CHA‑3.H (LO)
,
CHA‑3.H.1 (EK)

## Video transcript

so I have a vector-valued function each year when I say vector-valued it means you give me a t it's a function of T and so you give me a T I'm not just going to give you a number I'm gonna give you a vector and as we'll see you're gonna get a two-dimensional vector you could view this as the X component of the vector and the Y component of the vector and you 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 you viewed as engineering notation here where the X component is being multiplied by the horizontal Communitech Terr so you might see something like that where that's the unit vector plus the Y component for T to the fourth plus 2 T plus 1 is multiplied by the vertical unit vector so these are both representing the same thing it just has a different notation and sometimes you'll see a vector valued functions with an arrow on top to make it explicit that this is a vector valued function sometimes we'll people 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 other 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 let's 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 take the derivative of the respective components with respect to T so the X component with respect to T if you were to take the derivative so with respect to T what are you going to get well we're gonna use the power rule right over here 5 times the negative 1 or times a negative you're gonna get negative 5 times T to the 5 minus 1 power so T to the 4th power the derivative with respect to T of negative 6 well that's just 0 so that's the rate of change of the X component with respect to T and now we go to the Y component so we're gonna do the same thing derivative with respect to T is going to be and then once again we just use the power rule 4 times 4 is 16 T to the third power of Tootsie is just too and then derivative of a constant well that's zero we've already seen that so there you have it so 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 and one way to do it and you know vectors can represent many many many different things but the the type of a two-dimensional vector like this you could imagine this being H of T being a position vector in two dimensions and then if you're looking at the rate of change of position with respect to time well then this would be the velocity vector and then 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 four times negative 5 is equal to negative 20 T to the 4 minus 1 so T to the third power and then we have three times 16 is 48 T squared and then the derivative of 2 is just zero and so there you have it for any if you view T is time for any time if you view this one as a position this one is 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
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