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### Course: Multivariable calculus>Unit 2

Lesson 4: Differentiating parametric curves

# Vector valued function derivative example

Concrete example of the derivative of a vector valued function to better understand what it means. Created by Sal Khan.

## Want to join the conversation?

• at , Sal says that the path is the same. Yes, the paths appears to look the same, but how can they truly be the same if the graphs use different scalings (in x axis). If the graph on the right were scaled the same way as the graph on the left (each "notch" on the x and y axis represents 1 unit), the graph on the right would be much steeper and thus the "path" would -not- be the same.
• He is not using a different scalar multiple. Think of the vector r as (x, (y)^2). Now what he did was multiply r by 2 (2*r) which would multiply x and y in the same fashion (2*x, (2*y)^2), However (2*y)^2 = 4*y^2. Remember order of operation is parenthesis then exponents.
• I am new to this site, guys - I would like to find exercises associated with the lectures, and I know there are some on the site, but I have no idea how to identify / access them. Can anybody help and show me where to go (for example if I wanted to find an exercise on the exact content of this video, how would I do it)? Sorry for being thick.
• I assume you already figured this out since it has been 8 months, but if not no worries. The practice problems are only for certain sections of math. While they haven't added many for Multivariable calculus, they do have practice problems for Integration, Differentiation, and pre-k12 math (up to Pre calc.) I hope they add problems for all math and science related subjects in the future, but as of 7/6/2016 they do not have practice questions for everything. As for accessing them, go to the section you desire in the subjects tab at top and click on it in the drop down menu. Then it should have an option on that page to do practice problems. It will say something like continue or start your mission.
• Hello, I have a question. If I decide to plot this path as a function F(t) perpendicularly to the xy plane, wouldn't the derivative of r(t) behave just like the gradient of the F(t)? For me it looks so much like a gradient of some function, given that it would be directed perpendicularly to it and point in the direction of highest increase in its value. I might be confused though. Any help?
• A gradient implies you're dealing with a surface or a function that has 2 inputs, but in this case we only really have 1 input that changes everything, which is t. Since there's only the input t, the parametric system forms a path like a line rather than a surface.
• Umm this probably gonna be a stupid one but why does the vector originate from the point (1,1) and not from the origin of the graph (0,0)?
• Sal has the vector start at (1,1) to better visualize the movement of the particle at that specific point. Yes, we usually start vectors at the origin, but we don't have to, and in this case it wouldn't make sense to talk about the movement of a particle at (1,1) and not put our vector (which represents that movement) at the specific point we are talking about. Hope this helps!
• the velocity vector for the second graph is wrong, it should be double the length that is drawn.
• For the second r vector function. The graphical representation of the slope vector does not mirror the notation I believe. If we using I hat and j hat, then those are unit vectors. And since the scale of the graph is different to the first one, when Sal is moving 2 units to the right and 4 units up, he is really moving 1 whole i hat unit to the right and 2 whole j hat units up. I believe.
• The graphs and the notation agree. There is nothing wrong with using î and ĵ to represent a slope vector. You can use a linear combination of î and ĵ to represent any two-dimensional vector, including slope vectors. Also, while Sal did not go all the way and give the slope vector the full length, it was definitely larger than 1î + 2ĵ.
• Aren't the two curves just look the same because they have different x scales? I don't understand why Sal said they are essentially the same.
• Is it safe to say that the magnitude of a velocity vector is synonymous as the slope in 2-dimensional analysis?
• lots of people are getting confused and asking if the "scale" of the second graph is correct or not. I don't think they realize that Sal said t = 1/2 at , they think t ϵ {0, 1, 2} in the second graph when it's actually t ϵ {0, 1/2, 1}
(1 vote)
• why derivative of constant vector valued functionals i
s not zero?i could not grasp at this concept for years.plz help
(1 vote)