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Problem 44 Easy Difficulty

Find a vector function that represents the curve of intersection of the two surfaces.

The paraboloid $ z = 4x^2 + y^2 $ and the parabolic cylinder $ y = x^2 $


$\mathbf{r}(t)=<t, t^{2}, 4 t^{2}+t^{4}>$

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Video Transcript

and this problem, we are beginning to build or intuit intuition of vector equations and vector functions. And this is going to come up quite often in your later calculus courses. Or maybe in this course, because we're building up to using vectors in calculus, using vectors in integration and differentiation. So this is a great precursor to understanding those applications of the vector. So in this case, what we're doing essentially is we're finding the intersection off two functions to shapes in a plane, and we have to define that intersection using a vector function. So we have the equations. For those two shapes we have Z equals four X squared plus y squared, and why equals X squared. So what we need to dio is we're going to define a parametric representation in order to determine our vector function. So we're going to say let X equal t Well, that must mean that why which equals X squared has to be t squared were given by that equation. Now Z, we know is four x squared plus y squared. So let's just plug in T for X. We have fourty squared plus t squared squared, so we would get fourty squared plus t to the fourth. Now, if you're confused as to why we plugged in T Square here, that's because we defined why previously as t squared. So what does that mean? We haven't X, y and Z, so we can define our vector function are vector function which will car call Pardon me are of tea Our vector would be t t squared fourty squared plus t to the fourth. That is just our functions for X, y and Z. So this was not required in the problem. But just to build our intuition of the domain of these functions, T could be any real number and also ex could be any real number. And then Y and Z must be greater than or equal to zero. And these would still hold for it's it's, um, functions of Tia's Well, So I hope that this problem helped you understand a little bit more about vector functions and how we can define a vector function using our knowledge of the intersection of two functions in a plane with three dimensions