I'm working on a practice problem for my Calculus 3 course. It gives the function: $z=x^2+y^2$, and asks to graph the contours for $c=1,2,3$. Than asks to calculate the gradient at point $(2,1)$ and graph the result.
I'm fine with the first part. I'm letting z=c and solving for y, then graphing the result.It's the gradient portion I'm having issues with.
The gradient I came up with is: $\nabla(x^2+y^2)=\langle 2x,2y\rangle$, at $(2,1)$, = (4,2)
I'm not exactly sure how to graph this. Am I supposed to graph a line from $(2,1)$ to $(4,2)$ or what? Any help/explanation would be very much appreciated!
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$\begingroup$Gradients are drawn from the point that they're taken at. This shows where gradients are taken from, and allows gradients to be perpendicular to level curves. Since the gradient was taken at the point $(2,1)$, the vector $\langle 4,2 \rangle$ should be drawn from $(2,1)$ pointing to the point $(6,3)$ because $(2,1) + (4,2) = (6,3)$.
The gradient is a vectorfield, i.e. a vector attached to every point of you space. The most clear way to draw it is to draw an arrow of length (4,2) starting from the point (2,1).
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