Question

This is the question from Exam--2 of the course 18.02.

Q: On the contour plot below, mark the portion of the level curve f = 2000 on which ∂f/∂y≥ 0.

I will post the figure below.. Wait..

Answer 1

I couldn't get that figure here so I uploaded the exams and its solutions. The question is Problem 2.

I have both but I have great difficulty understanding this question... and contour plot in general...

Can someone explain the solution to me?

Answer 2

Have a look at the solution. A tangent line to any contour line corresponds to where the directional derivative of the function f(x,y) is zero. Basically, if we walk around a contour line we are staying at a constant value of the function (f(x,y)=2000 in this case). Since the value of the function never changes, the directional derivative is zero. In general, the directional derivatives will have contributions from both the x and y variables. The exceptions to this are when the direction derivatives are either horizontal (moving only in the x direction) or vertical (moving only in the y direction). If we are only moving in the x direction, keeping y constant, this is the partial derivative of f(x,y) with respect to x. If we are only moving in the y direction it means we keep x constant, which is the partial derivative of f(x,y) with respect to y. On the solution to this problem, where the dots are drawn, the tangent line to the contour f(x,y)=2000 is vertical. The vertical direction corresponds to the y direction, and we are keeping x constant. So, the partial derivative of f(x,y) with respect to y is zero at these points because we are ONLY moving in the y direction and we are staying on the curve f(x,y)=2000 (which is a constant). Now, where is the partial wrt y greater than zero? Well, pick any point on the coutour line f(x,y)=2000 and check the following: If you move from this point going up ONLY, are you going towards contours that are greater than 2000? If yes, then the partial wrt y at this point is greater than zero. If you are "walking" to values of lower potential then the partial wrt y will be less than zero at that point.

Answer 3

@eseidl .. thank you Sir/Madam. Wow!! that was great explanation from you. Suddenly the whole solution and the relation of Contour plot vs Partial derivative started making sense to me.

Answer 4

no problem, glad to help :)