Question

You are walking on the graph of f(x, y) = y cos(πx) − x cos(πy) + 16, standing at the point (2, 1, 19). Find an x, y-direction you should walk in to stay at the same level.

Answer 1

To walk and stay at the same level on the graph of the function f(x, y), one must find and move in a direction perpendicular to the gradient of the function at that point. This is similar to following a contour line at a constant altitude on a topographic map.

Step-by-step explanation:

The student is asking about walking on a level curve of the function f(x, y) = y cos(πx) − x cos(πy) + 16, which represents a three-dimensional surface.

To stay at the same level, one must move along a path that is entirely contained within a single contour line of the function. This is similar to following a path along a constant altitude on a topographic map.

This situation can be compared to walking on a two-dimensional field according to certain vector displacements.

For instance, if you walk 18.0 m straight west and then 25.0 m straight north, we can calculate how far you are from your starting point and the compass direction of your final position using vector addition.

This process involves finding the resultant R = A + B, where A and B are the two legs of the walk represented as vectors.

The length of the resultant vector R can be found using the Pythagorean theorem, R = √(Rx² + Ry²), and the direction using the arc tangent function, θ = tan⁻¹ (Ry/Rx).

However, the student's scenario specifically involves a three-dimensional function f(x, y), and so the approach must consider the gradient of this function (which indicates the direction of greatest increase) and seek out a direction perpendicular to this gradient, which will be the direction where the level remains constant.

Answer 2

To stay at the same level on the graph, one must walk in a direction perpendicular to the gradient at the point. The gradient at the point (2, 1) is <-1, -1>, so a direction perpendicular to this could be <1, -1>.

Step-by-step explanation:

To find an x, y-direction you should walk in to stay at the same level on the graph of f(x, y) = y cos(πx) − x cos(πy) + 16, standing at the point (2, 1, 19), we must consider the level curves (contour lines) of this function. To remain at the same level, a person must move in a direction where the function's value does not change, which is tangent to the level curve passing through the point (2, 1).

Mathematically, this is equivalent to finding the gradient of f(x, y) and moving in a direction perpendicular to it. The gradient is a vector pointing in the direction of the steepest ascent on the graph. Therefore, to stay at the same level, you need to move in a direction orthogonal to the gradient since this represents no change in f(x, y)'s value.

Calculating the gradient at the point (2, 1), we get the vector:

grad f(2, 1) = \∇f(2, 1) = <−cos(π), −cos(2π)> = <−1, −1>.

To stay on the same level, you need to walk in a direction perpendicular to <−1, −1>. One such direction is <1, −1>, which means you should walk in a direction where you increase your x-coordinate by the same amount that you decrease your y-coordinate.