Find the directions of maximum and minimum change of f at the given point, and the values of the

Find the directions of maximum and minimum change of f at...

Key Concepts

Maximum and Minimum Rates of Change

The maximum rate of change of a function at a given point is the magnitude of the gradient vector, and it occurs in the direction of the gradient. Conversely, the minimum rate of change is the negative of the gradient's magnitude, and it occurs in the direction opposite to the gradient. These rates indicate the steepest ascent and descent respectively.

Unit Vector

A unit vector is a vector with a magnitude of 1. When determining directions in the context of the directional derivative, it is important to use a unit vector in the direction of interest so that the magnitude of the change is accurately represented by the dot product with the gradient.

Gradient Vector

The gradient vector of a function f(x, y) is a vector comprised of the first partial derivatives with respect to x and y. It points in the direction of the steepest increase of the function, and its magnitude represents the rate of the increase at that point. The gradient is computed using the partial derivatives: ?f = (?f/?x, ?f/?y).

Directional Derivative

A directional derivative measures the rate at which a function changes at a point in a specified direction. It is computed as the dot product of the gradient vector and a unit vector pointing in the given direction. This concept generalizes the idea of a derivative to any direction in the domain rather than along the coordinate axes.

Transcript

00:01 In this video, we need to determine the directions of max and min change of f at the following points, and we also have to find the values of those max and min changes.

00:12 In both cases, we need to find the gradient, because the gradient will produce the direction of max change, and its opposite will be the direction of minimum change.

00:20 So we have to either way find our gradient.

00:24 So our gradient is a function of x and y.

00:26 Recall is the partial derivative with respect to x, partial with respect to y.

00:31 That's 2x squared minus y to the one half so that's partial derivative of the respect to x is is this times a 4x and then with respect to y it's similar okay except for the negative one so cleaning that up we'll get that our gradient it's actually rated as coordinates so it becomes in the end so cleaning that up you get 2x divided by the square root of 2x squared minus y and then you get a negative one divided by the square root of 2x squared minus y okay so let's evaluate this now for part a at the point three comma two which is a rast okay so the gradient at three comma two is going to be six two x squared minus y is two times nine minus two which is the square of 16 which is four and here i get a negative one over and i forgot a two.

01:50 There's a two down here that i missed so that ends up being a two times the square root of 16 which is two times four which is eight.

02:02 Okay so in the end we can simplify that to three halves comma negative one eighth.

02:10 Okay so the direction of maximum increase is three halves negative one eighth.

02:27 Okay and the max change, i should use the language, max change...

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