Finding Gradient of a Scalar Function

Find the gradient of a scalar function $z = f(x, y) = x^2 + y^2$, and evaluate it at the points (2, 1) and (-1, -1).

The gradient of a scalar function is a vector that captures both the direction of the greatest rate of increase of the function and the rate of increase itself. For a function of two variables, like the scalar function given here, the gradient can be found by taking the partial derivatives with respect to each of the variables involved. This results in a vector where each component is the partial derivative of the function with respect to one of the variables. Conceptually, this gradient vector points in the direction in which the function increases fastest. Moreover, the length of this vector indicates how steep the ascent is in that direction.

Evaluating the gradient at specific points provides insight into how the function changes in the vicinity of these points. At point (2,1), the gradient will tell us the direction and rate of change of the function right at that location in the xy-plane. Similarly, evaluating at (-1,-1) gives us information specific to that location. Understanding how to compute and interpret the gradient is essential in fields like optimization, where you need to find maximum or minimum values of functions, and in physics, where gradients represent fields such as temperature or pressure changes over space.

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