Function Gradients and Level Curves: Insights and Applications

Putting it together The graph below shows the function x squared plus y squared. We have seen before that the gradient of a function is always perpendicular to its levelcurves. And we see that it can be both big and small in some places. The value of the function is greater in areas where the gradient is steep; these are placeswhere the slope is steep. For comparison, here is a three-dimensional graph of the function x2 + y2. The curve will match up with the corner in the level-curve picture. That is this cornerlocated in the 3D picture. In the level curve picture, this point in the middle represents the bottom of the bowl.

At the bottom of the bowl, the slope is shallow. The arrows are short near the edges andlong in the middle, indicating that the slope is steepest at those points. The arrows areshort near the edges and long in the middle, indicating that the slope is steepest at thosepoints. That is the meaning of the arrows' length.