maximum rate of change multivariable calculus

That’s more fun, right?įirst, when we reach the hottest point in the oven, what is the gradient there? Don’t eat that cookie!īut before you eat those cookies, let’s make some observations about the gradient. Every time we nudged along and follow the gradient, we’d get to a warmer and warmer location.Įventually, we’d get to the hottest part of the oven and that’s where we’d stay, about to enjoy our fresh cookies. We’d keep repeating this process: move a bit in the gradient direction, check the gradient, and move a bit in the new gradient direction. This new gradient is the new best direction to follow. We get to a new point, pretty close to our original, which has its own gradient. The gradient is just a direction, so we’d follow this trajectory for a tiny bit, and then check the gradient again. Now, we wouldn’t actually move an entire 3 units to the right, 4 units back, and 5 units up. In this case, the gradient there is (3,4,5). Thus, we would start at a random point like (3,5,2) and check the gradient.

Remember that the gradient does not give us the coordinates of where to go it gives us the direction to move to increase our temperature. So, the gradient tells us which direction to move the doughboy to get him to a location with a higher temperature, to cook him even faster. In this case, our function measures temperature. The gradient at any location points in the direction of greatest increase of a function. He’s made of cookie dough, right? We place him in a random location inside the oven, and our goal is to cook him as fast as possible. Now suppose we are in need of psychiatric help and put the Pillsbury Dough Boy inside the oven because we think he would taste good.

The gradient is a direction to move from our current location, such as move up, down, left or right. The coordinates are the current location, measured on the $x,y,z$ axes. With me so far? We type in any coordinate, and the microwave spits out the gradient at that location.īe careful not to confuse the coordinates and the gradient. But this was well worth it: we really wanted that clock. Unfortunately, the clock comes at a price - the temperature inside the microwave varies drastically from location to location. The microwave also comes with a convenient clock. We can type any 3 coordinates (like “3,5,2″) and the display shows us the gradient of the temperature at that point. Suppose we have a magical oven, with coordinates written on it and a special display screen: I’m a big fan of examples to help solidify an explanation. Likewise, with 3 variables, the gradient can specify and direction in 3D space to move to increase our function. If we have two variables, then our 2-component gradient can specify any direction on a plane.

However, now that we have multiple directions to consider ($x$, $y$ and $z$), the direction of greatest increase is no longer simply “forward” or “backward” along the $x$-axis, like it is with functions of a single variable. The gradient of a multi-variable function has a component for each direction.Īnd just like the regular derivative, the gradient points in the direction of greatest increase ( here's why: we trade motion in each direction enough to maximize the payoff). The regular, plain-old derivative gives us the rate of change of a single variable, usually $x$. Now that we know the gradient is the derivative of a multi-variable function, let’s derive some properties. “Gradient” can refer to gradual changes of color, but we’ll stick to the math definition if that’s ok with you. Yes, you can say a line has a gradient (its slope), but using 'gradient' for single-variable functions is unnecessarily confusing. The term 'gradient' is typically used for functions with several inputs and a single output (a scalar field). Is zero at a local maximum or local minimum (because there is no single direction of increase).Points in the direction of greatest increase of a function ( intuition on why).The gradient is a fancy word for derivative, or the rate of change of a function.

Maximum rate of change multivariable calculus