 Hello and welcome to the screencast on section 10.4, linearization, tangent planes, and differentials. One of the central concepts in single variable calculus is that the graph of a differentiable function when viewed on a very small scale looks like a line. We call this line the tangent line and measure its slope with the derivative. In this section, we will extend this concept to functions of several variables. Let's see what happens when we look at the graph of a two variable function on a small scale. Consider the graph of this function f shown here. We are going to see what happens when we zoom in on the highlighted point where x equals one and y equals one. Just as the graph of a differentiable single variable function looks like a line when viewed on a small scale, we see that the graph of this particular two variable function looks like a plane as we zoom further and further in. In this section, we'll learn how to find the equation of this plane called the tangent plane. But first, we'll take a look at a few important definitions. It turns out that the existence of a tangent plane at a point relies on the graph looking like a plane as we zoom further and further in. We call this property locally linear. Aside from zooming in on the graph, how can we tell when a function of two variables is locally linear at a point? In single variable calculus, we needed the derivative to exist at a point for the function to be locally linear. However, it turns out that this is not enough for the partial derivatives to exist at a point for f, a function of two variables to be locally linear. Since these partials only tell us about the derivative in specific directions, we need a condition that's a bit stronger. We say that f is continuously differentiable at a point x naught y naught if both first order partial derivatives exist and are continuous on an open disk containing this point. Whenever f is continuously differentiable at some point, then we know that f has a tangent plane at that point. We are now ready to state the general formula for a tangent plane. If f has continuous first order partial derivatives, then the equation of the plane tangent to f at the given point has the following equation. Note that this equation relies on the equation of a plane formula we looked at in section 9.5. One important application of the tangent plane is linearization. The tangent plane to the graph of a function f at a point x naught y naught provides a good approximation of the function f near x naught y naught. We define the linearization L to be the function whose graph is this tangent plane. We can use the linearization to estimate values of f near this point x naught y naught. While the linearization enables us to estimate values of the function for points near the base point, sometimes we are more interested in the change in the function as we move from the base point to another point. A simple way to estimate the change in a function f given by delta f here is to use the differential, df, which represents the change in the linearization. We call the quantities dx, dy, and df differentials and we think of them as measuring small changes in the quantities x, y, and f.