Linear Approximations and Derivatives in Multivariable Functions

Linear approximations and tangent planes. Linear approximation: multivariable version The linear approximation for the function f of (x, y) is supposed to give a good approximation of how f behaves if we change x a little bit or we change y a little bit or both. When we start at a point (x0, y0) and then imagine adding a small amount of changeto x or y, we want to approximate the original value of f at (x0, y0). This is done byadding the changes that come from increasing delta x or decreasing delta y. When considering the derivatives of functions in multiple variables, we mustremember that there are two different things we could change. We could change thevalues of x, or we could change the values of y. Thus, there are two differentderivatives that describe the effect of these two different changes: one that describeshow delta x affects x and another that describes how delta y affects y. The derivative of x with respect to x describes how a function changes if we increasex by a small amount. The derivative of y with respect to y describes how the functionchanges if we change y by a small amount. Let's look at an example to see how it works. In our example, we will use the first function that you all discussed in recitationyesterday: f(x, y) = x2 + y2. In recitation, you computed its derivatives. So the xderivative.When taking a derivative of a function that includes both variables, it is important tothink of one variable as a constant. For instance, if you were taking the derivative of

x2 + 7x, the resulting function would have a constant value of 2x. And the derivativeof that would be 0. Let (x0, y0) be (negative 1, 1). Then f(x0, y0) = 2. If (x0, y0) is substituted into f(x),then f(−1) = −2. We can now write the linear approximation to the function byplugging in negative 1 plus delta x, 1 plus delta y, and finding that the linearapproximation is approximately 2.The first term to evaluate is the squared derivative of f(x) with respect to x, timesdelta x. This equals 2. Next, evaluate the derivative of f(x) with respect to y, timesdelta y; this is equal to negative 2 times delta y.

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