Taylor polynomials for multivariable functions

• SomeGuy
In summary, the linear approximation to a function of several variables is the equation of the tangent plane to the surface at a given point. When the base point is (0, 0, 0), the linear polynomial is 91 + 2x + 3y + 8z, but when the base point is (1, 2, 3), the linear polynomial is different due to the different slopes of the tangent plane. Translating the tangent plane to the origin does not necessarily make it tangent to the surface at that point. The general formula for the linear approximation is f(x_0,y_0,z_0)+ \frac{\partial f}{\partial x}(x_0,y_0,z_
SomeGuy
Ok there's something I don't get. I know for instance that the linear polynomial for say f = 91 + 2x + 3y + 8z + Quadratic(x, y, z) + Cubic(x, y, z) ... is 91 + 2x + 3y + 8z if the base point is (0, 0, 0). This is pretty clear. What I don't get is why when you take the base point to be say (1, 2, 3) all of a sudden 91 + 2x + 3y + 8z is no longer the linear approximation. I figure it's because we have to move the graph from (1, 2, 3) to the origin. But I did that, and that didn't seem to work since the constants didnt' work out. Any ideas on the mathematical and intuitive reasoning behind why the linear polynomial for (0, 0, 0) doesn't work? Thanks.

I'm not sure I get your drift here. The tangent plane at (0,0,0) is generally different from the tangent plane at (1,2,3). It's going to have different slopes, right?

Dick is exactly right. The linear approximation to a function of several variables is the equation of the tangent plane to the surface at that point. You can translate the tangent plane to the origin but you can't expect it to be tangent to the surface at that point.

The linear approximation to f(x,y,z) at $(x_0,y_0,z_0)$ is
$$f(x_0,y_0,z_0)+ \frac{\partial f}{\partial x}(x_0,y_0,z_0)(x-x_0)+ \frac{\partial f}{\partial y}(x_0,y_0,z_0)(y-y_0)+ \frac{\partial f}{\partial z}(x_0,y_0,z_0)(z-z_0)$$

What are Taylor polynomials for multivariable functions?

Taylor polynomials for multivariable functions are a way to approximate a multivariable function using a polynomial. They are similar to Taylor polynomials for single variable functions, but they take into account multiple variables.

What is the purpose of using Taylor polynomials for multivariable functions?

The purpose of using Taylor polynomials for multivariable functions is to approximate a complex function with a simpler polynomial function. This can make it easier to analyze and work with the function.

How are Taylor polynomials for multivariable functions calculated?

Taylor polynomials for multivariable functions are calculated using the same principles as Taylor polynomials for single variable functions. The coefficients of the polynomial are found by taking the partial derivatives of the function at a specific point, and the degree of the polynomial determines the accuracy of the approximation.

What is the difference between Taylor polynomials and Taylor series for multivariable functions?

The main difference between Taylor polynomials and Taylor series for multivariable functions is that Taylor polynomials are finite approximations, while Taylor series are infinite sums. Taylor series provide a more accurate approximation, but they are also more complex to calculate and work with.

How are Taylor polynomials for multivariable functions used in real-life applications?

Taylor polynomials for multivariable functions are used in many real-life applications, particularly in fields such as physics, engineering, and economics. They can be used to approximate complex functions in order to make predictions or solve problems, and they are also used in numerical analysis and computer graphics.

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