The Geometry of Linear Approximation

The geometry of linear approximation

• A linear approximation of a function is a polynomial that passes through the sample points and has the same slope at each of them.
• The linear approximation of a function is often used in statistics; many statistical methods assume that the data are generated by some underlying model, and the linear approximation provides an alternative expression for the model’s predictions.

Let’s go through it. So we take the x derivative of that. If we plug in x=1 and y=1, the equation becomes negative 4.

If y is the function of x and we take the derivative of y with respect to x, we obtain 2y. When we plug in for x and 1 for y: we get 2. f(x,y)=y^2-x^3-x F_x=-3x^2 -1 F_y=2y F(1,1)=-1 F_x(1,1)=-4 F_y(1,1)=2 One can then write the linear approximation to this function as the value of the function times its derivative multiplying by delta x.

In this box, there is a negative 4. And then, the derivative of y with respect to x multiplied by delta y, which equals 2 f(1+Δx,1+Δy)~~-1+ -4Δx+2Δy Now what is the other way to write it? The change in x is delta x. So let’s make a little image here. So here’s the point (1, comma 1). and take another point (x, comma y) near (1, comma 1). Delta x is the change in x, or x minus 1. The change in y is delta y, which is y minus 1.

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Then I can substitute this expression into the equation for delta x and delta y. x The function f(x, y) = -1 + 4x – 4y + 2 delta y is illustrated in the graph below. [Graph] We can group the terms of this equation to make it look like a times x plus a number of times y plus a number. And the only place that x’s can come from is here, so we have a negative 4x. The y’s can be derived from one source, which is here, so we have plus 2y. F(x,y)~~-1-4(x-1)+2(y-1) And then there are several places from which the constant can be obtained. From here, we see a negative 1 and from there a plus 4. From there is a minus 2. If these numbers are added together, we get a total of 1. F(x,y)~~-1-4(x-1)+2(y-1)=-4x+2y+1 The linear approximation written the other way is negative 4x plus 2y plus 1. F(x,y)~~-4x+2y+1 Here’s a visual representation of the functions that we just did, and then you’ll understand why we have two different ways to do them.