Linear Approximations and Tangent Planes. Linear Approximation Review

Linear approximations and tangent planes. Linear approximation: review

Given a function of one variable, g of x, linear approximation tells us how the function a changes if we change x by a small amount. If we look at the graph of g at x0 plus Delta x, we see that it is approximately the graph of g of x0 plus g prime of x0 times Delta x.

This approximation works well if Delta x is small.

Let’s do simple example.

Perhaps g of x is x squared, and x0 is 1. So we g get g of 1 plus Delta x. That’s approximately 1 plus 2 Delta x. 2 And there’s a slightly different way of writing this, which also is often helpful. If x is close to 1, then I could write x is equal to plus Delta x. And that means that Delta x equals the change in x away from 1 is x minus 1. And g then g of x is g of 1 plus Delta x, which is plus 2 times Delta x. And the Delta x is equal to x minus 1. 1 2 x So it’s 1 plus 2 times x minus 1, which if you do the algebra is 2x minus 1. Linear approx g(x) g(x_0+Δx) ~~ g(x_0)+g(x_0)Δx Ex g(x)=x^2 x_0=1 g(1+Δx) ~~ 1+2Δx If x new 1 x=1+Δx Δx=x-1 g(x)g(1+Δx)=1+2Δx=1+2(x-1)2x-1 [Graph] Here is the function g of x. Here is the derivative of g, which is simply 2x squared.

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And what is the linear approximation? One thing we could write is g is equal to two times x minus one. So it looks like that. So here in white is the actual function g,and in green is y equals two times x minus one. If we wanted to consider the 1 + Delta x version of our equation, we could substitute 1 + Delta x for g in our original equation. The linear approximation is a function that describes how the curve changes as the value of x approaches zero. The main point to remember is that when Delta x is small, the green line and white curve are nearly identical. This means that when Delta x is small, the linear approximation is accurate. When Delta is large, the linear approximation is not very good. And you can think of it as the line that hugs the curve as closely as possible. And because it’s trying to hug the curve as closely as possible, it’s tangent to the curve. So this here is a tangent line.