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Linear approximation is a method for estimating a value of a function near a given point using calculus. In this lesson, you’ll learn how to find a linear approximation and see an example of how it can be used.
Linear Approximation/Linearization
Linear approximation is a method of estimating the value of a function, f(x), near a point, x = a, using the following formula:
Linearization Formula
The formula we’re looking at is known as the linearization of f at x = a, but this formula is identical to the equation of the tangent line to f at x = a.
This lesson shows how to find a linearization of a function and how to use it to make a linear approximation. This method is used quite often in many fields of science, and it requires knowing a bit about calculus, specifically, how to find a derivative.
Tangent Lines and Linearization
Let’s review a basic fact about derivatives. The value of the derivative at a specific point, x = a, measures the slope of the curve, y = f(x), at that point. In other words, f ‘(a) = slope of the tangent line at a.
The tangent line to a function at a specific point.
Linear approximation as a tangent line
Now, the tangent line is special because it’s the one line that matches the direction of the curve most closely, at the specific x-value you are interested in. Notice how close the y-values of the function and the tangent line are when x is near the point where the tangent line meets the curve.
So, if the curve y = f(x) is way too complicated to work with, and if you’re only interested in values of the function near a particular point, then you could throw away the function and just use the tangent line. Well, don’t actually throw away the function. . . we may need it later!
Formula for Linearization
So, how do you find the linearization of a function f at a point x = a? Remember that the equation of a line can be determined if you know two things:
The slope of the line, m
Any single point that the line goes through, (a, b).
We plug these pieces of info into the point-slope form, and this gives us the equation of the line. (This is just algebra, folks; no calculus yet.)
y – b = m(x-a)
But, in problems like these, you will not be given values for b or m. Instead, you have to find them yourself. Firstly m = f ‘(a), because the derivative measures the slope, and secondly, b = f(a), because the original function measures y-values.
Putting it all together and solving for y:
Formula for linearization of a function
The last line is precisely the linearization of the function f at the point x = a. Now that we know where the formula comes from, let’s use it to find a linear approximation.
 
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