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0:00 // What is linear approximation?
0:44 // When do you use linear approximation?
1:28 // Estimating square roots using linear approximation
5:23 // Estimating trig functions using linear approximation
6:37 // How to find the error in a linear approximation
7:48 // Summary
Linear approximation, or linearization, is a method we can use to approximate the value of a function at a particular point. The reason liner approximation is useful is because it can be difficult to find the value of a function at a particular point.
Square roots are a great example of this. We know the value of sqrt(9); it’s 3. That’s easy to figure out. But we don’t know the value of sqrt(9.2). We can guess that it’s a little bit more than 3, since we know that sqrt(9) is 3, and 9.2 is a little bit more than 9, but other than that, we don’t know how to find a better estimate of sqrt(9.2). That’s where linear approximation comes in to help us.
Since we’re dealing with square roots, if we imagine the graph of the function sqrt(x), we know one point on that function is (9,3). If we find the tangent line to the function sqrt(x) through the point (9,3), then we can see that, since the tangent line is really close to the graph of the function around the area of (9,3), that the value of the function and the value of the tangent line will be pretty close to each other at x=9.2.
So to get an estimate for sqrt(9.2), we’ll use linear approximation to find the equation of the tangent line through (9,3), and then plug x=9.2 into the equation of the tangent line, and the result will be the value of the tangent line at x=9.2, and very close to the value of the function at x=9.2.
That’s why linear approximation is so helpful to us, because it’s a quick, simple method that let’s us estimate a value that would otherwise be very difficult to find.
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