In this section, the equation of the tangent line is very important! We are going to find an equation of a tangent line at a point, but then plug in another x or y value and use the tangent line at one point to approximate a point very close to it. 🤩
The picture above helps us visualize what we are doing when we use a tangent line to approximate value from a function. The closer x2 is to X1, the better the estimate will be. 🤓
If the function is concave up, the tangent line approximation will be an underestimate, because the whole tangent line will always be under the curve, making every value less than the actual function value.
What happens when the function is concave down? The line approximation will be an overestimate. Why? Try to write it down in your own words!
Let's take a closer look at that linearization formula: f(x2) - f(x1) = f'(x)(x2 - x1). That might look a little familiar to you from your algebra days as the point-slope form of a line! f(x2) and f(x1) are y2 and y1, f'(x) is the slope, and x2 and x1 are, well, x2 and x1! Knowing this can help you remember the linearization "formula" when you need it, and plus, now you understand it!
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4.6Approximating Values of a Function Using Local Linearity and Linearization