Here's some vocabulary:[br][quote]The line tangent to a function [math]f[/math] (that's differentiable) at [math]x=a[/math] is also called the [i]linearization [/i]of [math]f[/math] at [math]a[/math]. [/quote]You can use the linearization of a function [math]f[/math] at [math]a[/math] to approximate values of [math]f[/math] near [math]x=a[/math]. This technique is also called [i]tangent line approximation[/i].[br][br]Here's an example. Suppose someone asks you to estimate [math]\sqrt{125}[/math] (without a calculator).

But with calculus, we can do much better than that, even in our heads (or at least with paper and pencil only).[br][br]We're going to use the linearization of [math]f(x)=\sqrt{x}[/math] at [math]121[/math] (because [math]121[/math] is the nearest perfect square to [math]125[/math]). Determine this linearization and input your answer in the applet below as the function [math]l(x)[/math].[br]

In other words, plugging [math]125[/math] into [math]l(x)[/math] (which you can do without a calculator) should yield a result that is extremely close to [math]\sqrt{125}[/math] because [math]l(125)\approx f\left(125\right)[/math].[br][br]Use the scientific calculator below to compute these two quantities.