Linearization and Tangent Line Approximation

Here's some vocabulary:
The line tangent to a function (that's differentiable) at is also called the linearization of at .
You can use the linearization of a function at to approximate values of near . This technique is also called tangent line approximation. Here's an example. Suppose someone asks you to estimate (without a calculator).

Without using any calculus, you know is a number between which two consecutive integers? Why?

But with calculus, we can do much better than that, even in our heads (or at least with paper and pencil only). We're going to use the linearization of at (because is the nearest perfect square to ). Determine this linearization and input your answer in the applet below as the function .

How does what you observed in the applet related to Good Definition #2 of a tangent?

In other words, plugging into (which you can do without a calculator) should yield a result that is extremely close to because . Use the scientific calculator below to compute these two quantities.

Use the linearization of at to approximate .

Use tangent line approximation to estimate the cube root of . (This means you'll have to determine what function and linearization to use.)