O que seria uma aproximação linear?

A Polynomial Approximation is what it sounds like: an approximation of a curve with a polynomial. Here's an example:

We have the curve $f(x)=e^x$ in blue, and a Polynomial Approximation with equation $g(x)=1+x+\dfrac{1}{2}x^2+\dfrac{1}{6}x^3+\dfrac{1}{24}x^4+\dfrac{1}{120}x^5$ in green.

Note how the polynomial is extremely accurate in approximating the right half of $e^x$! In fact, if we use this polynomial to approximate the value of $e$, then: $$f(1)=e^1\approx 2.718$$ $$\begin{align*}g(1)&=1+(1)+\dfrac{1}{2}(1)^2+\dfrac{1}{6}(1)^3+\dfrac{1}{24}(1)^4+\dfrac{1}{120}(1)^5\\ &= \dfrac{163}{60}\\ &\approx 2.717 \end{align*}$$

This is an error of less than $0.05\%$!

In further topics you will learn how to find these polynomials, known as Taylor Polynomials or Taylor Series.

A common problem in mathematics is that we don't actually know how to calculate certain functions. For example, it's not too easy to calculate $\sin(1)$ - while we can look up its value in a table, or draw a triangle and perform measurements, these solutions aren't ideal. To solve this problem, mathematicians like to develop very good approximations to these functions - related functions which are very close to the function of interest, but much easier to calculate.

A similar problem is that of "predicting the future." It makes sense, intuitively, that if I've just thrown a ball, and I know its position, velocity, acceleration, the rate of change of its acceleration, and so on forever, I should be able to predict exactly where it lands. But how to actually do this isn't yet clear.

Probably the easiest functions to calculate are the polynomials - sums of powers of $x$, with some coefficients. Since these involve only addition and multiplication, we can generally compute these with ease. It'd be nice, then, if we could approximate functions like $\sin$ with polynomials! Taylor series, and more generally Taylor's theorem, allow us to do just that!

An example is below. The function $\sin(x)$ is plotted below in red, along with various polynomial approximations (in blue, green and orange).

In the next few sections, you'll learn how to calculate these approximations!