Here's the graph showing the Taylor series approximations for the function $\sin(x)$ at $x = 0$. The black curve is the original function, while the dashed lines represent the approximations up to degree 0, 1, 2, and 3. As you can see, the more terms we include, the better the approximation around the point $x = 0$.

Would you like more details or have any questions?

Here are 5 related questions:

1. How does the Taylor series change if expanded at a different point $a$?
2. Why does the Taylor series provide better approximations near the point of expansion?
3. Can all functions be represented by a Taylor series?
4. What is the difference between a Maclaurin series and a Taylor series?
5. How do higher-order terms affect the accuracy of a Taylor series approximation?

Tip: For functions with well-behaved derivatives, the Taylor series provides an increasingly accurate approximation as more terms are added.