Here are some Taylor series-related practice questions extracted and derived from the material:

1. Expand $e^x$ using the Taylor series up to the fifth term around $x = 0$.

2. Derive the Taylor series for $\sin(x)$ centered at $x = 0$ up to the fourth degree.

3. Using Taylor series, approximate $\ln(1+x)$ up to the fourth term for small values of $x$.

4. Find the Taylor series expansion of $f(x) = 1/(1-x)$ around $x = 0$ and specify its radius of convergence.

5. Show that the Taylor series for $\cos(x)$ around $x = 0$ has only even powers of $x$. Compute the expansion up to $x^6$.

6. Use Taylor's theorem to approximate $e^{-x}$ at $x = 0.1$ up to three terms. Calculate the error bound.

7. Derive the Taylor series expansion for $\tan^{-1}(x)$ around $x = 0$ up to the fifth term.

8. Verify that the function $f(x) = \ln(x+1)$ satisfies its Taylor series expansion about $x = 0$ up to the third derivative.

9. Expand $\cosh(x)$ (hyperbolic cosine) in a Taylor series around $x = 0$ up to $x^4$.

10. Use Taylor expansion to approximate $\sqrt{1+x}$ up to the fourth term for small values of $x$.

If you would like detailed solutions or explanations for any of these, let me know. Here are some further steps and related tips to dive deeper:

Related Questions:

1. What is the general formula for the Taylor series expansion of $f(x)$ about $x = a$?
2. How can you use the Taylor series to approximate definite integrals numerically?
3. Can you derive the Maclaurin series as a special case of the Taylor series?
4. Explore how the error in a Taylor series approximation is estimated using the Lagrange remainder term.
5. Investigate how the Taylor series changes if expanded around a non-zero point.

Tip:

Always cross-verify the radius of convergence for a Taylor series, especially when applying it to approximate functions over a specified domain.