I see that you've uploaded a tutorial sheet with a variety of questions focused on Taylor and Maclaurin series, power series, and limits. Let’s go through each question and its requirements:

1. Taylor Polynomials (Question 1):

   • You are asked to find Taylor polynomials up to the $x^3$ term about a specific point $x = a$ for two functions. This will involve calculating derivatives and evaluating them at the given point $a$.

2. Maclaurin Series (Question 2):

   • This section asks for the Maclaurin series of $e^{-x}$ and $\sin 3x$, expressed in sigma notation. Maclaurin series are Taylor series centered at $x = 0$, so you would use the derivatives of each function at $x = 0$.

3. Applying Standard Series (Question 3):

   • Here, you need to find the Maclaurin series of various functions using known series expansions. Some of these functions, like $\cos \sqrt{x}$ and $\frac{x^2}{1 - 2x}$, may require manipulation of basic series such as those for $e^x$, $\sin x$, and $\cos x$.

4. Summing Series (Question 4):

   • This question requires using known Maclaurin series to evaluate the sums of given series. For example, (a) is a power series expansion, while (b) and (c) involve specific trigonometric and exponential series.

5. Estimating an Integral (Question 5):

   • For this, you're instructed to use the first four non-zero terms in the Maclaurin series for $\ln(1 - 4x^2)$ to approximate an integral from 0 to 0.3. This involves substitution into the series and integrating term by term.

6. Evaluating Limits (Question 6):

   • The final question focuses on limits that might benefit from series expansion techniques. Using power series could help simplify the expressions and evaluate the limits as $x \to 0$.

Would you like detailed solutions or guidance for any specific question or part of the tutorial?

Here are some additional questions to deepen understanding:

1. What are the steps for deriving the Taylor series for functions not centered at zero?
2. How can we use a Maclaurin series for one function to approximate similar functions?
3. What is the significance of the remainder term in Taylor series approximations?
4. How do the radius and interval of convergence affect the usefulness of a Maclaurin series?
5. What methods exist for summing series when direct evaluation isn't possible?

Tip: When deriving Taylor series, always check for simplifications in derivatives at the expansion point to avoid unnecessary complexity.