The image presents a mathematical problem related to Taylor series and cubic interpolation for the function $f(x) = e^{-x^2}$. The problem is divided into several parts. Here's a breakdown of each question:

Problem 6: Taylor series and cubic interpolation

Given function:

$f(x) = e^{-x^2}$

(a)

By calculating $f(0)$, $f'(0)$, $f''(0)$, $f'''(0)$, and $f^{(4)}(0)$, obtain the Taylor-Maclaurin series for $f(a + h)$ about the point $a = 0$, up to and including terms involving $h^4$. [3 marks]

(b)

Show that you can also obtain this series using the standard Taylor series expansion for $\exp(x)$. [3 marks]

(c)

Obtain the Taylor series for $f(a + h)$ with $a = 1$, up to and including terms in $h^4$. [3 marks]

(d)

Given the data points for $f(x)$:

$(x_1, y_1) = (0, 1), \quad (x_2, y_2) = (1, e^{-1}), \quad (x_3, y_3) = (2, e^{-4}), \quad (x_4, y_4) = (4, e^{-16})$

Write down the cubic Lagrange polynomials $L_1(x)$, $L_2(x)$, $L_3(x)$, and $L_4(x)$ for this set of $x_i$ values. [You need not expand these out!] [6 marks]

Finally, write down the cubic polynomial that interpolates this set of data points. [You need not expand this out!] [6 marks]

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Here are 5 related questions that expand on the topics in the image:

1. How can we generalize the Taylor series to functions other than exponentials?
2. What is the geometric interpretation of Lagrange interpolation polynomials?
3. Why is the cubic polynomial interpolation particularly useful for small datasets?
4. What are the main differences between Maclaurin and Taylor series?
5. How does the convergence of the Taylor series depend on the function?

Tip: When computing derivatives for Taylor series, it's important to evaluate the pattern in higher derivatives early on to simplify calculations!