The question involves finding the Taylor polynomial of a function and estimating the error of approximation. Here's the step-by-step solution:

Given the function:

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

Part (a): Find the Taylor polynomial $p_3(x)$ of order 3 at $x_0 = 0$.

The Taylor series of a function $f(x)$ around $x_0 = 0$ is given by:

$f(x) = f(0) + f'(0)x + \frac{f''(0)x^2}{2!} + \frac{f'''(0)x^3}{3!} + \cdots$

We need the first three derivatives of $f(x) = e^{2x}$:

1. $f(x) = e^{2x}$
2. $f'(x) = 2e^{2x}$
3. $f''(x) = 4e^{2x}$
4. $f'''(x) = 8e^{2x}$

Evaluating these at $x = 0$:

1. $f(0) = e^0 = 1$
2. $f'(0) = 2e^0 = 2$
3. $f''(0) = 4e^0 = 4$
4. $f'''(0) = 8e^0 = 8$

Thus, the Taylor polynomial of order 3 is:

$p_3(x) = 1 + 2x + \frac{4x^2}{2!} + \frac{8x^3}{3!}$

Simplifying:

$p_3(x) = 1 + 2x + 2x^2 + \frac{4x^3}{3}$

Part (b): Write the expression for the remainder $R_3(x)$. Estimate, by bounding the remainder, the error when calculating $f(0.4)$ using $p_3(0.4)$.

The remainder term of the Taylor series is given by:

$R_3(x) = \frac{f^{(4)}(c) x^4}{4!}$

where $c$ is some value between 0 and $x$. The fourth derivative of $f(x)$ is:

$f^{(4)}(x) = 16e^{2x}$

To bound the error, we take the maximum value of $f^{(4)}(c)$ for $c \in [0, 0.4]$:

$|f^{(4)}(c)| \leq 16e^{2 \times 0.4} = 16e^{0.8}$

Thus, the bound for the remainder when $x = 0.4$ is:

$|R_3(0.4)| \leq \frac{16e^{0.8} \times (0.4)^4}{4!}$

Calculating this will give the estimated error. Let me know if you would like the numerical estimation of this error!

Do you want details on the numerical estimation or have any questions? Here are some related questions for further practice:

1. How do you find a Taylor polynomial of a different order for $e^{2x}$?
2. What changes if we expand the Taylor series around a different point, such as $x = 1$?
3. How does the error estimation differ if a higher-order term is considered?
4. How do you calculate the error for different values of $x$?
5. What are the implications of using Taylor polynomials in approximations?

Tip: Always check the interval over which you're estimating the error to ensure the error bound calculation is accurate.