Calculus II

To solve this problem, we will first need to determine the n-th derivative of the function \( f(x) = \cos(x) + e^{2x} \) evaluated at zero. Then, we will write a script to calculate the coefficients of the Maclaurin series up to a given N and store them in an array. Finally, we will use the script to approximate \( f(x) \) for different values of N and plot the results. ### Step 1: Determine the n-th derivative of \( f(x) \) at zero The function \( f(x) = \cos(x) + e^{2x} \) has two parts. The derivatives of \( \cos(x) \) at zero are well-known and follow a pattern: \( \cos(0) = 1 \), \( -\sin(0) = 0 \), \( -\cos(0) = -1 \), \( \sin(0) = 0 \), and so on. The derivatives of \( e^{2x} \) at zero are also straightforward: \( e^{2x} \) remains \( e^{2x} \) after differentiation, and evaluated at zero, it is always 1, but with a factor of \( 2^n \) from the chain rule. Thus, the n-th derivative of \( f(x) \) at zero is: - For even n: \( f^{(n)}(0) = (-1)^{n/2} + 2^n \) - For odd n: \( f^{(n)}(0) = 0 + 2^n \) ### Step 2: Write a script to calculate the coefficients Here is a Python script that calculates the coefficients of the Maclaurin series for \( f(x) = \cos(x) + e^{2x} \) up to a given N: ```python import math import numpy as np import matplotlib.pyplot as plt # Define the number of terms in the Maclaurin series N = 10 # Initialize an array to store the coefficients coefficients = [] # Calculate the coefficients for n in range(N + 1): if n % 2 == 0: # even n coeff = (-1)**(n//2) + 2**n else: # odd n coeff = 2**n coefficients.append(coeff / math.factorial(n)) # Print the coefficients print("The coefficients up to N =", N, "are:", coefficients) ``` ### Step 3: Approximate \( f(x) \) for different values of N and plot the results Now, we will use the script to approximate \( f(x) \) for N = 2, 5, 10, 20 and plot these approximations alongside the actual function. ```python # Define the function f(x) def f(x): return np.cos(x) + np.exp(2*x) # Define the range of x values x_values = np.linspace(0, 2, 200) # Plot the actual function plt.plot(x_values, f(x_values), label='f(x)', color='black') # Function to calculate the Maclaurin series approximation def maclaurin_approximation(x, coefficients): return sum(coef * x**n for n, coef in enumerate(coefficients)) # Plot the approximations for different values of N for N in [2, 5, 10, 20]: # Calculate the coefficients for the current N coefficients = [] for n in range(N + 1): if n % 2 == 0: # even n coeff = (-1)**(n//2) + 2**n else: # odd n coeff = 2**n coefficients.append(coeff / math.factorial(n)) # Calculate the Maclaurin series approximation for the current N y_approx = [maclaurin_approximation(x, coefficients) for x in x_values] # Plot the approximation plt.plot(x_values, y_approx, label=f'N={N}') # Add labels and legend plt.xlabel('x') plt.ylabel('f(x)') plt.legend() plt.title('Maclaurin Series Approximations of f(x)') # Show the plot plt.show() ``` This script will produce a plot with the actual function \( f(x) \) and its Maclaurin series approximations for N = 2, 5, 10, 20, each with a different color. The legend will indicate which line corresponds to which approximation, and the x and y axes will be labeled appropriately.

Math

The Maclaurin series (special case of the Taylor series) is a power series where the coefficients are expressed in terms of the derivatives evaluated at a specific point (at zero for a Maclaurin series): \[ f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n \] Where \( f^{(n)}(0) \) is the n-th derivative evaluated at zero. Write a script which evaluates the coefficients of this series into an array up to a parameter N, which should be prescribed at the top of the script, for the function \( f(x) = \cos(x) + e^{2x} \). Observe that the n-th derivative of this function evaluated at zero can be written explicitly in terms of n. Produce an array of coefficients for N=10. By using your script from above find the approximations for \( f(x) \) for values of N = 2, 5, 10, 20 and plot all of these on the same axis alongside the function \( f(x) \) itself. Plot this for \( x \in [0, 2] \), have a legend, color coding for each line and axis labels.

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