The problem involves finding the x-coordinates ($x$-values) of points $A$ and $B$ in the given graph of the function $y = e^{-x/2} \sin(4x)$. These points correspond to the extrema of the function, where the derivative $y'$ is zero. The derivative is given as:

$y' = \frac{\mathrm{d}}{\mathrm{d}x} \left( e^{-x/2} \sin(4x) \right).$

Using the product rule:

$y' = \left( \frac{\mathrm{d}}{\mathrm{d}x} e^{-x/2} \right) \sin(4x) + e^{-x/2} \frac{\mathrm{d}}{\mathrm{d}x} \sin(4x).$

The derivatives are: $\frac{\mathrm{d}}{\mathrm{d}x} e^{-x/2} = -\frac{1}{2} e^{-x/2}, \quad \frac{\mathrm{d}}{\mathrm{d}x} \sin(4x) = 4 \cos(4x).$

Thus: $y' = -\frac{1}{2} e^{-x/2} \sin(4x) + e^{-x/2} (4 \cos(4x)).$

Factoring out $e^{-x/2}$: $y' = e^{-x/2} \left( 4 \cos(4x) - \frac{1}{2} \sin(4x) \right).$

To find the critical points (extrema), solve: $4 \cos(4x) - \frac{1}{2} \sin(4x) = 0.$

Rearranging: $4 \cos(4x) = \frac{1}{2} \sin(4x),$

or: $8 \cos(4x) = \sin(4x).$

Dividing both sides by $\cos(4x)$ (assuming $\cos(4x) \neq 0$): $8 = \tan(4x).$

This simplifies to: $\tan(4x) = 8.$

The solutions for $4x$ are: $4x = \arctan(8) + n\pi, \quad n \in \mathbb{Z}.$

Dividing by 4 to solve for $x$: $x = \frac{\arctan(8)}{4} + \frac{n\pi}{4}, \quad n \in \mathbb{Z}.$

Using this equation, substitute $n$ to find the $x$-values corresponding to points $A$ and $B$. Let me calculate approximate values.import numpy as np

Calculate the principal value of arctan(8)

arctan_8 = np.arctan(8)

Define the general solution for x

n_values = np.array([0, 1]) # To calculate the first two solutions (points A and B) x_values = (arctan_8 / 4) + (n_values * np.pi / 4)

x_valuesarray([0.36161033, 1.1470085 ])array([0.36161033, 1.1470085 ])successThe approximate $x$-coordinates for the points $A$ and $B$ are:

• Point $A$: $x \approx 0.362$
• Point $B$: $x \approx 1.147$

These values correspond to the extrema of the function $y = e^{-x/2} \sin(4x)$.

Would you like me to provide further steps or verification details? Here are some related questions to explore further:

1. What are the corresponding $y$-values for points $A$ and $B$?
2. How does damping influence the behavior of this function over time?
3. Can you graph the derivative to visually confirm these extrema?
4. What happens to the function as $x \to \infty$?
5. How do we compute higher-order extrema for this function?

Tip: Always verify extrema by checking the sign change of the derivative $y'$ around the critical points.