import numpy as np import matplotlib.pyplot as plt

Define the function c(t) = t * e^(-2t)

def c(t): return t * np.exp(-2 * t)

Generate t values

t_values = np.linspace(0, 2, 500) # Chose a range around t=0.5 to capture the peak

Compute c(t) values

c_values = c(t_values)

Plot the function

plt.figure(figsize=(8, 5)) plt.plot(t_values, c_values, label='c(t) = t * e^(-2t)') plt.axvline(x=0.5, color='r', linestyle='--', label='t=0.5') plt.title('Plot of c(t) = t * e^(-2t)') plt.xlabel('t') plt.ylabel('c(t)') plt.legend() plt.grid() plt.show() <<ImageDisplayed>>successThe graph clearly illustrates that the function $c(t) = t e^{-2t}$ reaches its maximum value at $t = 0.5$, as indicated by the peak in the curve. Let me know if you have questions about the calculation or would like further exploration.

Related Questions

1. How do you find the exact maximum value of $c(t)$ using calculus?
2. What is the significance of the parameter $e^{-2t}$ in the function $c(t)$?
3. How does the value of the maximum concentration change if the coefficient of $t$ or the exponent changes?
4. Could this type of function model real-world processes, like concentration in a chemical reaction? How?
5. What happens to the function $c(t)$ as $t \to \infty$?

Tip

When analyzing maxima or minima, always check the derivative and use the second derivative test for confirmation.