To find maxima, take derivative: $A'(t) = 1000 [ -0.03 e^{-0.03t} \sin(0.1\pi t) + e^{-0.03t} (0.1\pi) \cos(0.1\pi t) ]$.

["Finding Maxima by Taking Derivatives: A Step-by-Step Guide Using the Derivative of an Exponential-Trigonometric Function", "When analyzing functions that model real-world phenomena—like population growth, oscillating systems, or financial trends—it’s crucial to determine their maxima to understand peak behaviors. One powerful method for finding maxima is through calculus, specifically by computing and analyzing the derivative. In this article, we’ll explore the derivative of a key function often used in growth and oscillation modeling, walk through its interpretation, and explain how to use it to find maxima.", "---", "### Understanding the Function and Its Derivative", "Consider the function modeling damped harmonic motion or modified exponential growth:", "[\nA(t) = 1000 \left[ -0.03 e^{-0.03t} \sin(0.1\pi t) + e^{-0.03t} (0.1\pi) \cos(0.1\pi t) \right]\n]", "This formula combines exponential decay ((e^{-0.03t})) with sinusoidal components, simulating systems influenced by both damping and periodic inputs—common in physics, engineering, and finance.", "The first step to finding maxima is computing (A'(t)), the derivative of (A(t)) with respect to time (t). Differentiating this expression involves applying the product rule and chain rule due to composite trigonometric and exponential terms.", "The derivative is:", "[\nA'(t) = 1000 \left[ -0.03 e^{-0.03t} \sin(0.1\pi t) + e^{-0.03t} (0.1\pi) \cos(0.1\pi t) \right]\n]", "Factoring out (e^{-0.03t}) highlights the function’s damping behavior—this exponential term steadily decays, controlling the amplitude of oscillations.", "---", "### Why Derivatives Matter for Finding Maxima", "To find maximum values of (A(t)), we locate critical points by solving (A'(t) = 0). Setting the derivative equal to zero:", "[\n-0.03 e^{-0.03t} \sin(0.1\pi t) + e^{-0.03t} (0.1\pi) \cos(0.1\pi t) = 0\n]", "Since (e^{-0.03t} > 0) for all real (t), we can divide both sides by it, simplifying to:", "[\n-0.03 \sin(0.1\pi t) + 0.1\pi \cos(0.1\pi t) = 0\n]", "Rearranging:", "[\n0.1\pi \cos(0.1\pi t) = 0.03 \sin(0.1\pi t)\n]", "Dividing both sides by (\cos(0.1\pi t)) (where defined):", "[\n\ an(0.1\pi t) = \frac{0.1\pi}{0.03} = \frac{10\pi}{3}\n]", "This equation reveals that maxima occur where the tangent of the argument matches this constant value, meaning rhythmic peaks align with exponential decay trends.", "---", "### Solving for Critical Times and Testing for Maxima", "Solving (\ an(0.1\pi t) = \frac{10\pi}{3}) gives:", "[\n0.1\pi t = \ an^{-1}\left( \frac{10\pi}{3} \right) + n\pi, \quad n \in \mathbb{Z}\n]", "Thus,", "[\nt = \frac{1}{0.1\pi} \left[ \ an^{-1}\left( \frac{10\pi}{3} \right) + n\pi \right]\n]", "Each solution corresponds to a critical point. To confirm a maximum, use the second derivative test or evaluate (A'(t)) around these points. Because (e^{-0.03t}) is monotonically decreasing and positive, the behavior of the trigonometric ratio near critical points strongly influences sign changes in (A'(t)).", "---", "### Practical Applications and Insights", "Understanding how to derive and analyze such functions empowers engineers, economists, and scientists to predict peak behaviors—such as maximum profits, oscillation peaks in mechanical vibrations, or optimal timing for interventions. The presence of exponential damping reflects real-world systems losing energy over time, making peak detection both timely and finite.", "---", "### Conclusion", "Finding maxima in oscillatory-damped systems starts with computing the derivative—here derived as:", "[\nA'(t) = 1000 \left[ -0.03 e^{-0.03t} \sin(0.1\pi t) + e^{-0.03t} (0.1\pi) \cos(0.1\pi t) \right]\n]", "Solving (A'(t) = 0) identifies critical times, and analyzing the derivative’s sign around these points confirms maxima. This calculus-based approach combines exponential decay and sinusoidal motion, offering deep insight into dynamic systems across science and industry.", "---", "### Key Takeaways", "- Use derivatives to locate critical points where the function may peak.\n- Simplify derivative expressions by factoring common terms.\n- Solve (A'(t) = 0) and use secondary tests to identify maxima.\n- The decaying exponential ensures damped oscillations with finite maxima.", "For anyone modeling time-dependent phenomena, mastering derivative analysis is indispensable—turning equations into actionable peak insights."]