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In calculus, the second derivative of a function provides insights into the behavior and characteristics of the original function, particularly its concavity and points of inflection. If you have a function \( y(t) \), its first derivative \( y'(t) \) represents the rate of change of \( y \) with respect to \( t \). Differentiating \( y'(t) \) again gives the second derivative \( y''(t) \).

• Increasing vs Decreasing: If \( y''(t) > 0 \), the function \( y(t) \) is concave up, suggesting that the function could be at a minimum or is part of an upward parabola structure.
• Concave Up vs Concave Down: Conversely, if \( y''(t) < 0 \), the function is concave down, indicating a possible maximum or a downward-facing parabola.

In the given exercises, computing the second derivative is critical for substituting into the differential equation. For example, with a function like \( y(t) = 3 \cos 5t \), differentiating twice shows that \( y''(t) = -75 \cos 5t \). Such insights help solve for \( \omega^2 \) when contrasted with the rest of the differential equation.

Trigonometric functions, central to periodic phenomena, are indispensable in differential equations, especially in modeling oscillatory behavior. When you see expressions involving \( \sin \) or \( \cos \), angle manipulation often predicts how functions behave periodically.

• Cosine and Sine Basics: Both \( \sin t \) and \( \cos t \) are periodic with a fundamental period of \( 2\pi \). This means that they repeat their values after an interval, integral to modeling cycles such as waves or repetitive movements.
• Derivatives of Trigonometric Functions: In calculus, the derivative of \( \cos(t) \) is \(-\sin(t)\), and the derivative of \( \sin(t) \) is \( \cos(t) \). These derivatives frequently show up in differential equations and form the cycle of periodicity in calculus operations.

When solving the problem, understanding these derivatives becomes evident. For \( y(t) = 2 \sin 3t + 5 \cos 3t \), differentiating gives terms like \( 6\cos 3t - 15\sin 3t \), which then lead to differential expressions useful for finding \( \omega^2 \).

The product rule is a fundamental principle in calculus used when differentiating expressions where two or more functions are multiplied together. Let's say you have two functions \( u(t) \) and \( v(t) \). If they are multiplied, their derivative is computed as:\[(uv)' = u'v + uv'\]This formula ensures each part of the product is considered in the derivative process.

• Application: This rule becomes crucial in problems where exponential and trigonometric functions are combined, as seen in \( y(t) = 3 e^{-t} \cos 5t \). Applying the product rule helps in calculating derivatives like \( y'(t) = -3 e^{-t} \cos 5t - 15 e^{-t} \sin 5t \).
• Simplification: After finding the first derivative, applying the product rule again if necessary for the second derivative yields the results needed to solve differential equations involving products of functions.

By using the product rule effectively, problems with multiple terms, especially in exponential and trigonometric combinations, become manageable and straightforward to interpret in the context of differential equations.