1. **Problem:** Find the function $f(t) = 3t + 4 \sin(2t)$ and understand its components. 2. **Formula and rules:** This function is a sum of a linear term $3t$ and a sinusoidal term $4 \sin(2t)$. The linear term grows without bound as $t$ increases, while the sinusoidal term oscillates between $-4$ and $4$ with frequency $2$. 3. **Intermediate work:** - The linear part is straightforward: $3t$. - The sinusoidal part: $4 \sin(2t)$ oscillates with period $\frac{2\pi}{2} = \pi$. 4. **Explanation:** The function $f(t)$ combines steady growth with oscillations. At any time $t$, $f(t)$ is the sum of $3t$ and the sine wave scaled by 4. 5. **Final answer:** The function is $f(t) = 3t + 4 \sin(2t)$. This completes the solution for the first problem.