Problem
(a) What is the derivative of $\sin t$ ? \[ \frac{d}{d t} \sin t= \] (b) The velocity of a particle at time $t$ is $v(t)=\cos t$. Use the Fundamental Theorem of Calculus to find the displacement of the particle between $t=0$ and $t=\frac{15 \pi}{2}$. \[ \int_{0}^{\frac{15 \pi}{2}} \cos t d t= \]
Solution
Step 1 ：\(\frac{d}{d t} \sin t = \cos t\)
Step 2 ：\(\int_{0}^{\frac{15 \pi}{2}} \cos t d t = \sin t \Big|_{0}^{\frac{15 \pi}{2}}\)
Step 3 ：\(= \sin\left(\frac{15 \pi}{2}\right) - \sin(0)\)
Step 4 ：\(= -1 - 0\)
Step 5 ：\(\boxed{-1}\)
