Calculus 10th Edition

Average rate of change for interval $[0, \frac{\pi}{6}]$: $\frac{3}{\pi}$ Instantaneous rate of change for endpoint 0: $1$ Instantaneous rate of change for endpoint $\frac{\pi}{6}$: $\frac{\sqrt 3}{2}$
Average rate of change is given by the formula $$\frac{f(x_2) - f(x_1)}{x_2 - x_1}$$ for $(x_1, f(x_1))$ and $(x_2, f(x_2))$. In this case, for endpoints $[0, \frac{\pi}{6}]$: $$f(x) = sin(x)$$ $$f(0) = sin(0) = 0$$ $$f(\frac{\pi}{6}) = sin(\frac{\pi}{6}) = \frac{1}{2}$$ Therefore, the average rate of change for the interval $[0, \frac{\pi}{6}]$ is: $$\frac{\frac{1}{2} - 0}{\frac{\pi}{6} - 0} = \frac{6}{2\pi} = \frac{3}{\pi}$$. Instantaneous rate of change is given by the first derivative of a function: $$f(x) = sin(x)$$ $$f'(x) = cos(x)$$ and for the endpoints of the interval $[0, \frac{\pi}{6}]$: $$f'(0) = cos(0) = 1$$ $$f'(\frac{\pi}{6}) = cos(\frac{\pi}{6}) = \frac{\sqrt 3}{2}$$