What is the average rate of change over this interval? A boat at anchor is bobbing up and down in the sea. The vertical distance, $y$, in meters, between the sea floor and the boat is given as a function of time, $t$, in hours, by $$6 + \operatorname{sin} ((\frac{\pi t}{3}) + 1.5).$$ What is the average rate of change of $y$ over the 4 hour interval $1 \leq t \leq 4?$ So I think I know what to do. I found the derivative of the function which is $\operatorname{cos} ((\frac{\pi t}{3}) + 1.5) \cdot \frac{\pi}{3}.$ Do I plug in $1,2,3,4$ and then find and then add the answers and divide by four to find the mean? Thanks!

The average rate of change of $f(t)$ on the interval $a\le t\le b$ is just ${f(b)-f(a)\over b-a}$. So here---assuming I am interpreting your function correctly with the extra set of parentheses---you get:

$${f(4)-f(1)\over 4-1}\approx -0.373338.$$

To understand this geometrically, just visualize the secant line between the points $(1,f(1))$ and $(4,f(4))$ shown in black. The number above is the slope of this line. The original function is shown in blue. (Note how the axes are scaled though.)

!Mathematica graphics