Activity 2.2.2. Consider the function $f(x) = \sin(x)$, which is graphed in
Figure 2.2.2 below. Note carefully that the grid in the diagram does not have
boxes that are 1 Ă— 1, but rather approximately 1.57 Ă— 1, as the horizontal
scale of the grid is $\frac{\pi}{2}$ units per box.
Figure 2.2.2. At left, the graph of $y = f(x) = \sin(x)$.
a. At each of $x = -2\pi, -\frac{3\pi}{2}, -\pi, -\frac{\pi}{2}, 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi$, use a straightedge
to sketch an accurate tangent line to $y = f(x)$.
b. Use the provided grid to estimate the slope of the tangent line you drew
at each point. Pay careful attention to the scale of the grid.
c. Use the limit definition of the derivative to estimate $f'(0)$ by using small
values of $h$, and compare the result to your visual estimate for the slope
of the tangent line to $y = f(x)$ at $x = 0$ in (b). Using periodicity, what
does this result suggest about $f'(2\pi)$? about $f'(-2\pi)$?
d. Based on your work in (a), (b), and (c), sketch an accurate graph of
$y = f'(x)$ on the axes adjacent to the graph of $y = f(x)$.
e. What familiar function do you think is the derivative of $f(x) = \sin(x)$?
