Activity 2.2.2.
Consider the function \(f(x) = \sin(x)\text{,}\) whose graph is provided in the lefthand side of the given figure. Note carefully that the grid in the diagram does not have boxes that are \(1 \times 1\text{,}\) but rather approximately \(1.57 \times 1\text{,}\) as the horizontal scale of the grid is \(\pi/2\) units per box.
(a)
At each of \(x = -2\pi, -\frac{3\pi}{2}, -\pi, -\frac{\pi}{2}, 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi\text{,}\) use a straightedge to sketch an accurate tangent line to \(y = f(x)\text{.}\)
(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 \(h\text{,}\) 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 \(f'(2\pi)\text{?}\) about \(f'(-2\pi)\text{?}\)
(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 \(y = f(x)\text{.}\)
(e)
What familiar function do you think is the derivative of \(f(x) = \sin(x)\text{?}\)
