To start thinking about this problem, experiment with the sketch above.

Drag point `x` and observe how point `sin x` moves as you vary `x`.

How does the slope of the arrow relate to the speed and direction of `sin x`?

What can you say about the slope of the arrow when `sin x` is increasing?

What can you say when `sin x` is decreasing?

What’s the arrow’s slope when `sin x` pauses briefly in its up-and-down movement, neither increasing nor decreasing?

Object: Drag `m` to line the arrow up with the graph, so that the arrow’s slope matches the rate of change of the function.

Warm Up: Drag `x` slightly to the right and again adjust `m` to line the arrow up. Do this several times to get the idea.

Play: Press Go. In two seconds `x` starts moving. Drag `m` to keep the arrow lined up.

Landmarks: What must the value of `m` be for these values of `x`:
`-(5pi)/2, -2pi, -(3pi)/2, -pi, -pi/2, 0, pi/2, pi, (3pi)/2, 2pi`, and `(5pi)/2`?

Change Level: When Level 1 is easy, press Reset and move the slider to Level 2. (Higher levels require greater accuracy.)

Object: Drag `m` to match the function's rate of change. (The traced segment is green when `m` is correct.)

Warm Up: Drag `x` and again adjust `m` to make the segment green. Do this several times to get the idea.

Play: Press Go. Drag m to match the graph. In two seconds (when `x` starts moving), drag `m` to keep the segment green.

Change Level: When Level 1 is easy, press Reset and move the slider to Level 2. (Higher levels require greater accuracy.)

Use the tools in this sketch to graph `f(x) = sin x` and then construct a secant line (a line with both its defining points on the graph). Measure the slope of the secant line, and create a button to animate these two points along the graph. What do you notice when you animate the points? What do you wonder?

Stop the animation, measure the abscissa of one of the points, and plot the point defined by the measured abscissa and slope. Animate the points again; what do you notice about the trace of the plotted point? Does it actually measure the slope of the sine graph? How could you determine the slope more accurately? Try out your ideas.

If you tried moving the two points closer and closer, what's the closest you were able to get them? Try making a Move button to move one even closer to the other. What happens? Why does this happen? What could you do about it?

Use the tools in this sketch to construct a more systematic way to measure the slope of the sine function.

Graph the sine curve and then construct a point `x` on the `x`-axis and its corresponding point on the graph, with coordinates `(x, f(x))`. Create a parameter `h` use it to plot the point `(x+h, f(x+h))`, and connect the two plotted points with a line. Animate point `x` and observe the behavior of the line.

Calculate the slope of the line, plot and trace the point `(x, slope)`, and observe the behavior of this traced point as you animate `x`.

Try different values of `h` observing the animation each time. For what values of `h` does the line come closer to following the sine graph? What do you think is the best value of `h` to use? Explain your reasoning. Are there values of `h` that you should not use? Explain your reasoning.

When you animate `x` using the very best value of `h` that you can, what shape do the traces make? Can you graph a function that matches the traces? If so, how close do you think the match really is?

In your report, write down the calculation you used to find the slope of your line. The explain how you adjusted `h` to make the line follow the sine graph as closely as possible. Say whether or not your line exactly matches the slope of the sine function, and explain how you know this.