1 Construct the Slope Function
In this lesson you will graph the sine function and measure its slope at various places along the graph. You’ll use this measurement to trace a graph that shows the slope of the sine at each point along the sine curve.
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.
How does the trace change if you put the points farther apart before animating them, or if you put them closer together? What do you notice, and what do you wonder?
How can you make your traces as accurate as possible?
Use a Move button to put the two points really really close together. Describe and explain what happened when you tried this.
2 How Close Can You Come?
The Warm-Up section is normally visible by default, so includes the fcVisible class in the fcSectionBody and uses data-url instead of data-delayed to specify the sketch json.
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))`.
To construct a second point on the graph, create a parameter to control the difference of their `x` values. Call the parameter `h`, and set its value to 0.5.
Use this parameter to plot the point `(x+h, f(x+h))`. Then connect the two plotted points with a secant line. Animate point `x` and observe the behavior of the line.
You don't have a slope tool, so use the Calculator to 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 secant line follow the sine graph as closely as possible. How closely do you think your traced line matches the slope of the sine function? Explain how you know this.
3 Objectives
To be written...
The Fine Print
Requirements:
These activities require web access using a browser that supports HTML5 and JavaScript. (That means almost any current browser.) No purchase is required, and there’s no advertising anywhere.
Release Information
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24 Jan 2018: Created this lesson template.
This web site is based in part upon work supported by the National Science Foundation under NCSU IUSE award 1712280 (July 2017 through December 2020) and KCP Technologies DRK-12 award ID 0918733 (September 2009 through August 2013). Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.
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