## Interactive Calculus Tool

9-12
Standards:
Math Content:
Algebra

Master calculus concepts in an interactive environment. Explore graphs of polynomial functions. Approximate tangent lines, derivative curves, and areas. Then, see the actual result of each. Modify graphs and parameters as you work to see if you can improve your approximations.

Some helpful instructions are given at the bottom right for creating various features on the graph.

### Coordinate Plane Tools

• Zoom In/Zoom Out/Reset Zoom: Selecting any of these tools performs the indicated function.
• Move: After selecting the tool, click and drag on the coordinate plane to move it in the view screen. This tool remains active until another tool is selected.
• Trace: A graph must be displayed to enable the tool. Click on any point along a graph to view the coordinates of that point. Drag along the graph to view other coordinates. This tool remains active until another tool is selected.

### Graphing Tools

• Draw Graph: You must begin by creating a polynomial graph on the coordinate plane.
1. After clicking Draw Graph, select up to five points on the x-axis. These points are the roots of the function. If you click on the same point a second time, this will create a double root. Click three times for a triple root, etc.
2. Next, click Stretch Curve. A line will appear on the x-axis. Click and drag the line up or down to stretch your curve. You can also drag the roots to change their position.
3. When you are satisfied with your graph, click Done Drawing to finish.
• Modify Graph: This tool allows you to move the roots and change the stretch of your graph. Click once to begin, and click Done Modifying to end.
• Clear Graph: This tool clears the entire workspace.

### Tangent

• Approximate: Select this tool to create an approximate tangent. (Note: You cannot create an approximate tangent if the actual tangent is displayed.)
1. Click the curve to select the point of tangency.
2. Click and drag the yellow point or blue line to rotate the line about the blue point. You can also click and drag the blue point to change the point of tangency.
• Actual: Select this tool to display the tangent at a point. If no tangent is displayed, you will need to select one.
• Clicking either slope button will show/hide the respective slope.

### Derivatives

• Approximate: Select this tool to create approximate derivative curves. Depending on your original graph, you can create up to a fourth derivative. (Note: You cannot create an approximate derivative if the corresponding actual derivative is displayed.)
1. Once you click Approximate, you will automatically begin creating the first derivative curve. As with the Draw Graph tool, select up to five roots on the x-axis.
2. Next, click Stretch Curve. A line will appear on the x-axis. Click and drag the line up or down to stretch your curve. You can also drag the roots to change their position.
3. When you are satisfied with your graph, click Done Drawing to finish.
• y', y'', etc.: To draw the second, third, etc. derivatives, click on the corresponding button. You can also click any button to hide/show that graph.
• Actual: Select this tool to display the graph of the first derivative.
• y, y', y'', etc.: Click any button to show/hide that graph.

### Area

• Approximate: Select this tool to approximate the area under the curve using rectangles. (Note: You cannot create an approximate area if the actual area is displayed.)
1. Select two points on the x-axis as the upper and lower limits.
2. Specify the number of rectangles in the Rectangles field. You can use the arrows to increase/decrease the value or type in a value directly.
3. Drag away from the x-axis to adjust the height of the rectangles. Rectangles will snap to the corners or midpoint when near the curve. At any point you can also move the limits or change the number of rectangles.
• Actual: Select this tool to display the area between two points. If limits are not selected, you will need to select them.
• Clicking either area button will show/hide the respective area.

Create a graph with roots approximately at –12, –1, 10, and 10. This means there will be a double root at (10,0). Stretch the graph so the y-intercept is approximately 1.

• What are some key features of the graph?

Tangent

One at a time, draw approximate tangents where x is approximately –8, –3, and 5.

• What did you notice about the yellow and blue points when you created your approximations?
• Compare the three tangents.
• Compare each approximate tangent to the actual tangent at that point. Was your approximation close? What could you do to make your approximation better?

Derivatives

Draw approximations for all possible derivatives.

• How many non-zero derivatives are there for this function? How can you know this by looking at the original graph?
• How did you select where to place the roots of your derivative graphs?
• How did you select how much to stretch the graph of your derivative graphs?
• Compare each approximate derivative curve to the corresponding actual derivative. (Hint: Hide the other curves to make it easier to compare.) Were your approximations close? What could you do to make your approximations better?

Area

Select –13 and –3 as your lower and upper limits, respectively. Then, change the number of rectangles to 5. Create each of the following approximations:

1. The left corner of each rectangle is touching the graph.
2. The right corner of each rectangle is touching the graph.
3. The midpoint of the side of each rectangle is touching the graph.
4. Adjust all rectangles to create your best approximation.
• Display the actual area. Which approximation came closest? What else can you observe about the various approximations?
• Increase and decrease the number of rectangles. Adjust the rectangles. What happens to the accuracy of your approximations?

### Trigonometric Graphing

9-12
Explore the amplitude, period, and phase shift by examining the graphs of various trigonometric functions.