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This Exploration uses the interactive Java applet MIC GradientGrapher (below) to investigate the equation for the gradient function of f(x) = x2, as referenced on page 9-7 of your text.
Instructions
- Please be patient while the program is loading. It can take up to 30 seconds to load and then run, depending on the available bandwidth. It runs much faster if you download it to your hard drive.
- In the box following "f(x) = " enter the equation "x^2" and check the box to its immediate right.
- Click the "Graph" button.
- The graph of f(x) = x2 will be drawn (in red) on the grid.
- Using the mouse, click anywhere along the x-axis under the curve. The tangent line will be drawn and the value for x and the gradient of the tangent m will be shown. Also a blue dot representing this value for m will be placed on the grid.
- Continue clicking under the curve until the shape of the gradient function becomes apparent.
- Try to determine the equation for the gradient function. Enter the equation you believe to be the gradient function into the box following "f ' (x) = " and check the box to its immediate right and click the "Graph" button again.
- The program will draw the graph that you have suggested (in magenta). Clicking under the curve for f(x) = x2 will get the values for the gradient of the tangent again, so that you can compare the "fit".
Any equation can be entered into the "f(x) = " box and its derivative funtion inferred. Of course, the correct angle measure has to be selected before drawing trigonometric functions.
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