Ideas for Extension |
The following ideas for extending this topic require the full version of Autograph. |
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Idea 1 – Estimating the Area under a Curve |
Download 1. Estimating.agg |
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There are three different ways of estimating the area under a curve using Autograph |
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Challenge your students to calculate  |
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Is the estimate of the area using rectangles an over estimate or an under estimate? Can you explain why? |
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What will happen to the estimate if we increase the number of rectangles? |
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Click on Animation Controller and experiment by changing the number of rectangles |
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When you are ready, double-click on the rectangles and choose Trapezium Rule. |
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Can you see how this calculates and estimate of the area? |
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Is it more or less accurate than using rectangles? Why? |
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Using the Animation Controller experiment with what happens when you change the number of strips. |
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What happens to the estimate if you move the two points to new positions? Predict, and then drag the points to new positions to find out! |
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Finally, try experimenting with Simpson’s Rule for estimating the area under the curve. Can you see how this technique works? |
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You can also double-click on the curve itself and experiment with different equations. |
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Idea 2 – Infinite and Improper Integrals |
Download 2. Unbounded.agg |
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We can also use Autograph to investigate integrals that have undefined or unbounded limits |
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Before opening the Autograph file, challenge your students to work out the following:  |
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How about this?  |
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Why is this? What is the difference between this and the first question? |
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Challenge your students to sketch the situation, and then open up Autograph. |
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Select the point at x = 1 and click on Animation Controller. Move the value closer to 1, adjusting the step size as you go. |
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What is happening to the size of the area? Will it reach a limit? Why? |
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When your students are happy with this, return the point back to x = 1. |
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Challenge your students to think what the answer to this will be:  |
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Select the point at x = 2 and click on Animation Controller. Increase the value, adjusting the step size as you go. |
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What is happening to the size of the area? Will it reach a limit? Why? |
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When you are happy, try the same thing with:  |
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Idea 3 – Area between functions |
Download 3. Between Curves.agg |
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Autograph can also work out the area between two functions. |
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Challenge your students to work out the area between the line and the curve. |
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How many different ways can they think to do this?
– Using integration and the area of a triangle
– Calculating two integrations and then subtracting the answer
– Can they do it using a single integration? |
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What happens if you do the subtraction the other way around? |
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Once students have reached an answer, click on the shaded area and select Text Box to display the size of the area. |
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You can double click on either of the lines to change the equation and test your students on other situations. |
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Idea 4 – Volume of Revolution |
Download 4. Volume of Revolution.agg |
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The way Autograph brings the concept of volume of revolution to life in 3D is quite something! |
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Challenge your students to picture the solid that will form if we rotate the shaded area 360° around the x-axis. |
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When you are ready, select Slow Plot, left-click on the shaded area, right-click and choose Find Volume |
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Hold down left-click and drag the cursor around the screen to see the solid emerge. |
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Now you can challenge your students to work out the volume. |
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A good way to start is to select the point at x = 3 and drag it closer to the other point (see Handy Autograph Tip below). |
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Students should hopefully see that the solid begins to resemble a cylinder the closer it gets, and following on from this the entire volume itself can be thought of as the sum of lots of these thin cylinders. |
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