Using Autograph to Teach Calculus
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1. The document outlines a workshop on using Autograph software to teach calculus topics. 2. The workshop consists of 5 mini-sessions on sketching graphs, numerical integration, volumes of revolution, differentiating trig functions, and sketching/differentiating y=ax. 3. Participants will get a short presentation then hands-on experimentation with the software during each mini-session.
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d?
R
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A
A¡¯B
S
C
rb
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- 2. Introduction to Workshop ? Five mini-sessions of 15 minutes each. 1.) Sketching and Transforming Graphs. 2.) Numerical Integration. 3.) Volumes of Revolution. 4.) Differentiating Trig Functions. 5.) Sketching and Differentiating y = ax. ? Short presentation (5 mins). ? Hands on experimentation (10 mins). 16/01/2012 2
- 3. 1.) Sketching and Transforming Graphs - Cartesian. - Polar. - Dynamic constant editor. - Slow plotting. 16/01/2012 3
- 4. Can you draw the ¡®Maths Man¡¯? What three plenary questions could you ask students to summarise their learning? 16/01/2012 4
- 6. Polar Graphs Use Autograph to experiment with drawing graphs of the form r = a + b sin (c¦È) where a, b, c ? Z Can you sketch these graphs? 16/01/2012 6
- 7. 2.) Numerical Integration - Use Autograph to establish principle of ¡®Trapezium Rule¡¯ etc. - y = x2 + 1 with domain 0 ¡Ü x ¡Ü 1 and range 0 ¡Ü y ¡Ü 2. - Two intervals? - Over or under estimate? - How do you improve the accuracy of the approximation? 16/01/2012 7
- 8. 16/01/2012 8
- 9. 3.) Volumes of Revolution - Visualising 3D shapes. - Using ¡®Jing¡¯ as a teaching tool. - Watch the hyperlinked clip. - http://screencast.com/t/0lkBztEAxvp - Now try yourself! - Can you drawn a ¡®rugby ball¡¯? 16/01/2012 9
- 10. Visualising an approximation of the volume of revolution generated when y = x2 is rotated 2¦Ð about the x-axis using rectangular strips 16/01/2012 10
- 11. Can you use Autograph to drawn the ¡®rugby¡¯ ball? 16/01/2012 11
- 12. 4.) Differentiating Trigonometric Functions - Plot y = sin x. - Plot dy/dx. - Describe the relationship between the curves. - Can you explain why? - Can you hypothesise the relationship between y = cos x and the associated dy/dx curve? 16/01/2012 12
- 13. 16/01/2012 13
- 14. 16/01/2012 14
- 15. Investigate Sketch the graphs of y = sinh(x), y = cosh(x) and their associated derivatives. Can you derive graphically the rules for the derivatives of y = sinh(x), y = cosh(x), y = sinh(ax) etc. 16/01/2012 15
- 16. 5.) Sketching and Differentiating y = ax - Sketch y = ax. - Sketch dy/dx. - What happens as a ¡ú e? 16/01/2012 16
- 17. 16/01/2012 17