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Using ‘Autograph’ to teach Calculus topics from AS and A2 Level Maths and Further Maths.

d.barton@poolehigh.poole.sch.uk

tinyurl.com/amaths

@mrdebarton

Darren Barton – Southampton University 16th January 2012

Darren Barton – Southampton Uni – 16/01/12

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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).

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1.) Sketching and Transforming Graphs

- Cartesian.- Polar.- Dynamic constant editor.- Slow plotting.

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Can you draw the ‘Maths Man’?

What three plenary questions could you ask students to summarise their learning?

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Extension!

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Polar Graphs

Use Autograph to experiment with drawing graphs of the form

r = a + b sin (cθ) where a, b, c ϵ Z

Can you sketch these graphs?

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2.) Numerical Integration

- Use Autograph to establish principle of ‘Trapezium Rule’ etc.

- y = x2 + 1 with domain 0 ≤ x ≤ 1 and range 0 ≤ y ≤ 2.

- Two intervals?- Over or under estimate?- How do you improve the accuracy of the

approximation?

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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’?

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Visualising an approximation of the volume of revolution generated when y = x2 is rotated 2π about the x-axis using rectangular strips

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Can you use Autograph to drawn the ‘rugby’ ball?

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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?

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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.

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5.) Sketching and Differentiating y = ax

- Sketch y = ax.- Sketch dy/dx.- What happens as a → e?

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