Using Autograph to Teach Concepts in the Calculus

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Using Autograph to Teach Concepts in the Calculus
A Dynamic approach to teaching Calculus

• Defining the slope of a curve at a point as the
  slope of the Tangent at that point.
• The limiting position of the slope of the secant.
• The Gradient function using the button on
  the toolbar.
• Demonstrate and investigate the Gradient
  function.
• The definition of f?(x) as a limit, and the
  animation of this limiting function.
• Some further ideas suggestions for Lessons.

Introducing Autograph - Jim Claffey
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Introducing Concepts in The CALCULUS - Slope

• Plot any curve yf(x) Here yx²
• Click on the cursor Button and place a point on
  the curve at A.
• With the point selected right click the mouse
• Select tangent from the menu. The equation of the
  tangent is given in the status bar at the bottom
  of the screen.

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The Slope of a Curve

• Click on the zoom button.
• Hold it over point A and left click on the mouse.
• Each click on the mouse zooms further in on the
  curve and the tangent at A.
• The axes for the graph are automatically rescaled
  as you zoom in on point A.
• At A the slope of the curve and the slope of the
  tangent are identical.

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The Tangent As the Limiting Position of the Secant

• Insert a cursor point on the curve at P then draw
  the tangent at P. Insert a second point at Q.
• While holding down the shift key select both P
  and Q.
• Right click the mouse. Select line from the menu.
  This draws a line through P and Q.
• Again with both P and Q selected right click on
  the Mouse. Select Gradient from the menu.
• Select the point Q and move the point Q towards
  point P.

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The Gradient Function Plotted in Autograph

• Press the ENTER key then type in the function
  yx³-13x12
• On the toolbar click on the gradient button
  This draws the gradient function without giving
  its equation.
• Click on the slow plot turtle button. From the
  dialogue box check the box Draw Tangent (You
  could check all three boxes).
• Click OK and watch as the tangent and the
  gradient function are drawn. Note what happens at
  the critical values.
• Use the spacebar to stop-start.

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Developing a Table of Values for the Gradient
Function f?(x)

• Place a point on the graph (say at x-5).
• With the point selected right click and select
  Tangent from the menu offered.
• The tangent is drawn, its equation is given in
  the status bar below the graph.
• Select the tangent point, hold down the ltShift gt
  key. Use the ?cursor key to move the tangent to
  the next x-value.
• The slope of the curve at this point is given by
  the slope of the tangent line given in the status
  bar.

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The Gradient Function Plotted and Investigated
in Autograph

• Press the ENTER key then type in the function y
  x² 5
• On the toolbar click on the gradient button
  This draws the gradient function without giving
  its equation.
• Click on the slow plot turtle button. From
  the dialogue box check the box next to Draw
  Tangent (You could check all three boxes).
• Click OK and watch the tangent and the gradient
  function as they are drawn.
• Use the spacebar to stop-start.

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The Gradient Function f?(x) Defined As a Special
Limit

• Click on the toolbar button.
• Enter a function eg f(x) x²-4x-3
• On the toolbar click on the gradient button
  to draw the gradient function.
• Press ltENTERgt and input the equation
  y(f(xh)-f(x))/h(The starting value for h is
  taken to be 1).
• Click on the graph just drawn in the last step.
• On the toolbar click on the Constant controller
  Button
• Study what happens as h approaches zero. The
  step size can be changed.

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Limits Continuity and Differentiability

• Piecewise functions can be entered quite easily.
• Determine any critical values of x where the
  function should be checked for(i) the existence
  of a limit(ii) Possible points of
  discontinuity(iii) Point-wise differentiability.
• Note the relationship between the graph of f?(x)
  and f(x).

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Limits Continuity and Differentiability
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The Chain Rule
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Differentiating Exponential Functions

• Enter the function yax
• Autograph sets the initial value of a at a1.
• On the toolbar click on the gradient button
  to draw the gradient function.
• click on the Constant controller Button
• Investigate what happens!
• For what value of a is yax the same function
  as its gradient function?

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Log Exponential Functions and Their Inverses
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Derivative of the Logarithmic function
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Investigate the Derivative of logx and lnx
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Numerical Integration Areas
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Numerical Integration AreasBound by f(x),
x-axis, xa, ab

• Enter the function yf(x).
• Select the curve then right click. Select Area
  from the screen menu offered.
• In the Edit Area box place the start value a, the
  end value b, then the number of divisions in your
  partition. The numerical approximation of the
  area is given in the status bar.
• If you place a cursor at A and B the Edit Area
  Window enters these as the default values.You
  can move either A or B on the curve. The area
  adjusts.

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Numerical IntegrationTwo Views of the Same Area
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Differential Equations 1st Order DEs.
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1st Order Differential Equations Relationship
between y1/x ylnx
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In My Humble Opinion

• Autograph will alter the way mathematics is
  currently taught.
• I believe Autograph will change present
  classroom dynamics.
• There are many concepts in the present High
  School Maths courses that could be better taught
  by using aids such as Autograph.
• Autograph is an excellent student resource as
  well as an excellent teaching tool. Its
  interactive animation feature aids understanding.
• Autograph lessons can be annotated, stored and
  improved upon. They can be sent or exchanged
  worldwide via e-mail or the internet.
• Autograph is in my opinion the best software
  world-wide for use in secondary Mathematics
  classrooms.
• Autograph has been designed by expert classroom
  practitioners.
• Autograph can be used with Office 2000 in
  preparing documents.