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Autograph

Using

Autograph

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

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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.

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7/08/2001

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Autograph

Introducing Concepts in The CALCULUS Slope

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Plot any curve y=f(x) Here y=x ² 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.

Introducing

Autograph

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Jim Claffey

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The Slope of a Curve

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Click on the zoom button.

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Hold it over point A and left click on the mouse.

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Each click on the mouse zooms further in on the curve and the tangent at A.

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The axes for the graph are automatically rescaled as you zoom in on point A.

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At A the slope of the curve and the slope of the tangent are identical.

Introducing

Autograph

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Jim Claffey 7/08/2001

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Autograph

The Tangent As the Limiting Position of the Secant

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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 selected right click on the Mouse. Select P and Gradient Q from the menu.

Select the point Q and move the point Q towards point P.

Introducing

Autograph

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Jim Claffey 7/08/2001

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Autograph

The Gradient Function Plotted in Autograph

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Press the ENTER key then type in the function y=x³-13x+12

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On the toolbar click on the gradient button This draws the gradient function without giving its equation.

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Click on the slow plot turtle button. From the dialogue box check the box Draw Tangent (You could check all three boxes).

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Click OK and watch as the tangent and the gradient function are drawn. Note what happens at the critical values.

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Use the spacebar to stop-start.

Introducing

Autograph

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Jim Claffey 7/08/2001

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Autograph

Developing a Table of Values for the Gradient Function f



(x)

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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 < Shift > 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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Introducing

Autograph

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Jim Claffey 7/08/2001

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Autograph

The Gradient Function Plotted and Investigated in Autograph

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Press the ENTER key then type in the function y = x² + 5

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On the toolbar click on the gradient button This draws the gradient function without giving its equation.

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Click on the slow plot turtle button. From the dialogue box check the box next to Draw Tangent (You could check all three boxes).

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Click OK and watch the tangent and the gradient function as they are drawn.

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Use the spacebar to stop-start.

Introducing

Autograph

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Jim Claffey 7/08/2001

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Autograph

The Gradient Function f

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(x) Defined As a Special Limit

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Click on the toolbar button.

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Enter a function: eg f(x) =x²-4x-3

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On the toolbar click on the gradient button to draw the gradient function.

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Press < ENTER > and input the equation y=(f(x+h)-f(x))/h (The starting value for h is taken to be 1).

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Click on the graph just drawn in the last step.

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On the toolbar click on the Constant controller Button

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Study what happens as h approaches zero. The step size can be changed.

Introducing

Autograph

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Jim Claffey 7/08/2001

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Autograph

Limits: Continuity: and Differentiability

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Piecewise functions can be entered quite easily.

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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.

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Note the relationship between the graph of f

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(x) and f(x).

Introducing

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Jim Claffey 7/08/2001

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Limits: Continuity: and Differentiability

10 Introducing

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Jim Claffey 7/08/2001

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The Chain Rule:

11 Introducing

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Jim Claffey 7/08/2001

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Differentiating Exponential Functions

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Enter the function y=a x

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Autograph sets the initial value of “a” at a=1.

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On the toolbar click on the gradient button to draw the gradient function.

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click on the Constant controller Button

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Investigate what happens!

For what value of “ a ” is y=a x the same function as its gradient function?

Introducing

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Jim Claffey 7/08/2001

Autograph

Log & Exponential Functions and Their Inverses 13 Introducing

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Jim Claffey 7/08/2001

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Derivative of the Logarithmic function 14 Introducing

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Jim Claffey 7/08/2001

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Investigate the Derivative of logx and

ℓ

nx

15 Introducing

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Jim Claffey 7/08/2001

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Numerical Integration Areas 16 Introducing

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Jim Claffey 7/08/2001

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Numerical Integration: Areas Bound by f(x), x-axis, x=a, a=b

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Enter the function y=f(x).

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Select the curve then right click. Select Area from the screen menu offered.

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In the Edit Area start value a box place the , the end value b , then the number of divisions in your partition. The numerical approximation of the area is given in the status bar.

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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.

Introducing

Autograph

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Jim Claffey 7/08/2001

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Numerical Integration: Two Views of the Same Area 18 Introducing

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Jim Claffey 7/08/2001

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Differential Equations: 1 st Order DEs .

19 Introducing

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Jim Claffey 7/08/2001

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1 st Order Differential Equations: Relationship between y=1/x & y=lnx 20 Introducing

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Jim Claffey 7/08/2001

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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.

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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. It’s 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.

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Autograph

can be used with Office 2000 in preparing documents.

Introducing

Autograph

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Jim Claffey 7/08/2001