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Autograph Introducing Autograph - Jim Claffey 7/08/2001 1 Using Autograph to Teach Concepts in the Calculus 1.Defining the slope of a curve at a point as the slope of the Tangent at that point. 2.The limiting position of the slope of the secant. 3.The Gradient function using the button on the toolbar. 4. Demonstrate and investigate the Gradient function. 5.The definition of f (x) as a limit, and the animation of this limiting function. 6.Some further ideas & suggestions for Lessons. A Dynamic approach to teaching Calculus Introducing Autograph - Jim Claffey

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Autograph Introducing Autograph - Jim Claffey 2 Plot any curve y=f(x) Here y=x² 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 Concepts in The CALCULUS - Slope

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Autograph Introducing Autograph - Jim Claffey 7/08/2001 3 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. The Slope of a Curve

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Autograph Introducing Autograph - Jim Claffey 7/08/2001 4 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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Autograph Introducing Autograph - Jim Claffey 7/08/2001 5 The Gradient Function Plotted in Autograph Press the ENTER key then type in the function y=x³-13x+12 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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Autograph Introducing Autograph - Jim Claffey 7/08/2001 6 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 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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Autograph Introducing Autograph - Jim Claffey 7/08/2001 7 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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Autograph Introducing Autograph - Jim Claffey 7/08/2001 8 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 and input the equation y=(f(x+h)-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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Autograph Introducing Autograph - Jim Claffey 7/08/2001 9 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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Autograph Introducing Autograph - Jim Claffey 7/08/2001 10 Limits: Continuity: and Differentiability

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Autograph Introducing Autograph - Jim Claffey 7/08/2001 11 The Chain Rule:

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Autograph Introducing Autograph - Jim Claffey 7/08/2001 12 Differentiating Exponential Functions Enter the function y=a x Autograph sets the initial value of a at a=1. 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 y=a x the same function as its gradient function?

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Autograph Introducing Autograph - Jim Claffey 7/08/2001 13 Log & Exponential Functions and Their Inverses

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Autograph Introducing Autograph - Jim Claffey 7/08/2001 14 Derivative of the Logarithmic function

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Autograph Introducing Autograph - Jim Claffey 7/08/2001 15 Investigate the Derivative of logx and nx

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Autograph Introducing Autograph - Jim Claffey 7/08/2001 16 Numerical Integration Areas

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Autograph Introducing Autograph - Jim Claffey 7/08/2001 17 Numerical Integration: Areas Bound by f(x), x-axis, x=a, a=b Enter the function y=f(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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Autograph Introducing Autograph - Jim Claffey 7/08/2001 18 Numerical Integration: Two Views of the Same Area

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Autograph Introducing Autograph - Jim Claffey 7/08/2001 19 Differential Equations: 1 st Order DEs.

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Autograph Introducing Autograph - Jim Claffey 7/08/2001 20 1 st Order Differential Equations: Relationship between y=1/x & y=lnx

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Autograph Introducing Autograph - Jim Claffey 7/08/2001 21 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.

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