Junior Math Tutorial - Spring

Alexis Doval, Apr 3, 2014

GeoGebra items for Integral Junior Math Tutorial

Table of Contents

  1. Archimedes
    1. Archimedes QP Prop. 1
    2. Archimedes QP Prop. 2
    3. Archimedes QP Prop. 3
    4. Archimedes QP Prop. 18
    5. Archimedes QP Prop. 19
    6. Archimedes Mechanical Demonstration
  2. Cavalieri
    1. Cavalieri - Area under curve
  3. Newton Lemmas
    1. Application of Newton's Lemma VI
    2. Application of Newton Lemma VI 2
    3. Application of Newton's Lemma VI - p. 91, prob. 7
  4. Leibniz
    1. Leibniz - Ellipse/Parabola
    2. Example of Latus Rectum
    3. Leibniz Parabola - Manual Page 97
    4. Leibniz Differential Calculus Example - Manual p. 99
    5. Leibniz Differential Calculus Example - Manual p. 100
    6. Leibniz Rules for Differential Calculus Addition/Subtraction
    7. Leibniz Rules for Differential Calculus Multiplication
    8. Leibniz Rules for Differential Calculus - Division
  5. Leibniz Application of Rules Problems
    1. Leibniz - Application of Rules Problem 1 a-e
    2. Leibniz - Application of Rules Problem 2 a-e
    3. Leibniz - Application of Rules Problem 3 a-d
    4. Leibniz - Application of Rules Problem 4 a-d
    5. Leibniz - Application of Rules Problem 5 a-d
    6. Leibniz - Application of Rules Problem 6 a-d
    7. Leibniz - Application of Rules Problem 7 a-d
  6. Trigonometric Applications
    1. Note 21 - Trigonometric Applications
    2. The unit circle and plotting y = sin θ
    3. Radians
    4. The unit circle and plotting y = sin θ / y = cos θ

1.

Archimedes

  • 1. Archimedes QP Prop. 1
  • 2. Archimedes QP Prop. 2
  • 3. Archimedes QP Prop. 3
  • 4. Archimedes QP Prop. 18
  • 5. Archimedes QP Prop. 19
  • 6. Archimedes Mechanical Demonstration

1.1.

Archimedes QP Prop. 1

Archimedes Quadrature of Parabola Prop. 1
Drag point V to get different chord lengths
– Alexis Doval

1.2.

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1.3.

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1.4.

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1.5.

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1.6.

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2.

Cavalieri

Theorems of Cavalieri

  • 1. Cavalieri - Area under curve

2.1.

Cavalieri - Area under curve

Cavalieri - Area under curve

Click on Areas checkbox to view rectangle and triangle in segment of parabola. Use the Function Inspector (set to -2 / 2) to see area under curve.
– Alexis Doval

3.

Newton Lemmas

  • 1. Application of Newton's Lemma VI
  • 2. Application of Newton Lemma VI 2
  • 3. Application of Newton's Lemma VI - p. 91, prob. 7

3.1.

Application of Newton's Lemma VI

Move point B toward A to see the eventual slope of the tangent to the curve [math]y=x^2[/math] at A. Do the same with points D and C for [math]y=x^3[/math]. Hold Shift and scroll to zoom in and out.
– Alexis Doval

3.2.

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3.3.

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4.

Leibniz

  • 1. Leibniz - Ellipse/Parabola
  • 2. Example of Latus Rectum
  • 3. Leibniz Parabola - Manual Page 97
  • 4. Leibniz Differential Calculus Example - Manual p. 99
  • 5. Leibniz Differential Calculus Example - Manual p. 100
  • 6. Leibniz Rules for Differential Calculus Addition/Subtraction
  • 7. Leibniz Rules for Differential Calculus Multiplication
  • 8. Leibniz Rules for Differential Calculus - Division

4.1.

Leibniz - Ellipse/Parabola

Move point B to extend focus C toward infinity. What happens when line BP is parallel to line VAC? Leibniz says that when line BP is all but parallel to line VAC, angle ACB is infinitely small. But I have added line B'C which will create and angle B'CA that is always less than angle BCA. Will angle B'AC be less than angle BAC at the "transition point" when the angle (BAC or B'AC?) is infinitely small?
– Alexis Doval

4.2.

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4.3.

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4.4.

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4.5.

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4.6.

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4.7.

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4.8.

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5.

Leibniz Application of Rules Problems

  • 1. Leibniz - Application of Rules Problem 1 a-e
  • 2. Leibniz - Application of Rules Problem 2 a-e
  • 3. Leibniz - Application of Rules Problem 3 a-d
  • 4. Leibniz - Application of Rules Problem 4 a-d
  • 5. Leibniz - Application of Rules Problem 5 a-d
  • 6. Leibniz - Application of Rules Problem 6 a-d
  • 7. Leibniz - Application of Rules Problem 7 a-d

5.1.

Leibniz - Application of Rules Problem 1 a-e

Leibniz - Application of Rules Problem 1 a-e

Check the box for each problem a-e to see the graph and slide the tangent point to show the differential at various points. You can check the values against your equations.
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5.2.

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5.3.

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5.4.

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5.5.

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5.6.

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5.7.

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6.

Trigonometric Applications

  • 1. Note 21 - Trigonometric Applications
  • 2. The unit circle and plotting y = sin θ
  • 3. Radians
  • 4. The unit circle and plotting y = sin θ / y = cos θ

6.1.

Note 21 - Trigonometric Applications

Trigonometric Applications - Application of Newton's Lemmas and Leibniz's Differentials Slide point B toward point A
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6.2.

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6.3.

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6.4.

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