Calculus

John Golden, Sep 22, 2014

Sketches for calculus.

Table of Contents

  1. Derivatives
    1. Derivative of a linear piecewise function
    2. Drawing Derivatives
    3. Tangent Zoom
    4. Slope and Tangent Lines
    5. Relating Tangent values to the derivative function
    6. Calculus of Trigonometry
    7. Function Composition
    8. Chain Rule
    9. Open Box
    10. Minimizing the total time spent on a hike
    11. Two Shapes from a Wire
    12. Leaving Fargo
    13. Taylor Approximation
    14. Baseball: Distance between two players
    15. Derivative Guess
    16. Derivative of Sine
    17. Integration by Substitution
    18. Related Rates (Rolling Carts Problem)
    19. Related Rates
    20. Finding Rate of Change at Points
    21. Off Road
    22. Cost of a Can
    23. USPS Package
    24. Filling Flasks
  2. Integrals
    1. Accumulating Penguins
    2. Accumulation Curve
    3. Growing Squares
    4. Pyramid Volume and Cross Section
    5. Solid by Cross Section
    6. Riemann Sum
    7. Riemann Sums and Integrals
    8. Slopefields and Euler's Method
  3. Limits
    1. The limit of a sequence

1.

Derivatives

  • 1. Derivative of a linear piecewise function
  • 2. Drawing Derivatives
  • 3. Tangent Zoom
  • 4. Slope and Tangent Lines
  • 5. Relating Tangent values to the derivative function
  • 6. Calculus of Trigonometry
  • 7. Function Composition
  • 8. Chain Rule
  • 9. Open Box
  • 10. Minimizing the total time spent on a hike
  • 11. Two Shapes from a Wire
  • 12. Leaving Fargo
  • 13. Taylor Approximation
  • 14. Baseball: Distance between two players
  • 15. Derivative Guess
  • 16. Derivative of Sine
  • 17. Integration by Substitution
  • 18. Related Rates (Rolling Carts Problem)
  • 19. Related Rates
  • 20. Finding Rate of Change at Points
  • 21. Off Road
  • 22. Cost of a Can
  • 23. USPS Package
  • 24. Filling Flasks

1.1.

Derivative of a linear piecewise function

On the left is a piecewise linear function. On the left is the derivative of that piecewise linear function.
1) In the left window, drag points A, B, C, D, and E to see how the derivative changes. [br]2) How is the graph formed?[br]3) Why are there open circles in the derivative? Should they be connected? Explain.
– cconrad

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1.8.

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1.10.

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1.11.

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1.12.

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1.13.

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1.14.

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1.15.

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1.16.

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1.17.

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1.18.

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1.19.

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1.20.

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1.21.

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1.22.

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1.23.

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1.24.

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2.

Integrals

  • 1. Accumulating Penguins
  • 2. Accumulation Curve
  • 3. Growing Squares
  • 4. Pyramid Volume and Cross Section
  • 5. Solid by Cross Section
  • 6. Riemann Sum
  • 7. Riemann Sums and Integrals
  • 8. Slopefields and Euler's Method

2.1.

Accumulating Penguins

In a math twitter discussion about calculus, the idea was brought up of penguins entering a room. [br][br]If this many entered each minute, how many penguins?
– John Golden

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3.

Limits

  • 1. The limit of a sequence

3.1.

The limit of a sequence

The limit of the sequence ([math]a_n[/math]) is [math]L[/math] if given any [math]\varepsilon>0[/math], there is an [math]N>[/math]0 such that [math]|a_n-L|<\varepsilon[/math] for all [math]n\ge N[/math]. [br][br]The yellow band is [math]L±\varepsilon[/math] and the blue region correspons to those [math]n\ge N[/math]. The green dots are within [math]\varepsilon[/math] of [math]L[/math] and the red dots and red x's are not. The limit is [math]L[/math] if for any given [math]\varepsilon[/math] you are able to move the blue region far enough to the right that there are no red x's.[br][br]Note: To move the window, hold the shift key down while you click and drag. To zoom in or out, hold down the shift key and scroll the mouse.
The limit of a sequence
– David Richeson

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