Calculus

Juan Carlos Ponce Campuzano, Oct 5, 2015

Geometrical representations of different concepts of calculus.

Table of Contents

  1. Differentiation
    1. Derivative
    2. Relationship between a function and its derivative
    3. Riemann sums
    4. Taylor and Maclaurin polynomials
  2. Integration
    1. Riemann sums
  3. Optimisation
    1. Mathematical optimization: Building a rectangular pen
    2. Box problem (right square prism)
    3. Cylinder container

1.

Differentiation

  • 1. Derivative
  • 2. Relationship between a function and its derivative
  • 3. Riemann sums
  • 4. Taylor and Maclaurin polynomials

1.1.

Derivative

Geometrical representation of the derivative.
Drag the red point. Change the value of h.
– Juan Carlos Ponce Campuzano

1.2.

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1.3.

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1.4.

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2.

Integration

  • 1. Riemann sums

2.1.

Riemann sums

The sum is calculated by dividing the region up into shapes (rectangles) that together form a region that is similar to the region being measured, then calculating the area for each of these shapes, and finally adding all of these small areas together. Because the region filled by the small shapes is usually not exactly the same shape as the region being measured, the Riemann sum will differ from the area being measured. This error can be reduced by dividing up the region more finely, using smaller and smaller shapes. As the number of shapes grows towards infinity, the sum approaches the Riemann integral.
– Juan Carlos Ponce Campuzano

3.

Optimisation

  • 1. Mathematical optimization: Building a rectangular pen
  • 2. Box problem (right square prism)
  • 3. Cylinder container

3.1.

Mathematical optimization: Building a rectangular pen

A farmer wants to build a rectangular pen for his sheep. [br]One side is a straight river. For the other three sides [br]he has 200m of fencing to use. [br][br]What is the maximum area of pen he can make?
Animate to see how the fence changes. Click on the Representation box to see the graphic representation of the relationship between one side of the fence and the Area.
– Juan Carlos Ponce Campuzano

3.2.

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3.3.

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Information