Making Sense of Calculus with Mapping Diagrams[2016 MAA]

Martin Flashman, Dec 8, 2015

Making Sense of Calculus with Mapping Diagrams MAA Mini-Course 2016

Table of Contents

  1. Introduction
    1. Mapping Diagrams and Cartesian Geometry
    2. Understanding functions:tables, mapping diagrams and graphs.
  2. Linear Functions
    1. Mapping Diagrams and Graphs of Linear Functions
  3. Limits and The Derivative
    1. Mapping Diagrams and Limits of Functions
    2. Mapping Diagrams & Graphs: Def'n of Derivative as Number
    3. A Mapping Diagram and Graph for Estimating The Derivative
    4. Bisection Method
    5. Newton's Method: Mapping Diagrams and Graphs
    6. Mapping Diagrams and Graphs of Functions: Continuity Definition
    7. Mapping Diagrams and Graphs of Functions: Uniform Contiuity Definition
    8. Uniform Continuity Definition: Mapping Diagrams and Graphs of Functions
    9. Uniform Continuity Definition: Mapping Diagrams and Graphs of Functions
    10. Mapping Diagrams Functions: Uniform Continuity and Compact Sets
  4. Differentials, Differential Equations, and Euler's Method
    1. The Differential: Mapping Diagrams and Graphs
    2. Differential Equations,Euler, Mapping Diagrams
  5. Integration and the Fundamental Theorem
    1. Euler's Method visualized with mapping diagram and graph
    2. Additive Property for Definite Integrals
    3. Integral Mean Value Theorem with Mapping Diagram
  6. Sequences
    1. Mapping Diagrams for Sequence Examples
    2. Limits for functions defined using sequences Version 0
  7. Series
    1. MacLaurin Polynomials, Mapping Diagrams and Graphs
  8. Multi-dimensional Calculus
    1. 3D Mapping Diagram for linear z=f(x,y) with focus point
    2. Mapping Diagram for z= f(x,y), its partial deriv's & diff'l

1.

Introduction

  • 1. Mapping Diagrams and Cartesian Geometry
  • 2. Understanding functions:tables, mapping diagrams and graphs.

1.1.

Mapping Diagrams and Cartesian Geometry

Two frames illustrating the connection between Cartesian Graphs and Mapping Diagrams.

Notice how the points on the graph is paired with the points on the mapping diagram.
– Martin Flashman

1.2.

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2.

Linear Functions

  • 1. Mapping Diagrams and Graphs of Linear Functions

2.1.

Mapping Diagrams and Graphs of Linear Functions

Visualizing linear functions using mapping diagrams and graphs with focus point.

Notice how points on the graph pair with arrows and points on the mapping diagram. Check the box to see the focus point for the function on the mapping diagram.
– Martin Flashman

3.

Limits and The Derivative

  • 1. Mapping Diagrams and Limits of Functions
  • 2. Mapping Diagrams & Graphs: Def'n of Derivative as Number
  • 3. A Mapping Diagram and Graph for Estimating The Derivative
  • 4. Bisection Method
  • 5. Newton's Method: Mapping Diagrams and Graphs
  • 6. Mapping Diagrams and Graphs of Functions: Continuity Definition
  • 7. Mapping Diagrams and Graphs of Functions: Uniform Contiuity Definition
  • 8. Uniform Continuity Definition: Mapping Diagrams and Graphs of Functions
  • 9. Uniform Continuity Definition: Mapping Diagrams and Graphs of Functions
  • 10. Mapping Diagrams Functions: Uniform Continuity and Compact Sets

3.1.

Mapping Diagrams and Limits of Functions

A web page illustrating the limits of functions using Mapping Diagrams and Geogebra.
Notice how the points on the graph is paired with the points on the mapping diagram.
– Martin Flashman

3.2.

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3.3.

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3.4.

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3.5.

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3.6.

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3.7.

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3.8.

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3.9.

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3.10.

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4.

Differentials, Differential Equations, and Euler's Method

  • 1. The Differential: Mapping Diagrams and Graphs
  • 2. Differential Equations,Euler, Mapping Diagrams

4.1.

The Differential: Mapping Diagrams and Graphs

A web page illustrating the differential with graphs of functions and Mapping Diagrams.

Notice how the points on the graph is paired with the points on the mapping diagram.
– Martin Flashman

4.2.

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5.

Integration and the Fundamental Theorem

  • 1. Euler's Method visualized with mapping diagram and graph
  • 2. Additive Property for Definite Integrals
  • 3. Integral Mean Value Theorem with Mapping Diagram

5.1.

Euler's Method visualized with mapping diagram and graph

Euler's Method visualized with mapping diagram and graph, showing the connection between the mapping diagram and the area of a region in the plane bounded by the graph of y=P(x) = f'(x), the X axis, X=a and X = b.

Move the sliders to change a,b, and N. You can also change the function P(x) = f'(x) by entering a new function in the box.
– Martin Flashman

5.2.

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5.3.

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6.

Sequences

  • 1. Mapping Diagrams for Sequence Examples
  • 2. Limits for functions defined using sequences Version 0

6.1.

Mapping Diagrams for Sequence Examples

Sequences visualized using mapping diagrams

Move the sliders to change the sequence example and the number of points and arrows displayed. Notice how the points on the graph are paired with the points on the mapping diagram.
– Martin Flashman

6.2.

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7.

Series

  • 1. MacLaurin Polynomials, Mapping Diagrams and Graphs

7.1.

MacLaurin Polynomials, Mapping Diagrams and Graphs

Understanding MacLaurin Polynomials using mapping diagrams and graphs.

Notice how data in the table relates to coefficients on the polynomial while the points and arrows on the mapping diagram for the MacLaurin Polynomial and the remainder function are influenced by the size of n.
– Martin Flashman

8.

Multi-dimensional Calculus

  • 1. 3D Mapping Diagram for linear z=f(x,y) with focus point
  • 2. Mapping Diagram for z= f(x,y), its partial deriv's & diff'l

8.1.

3D Mapping Diagram for linear z=f(x,y) with focus point

An example of a mapping diagram for a two variable function. Use the check boxes to see the focus point. Notice: Every plane through x, y, and the focus point will pass through f(x,y). Every plane through x, y, and f(x,y) will pass through the focus point .
– Martin Flashman

8.2.

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