Arizona State University

Mathematics Education Colloquium ¤

November 3, 2017

**Making Sense of Calculus with Mapping Diagrams:**

A Visual Alternative to Graphs

Martin Flashman

Professor of Mathematics

Humboldt State University

Link for these notes:

**http://flashman.neocities.org/Presentations/MD.ASU.11_3_17.html**

Mathematics Education Colloquium ¤

November 3, 2017

A Visual Alternative to Graphs

Martin Flashman

Professor of Mathematics

Humboldt State University

Link for these notes:

Abstract:

Mapping diagrams are an important and underutilized alternative to graphs for visualizing functions.

Starting from basics, Professor Flashman will demonstrate some of his assaults on the challenges of visualizing differential and integral calculus using mapping diagrams.

Background and References on Mapping Diagrams

1. Mapping Diagrams. ¤

What is a mapping diagram?

Introduction and simple examples from the past: Napiers Logarithm

Understanding functions using tables. mapping diagrams and graphs.

Functions: Tables, Mapping Diagrams, and Graphs

See also Underground Mathematics (University of Cambridge): mapping-a-function

2. Linear Functions. ¤

Linear functions are the key to understanding calculus.

Linear functions are traditionally expressed by an equation like : $f(x)= mx + b$.

Mapping diagrams for linear functions have one simple unifying feature-

Mapping Diagrams and Graphs of Linear Functions

3.1 Limits with Mapping Diagrams and Graphs of Functions

The traditional issue for limits of a function $f$ is whether $$ \lim_{x \rightarrow a}f(x) = L$$.

The definition is visualized in the following example.

mapping-a-derivative

This is the fundamental concept for the chain rule.

Notice how points on the graph are paired with the points and arrows on the mapping diagram.

The traditional analysis of the first derivative is visualized with mapping diagrams. Extremes and critical numbers and values connected. Time permitting- the Intermediate and Mean Value Theorems are visualized- along with Newton's Method for estimating roots to an equation.

4.1 First [and Second] Derivative Analysis.

The major connection between the derivative and the differential is visualized by a mapping diagram.

5.1 Mapping Diagrams for the Differential

Connecting Euler's method to sums leads to a visualization of the definite integral as measuring a net change in position in a mapping diagram and an area of the graph of the velocity.

$\int_a^b P(x)dx + f(a) = f(b)$

or

$\int_a^b P(x)dx = f(b) - f(a)$

where $f'(x) = P(x)$.

or

$\int_a^b P(x)dx = f(b) - f(a)$

where $f'(x) = P(x)$.

Thanks for participating

A Reference and Resource Book on Function Visualizations Using Mapping Diagrams

Martin Flashman, GeoGebra Book: Making Sense of Calculus with Mapping Diagrams[2016 MAA]

AMATYC Webinar M Flashman Using Mapping Diagrams to Understand Trig Functions (YouTube)

Martin Flashman ...Solving Linear Equations Visualized with Mapping Diagrams (YouTube)

Martin Flashman ...Partial Derivatives: An Introduction Using Mapping Diagrams (You Tube)

Martin Flashman ...Solving Linear Equations Visualized with Mapping Diagrams (YouTube)

Martin Flashman ...Partial Derivatives: An Introduction Using Mapping Diagrams (You Tube)