University of Nevada, Las Vegas

Mathematics Department Colloquium ¤

October 13, 2015

**Mapping Diagrams Take on Calculus and Complex Variables**

Martin Flashman

Professor of Mathematics

Humboldt State University

Link for these notes:

**http://flashman.neocities.org/Presentations/MD.UNLV.10_13_15.html**

Mathematics Department Colloquium ¤

October 13, 2015

Martin Flashman

Professor of Mathematics

Humboldt State University

Link for these notes:

Abstract:

Visualizing functions with mapping diagrams has been a recurrent theme and a central passion for me over many years.

In this presentation I'll demonstrate some of my more recent assaults on the challenges of differential and integral calculus and even the study of functions of a complex variable using mapping diagrams. Knowledge of at least one semester of calculus will be presumed.

Background and References on Mapping Diagrams

1.1. 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

1.2. 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

1.3.1 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.

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.

1.4.1 First [and Second] Derivative Analysis.

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

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

Mapping diagrams visualize functions of complex variables in new ways that illustrate some important functions.

Examples using linear functions, linear fractional (Moebius), and power functions.

Use the slider to change the radius of the circle, r, in the domain for the mapping diagram.

A complex

This example shows a linear function on a single complex number and points on a line in the complex plane.

You can move the point z

You can move a,b,and c in the Graphics window to change the complex parameters for the Moebius function.

The sliders adjust the height of the second complex plane and the radius of the circle for the mapping diagram arrow source.

2.1.4 Visualizing Complex Power Functions with Mapping Diagrams ¤

Thanks for participating

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)

A Reference and Resource Book on Function Visualizations Using Mapping Diagrams