MAA Minicourse #7 ¤

January 7 and 9, 2015

Martin Flashman

Professor of Mathematics

Humboldt State University

Link for these notes:

Abstract:

In this mini-course participants will learn how to use mapping diagrams (MD) to visualize functions for many calculus concepts. For a given function, f, a mapping diagram is basically a visualization of a function table that can be made dynamic with current technology. The MD represents x and f(x) from the table as points on parallel axes and arrows between the points indicate the function relation. The course will start with an overview of MD’s and then connect MD's to key calculus definitions and theory including: linearity, limits, derivatives, integrals, and series. Participants will learn how to use MD’s to visualize concepts, results and proofs not easily realized with graphs for both single and multi-variable calculus. Participants are encouraged to bring a laptop with wireless capability.

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

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.

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.

3.5.1 First [and Second] Derivative Analysis.

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

4.1 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.

Suppose $P(x)$ is a continuous function on $[a,b]$, then $\int_a^b P(x)dx $ exists.

$$\int_a^b P(x)dx + \int_b^c P(x)dx = \int_a^c P(x)dx$$

Geometry Interpretation:

See Figure V.B.1.

Figure V.B.1. |

These properties allow some determination of definite integrals based on knowledge of the components that make up the integrand. Although the calculus is not as easy as that of the derivative here are some examples of how the properties just discussed can be used for an elementary calculus for the definite integral.

End SC STUFF The additive property of the definite integral visualized with mapping diagram and graph, showing the connection between the mapping diagrams and the areas of regions in the plane bounded by the graph of

If $p$ is a continuous function on $[a,b]$ then there is an number $c_* $ between $a $ and $b $ where

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

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

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

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