University of Arizona
Mathematics Instruction Colloquium ¤
January 31, 2017
Making Sense of Pre-Calculus and 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/UA.MD.1_31_17.html

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 algebra, trigonometry, differential and integral calculus using mapping diagrams.

Background and References on Mapping Diagrams ¤

1. Mapping Diagrams. ¤

1.0. What is a mapping diagram?
Introduction and simple examples from the past: Napiers Logarithm

1.1. Warm up- Solve Equation: $2x+1 =5$.
Show Ignite on Linear Functions and Equations Meet Mapping Diagrams (.PPT)

1.2. Linear Functions. ¤ 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- the focus point, determined by the numbers
$m$ and $b$, denoted here by $[m,b]$
.
Mapping Diagrams and Graphs of Linear Functions
Visualizing linear functions using mapping diagrams and graphs.

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

3. Other Elementary Functions. ¤

3.1 Understanding elementary functions is the usual prelude to calculus-sometimes described as algebra and trigonometry.

The fundamental concept is to build a resource of functions based on core functions and processes to create more evolved functions.

The core functions are : constant functions, "x", exp, sine.

The processes are: addition, subtraction, multiplication, multiplicative inversion, composition, function inversion.

The class of elementary functions includes :
• (real) polynomials
• rational functions
• all exponential and logarithmic functions
• trigonometric and inverse trigonometric functions
• hyperbolic and inverse hyperbolic functions.

3.2. Solving equations connects to functions, composition, and mapping diagrams.¤
Mapping Diagrams from A(lgebra) B(asics) to C(alculus) and D(ifferential) E(quation)s.
A Reference and Resource Book on Function Visualizations Using Mapping Diagrams

4.Limits and The Derivative ¤
Mapping Diagrams Meet Limits and The Derivative
4.1 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 diagrams and graphs visualize how the definition of a limit works for real functions.

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

4.2 The derivative of $f$ at $a$ is a number, denoted $f'(a)$, defined as a limit of ratios ( average rates or slopes of lines). i.e., $$f'(a) = \lim_{x \rightarrow a} \frac {f(x)-f(a)}{x-a}.$$
The derivative can also be understood as the magnification factor of the best linear approximating function.¤

The derivative can be visualized using focus points and derivative "vectors" on a mapping diagram.

4.3 Mapping Diagrams for Composite (Linear) Functions ¤
This is the fundamental concept for the chain rule.

Visualizing the composition of linear functions using mapping diagrams and graphs.

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

5. 1st Derivative Analysis ¤
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.

5.1 First [and Second] Derivative Analysis.
Graphs of functions and mapping diagrams visualize first and second derivative analysis.
Notice how the points on the graph is paired with the points on the mapping diagram.

6. Differentials, Differential Equations, and Euler's Method ¤
The major connection between the derivative and the differential is visualized by a mapping diagram.

6.1 Mapping Diagrams for the Differential
Mapping Diagram for the Differential

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

6.2 Differential Equations, Euler, Mapping Diagrams ¤
Iterating the differential gives a numerical tool (Euler's Method) for estimating the solution to an initial value problem for a differential equation.
$P(x,y)= \frac {dy}{dx}, f(a)=c$

Estimate $f(b)$ given $y'= P(x,y)$ and $f(a)=c$ in N steps.
$\Delta x = \frac{b-a}N; f(b) \approx f(a) + \sum_{k=0}^{k=n-1} P(x_k,y_k)\Delta x$

7.Integration and the Fundamental Theorem
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.
Definition: As $N \rightarrow \infty$   $\sum_{k=0}^{k=n-1} P(x_k)\Delta x \rightarrow \int_a^b P(x)dx$

The Fundamental Theorem of Calculus.
Suppose $y = P(x) = f'(x)$ is a continuous function, then
$\int_a^b P(x)dx + f(a) = f(b)$
or
$\int_a^b P(x)dx = f(b) - f(a)$
where $f'(x) = P(x)$.

7.1 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. ¤

Thanks for participating
¤.

Questions?

The End!

References:¤

Mapping Diagrams from A(lgebra) B(asics) to C(alculus) and D(ifferential) E(quation)s.
A Reference and Resource Book on Function Visualizations Using Mapping Diagrams

The Sensible Calculus Program