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 Session G4 14:00-15:30
Wednesday, April 12, 2017
Making Sense of Functions with Mapping Diagrams:
From Algebra Basics to Calculus

Martin Flashman
Professor of Mathematics
Humboldt State University
Arcata, CA, USA

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A mapping diagram (MD) is an alternative to a Cartesian graph that visualizes a function. Like a table,  it can present finite data,  but  it also can work continuously and dynamically with technology. Participants will learn how to use MDs to make sense of linear functions and basic function concepts. Examples will connect MDs with graphs to make sense of differential and integral calculus with both estimations and theoretical results.
Background and examples are available at Mapping Diagrams from A(lgebra) B(asics) to C(alculus) and D(differential) E(equation)s.
http://flashman.neocities.org/MD/section-1.1VF.html  and at geogebra.org.

Background and References on Mapping Diagrams ¤

1. Mapping Diagrams. ¤

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

Beginning example: Linear Functions.
Understanding functions using tables. mapping diagrams and graphs.
Functions: Tables, Mapping Diagrams, and Graphs 

2. Linear Functions. ¤
2.1 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.

2.2 Solving linear equations as a model for visualizing the solution of other equations.
Every linear function can be considered a composition: $f(x)= mx+b = f_{+b}(f_{m*}(x))$ where  $f_{+b}(u) = u+b$ and $f_{m*}(x)= mx$. The composition allows the visualization of the traditional steps in solving a linear equation.

3.Limits and The Derivative ¤
Mapping Diagrams Meet Limits and The Derivative
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 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.

3.2 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.

3.3 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.

4. 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.

4.1 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.

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

5.1 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.

5.2 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 $  

6.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
$\int_a^b P(x)dx + f(a) = f(b)$
$\int_a^b P(x)dx = f(b) - f(a)$
where $f'(x) = P(x)$.

6.1 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


The End!


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