MAA Minicourse #7
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January 7 and 9, 2015
Making Sense of Calculus with Mapping Diagrams

Martin Flashman
Professor of Mathematics
Humboldt State University

http://flashman.neocities.org/Presentations/MD.JMM.Mini.1_7_16.html

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

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 $[m,b]$
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Mapping Diagrams and Graphs of Linear Functions
Visualizing linear functions using mapping diagrams and graphs.
This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com
Notice how points on the graph pair with arrows and points on the mapping diagram.

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 $$\lim_{x \rightarrow a}f(x) = L$$
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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.

3.4. Newton's method. An early application of the first derivative, Newton's method for estimating roots of functions is visualized with mapping diagrams.

Add GeoGebra Here for Newton's Method.
This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com

3.5. 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.

3.5.1 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.

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

4.1 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.

4.2 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.
$\Delta x = \frac{b-a}N; f(b) \approx f(a) + \sum_{k=0}^{k=n-1} P(x_k,y_k)\Delta x$

5 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: The definite integral of $P$ over the interval $[a,b]$, denoted $\int_a^b P(x)dx$, is defined as the number $I$, where as $N \rightarrow \infty$, $\Delta x \to 0$,   $$\sum_{k=0}^{k=N-1} P(x_k)\Delta x \rightarrow I \equiv \int_a^b P(x)dx, \ provided\ the \ limit, I , \ exists.$$
Theorem:
Suppose $P(x)$ is a continuous function on $[a,b]$, then $\int_a^b P(x)dx$ exists.

Theorem:  Suppose $y = P(x)$ is a continuous function on $[a,c]$ and $a<b<c$ then
$$\int_a^b P(x)dx + \int_b^c P(x)dx = \int_a^c P(x)dx$$

5.1 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.

5.2 SC STUFF [The Additive Property.] $\int_a^b P(x) dx = \int_a^c P(x) dx + \int_c^b P(x) dx$ for any $a, b,$ and $c$.
Geometry Interpretation:
If $P(x) > 0$ and $a < c < b$ , then the area of the regions above the $X$-axis and below the graph of $Y=P(X)$ between the lines $X=a$ and $X=b$, $\int_a^b P(x) dx$ , is the sum of the areas of the regions enclosed by the $X$-axis, the graph of $Y=P(X), X=a$ and $X=c$, $\int_a^c P(x) dx$ , and by the $X$-axis, the graph of $Y=P(X), X=c$ and $X=b$,  $\int_c^b P(x) dx$  .
See Figure V.B.1.

Motion Interpretation: Assume  $a < c < b$. The net change in position of an object moving with velocity $P$ between time $a$ and time $b$, $\int_a^b P(x) dx$, is the sum of the net changes in position between time $a$ and time $c$, $\int_a^c P(x) dx$, and between time $c$ and time $b$, $\int_c^b P(x) dx$. See Figure V.B.1.
 Figure V.B.1. This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com

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

$y = P(x) = f'(x)$, the X axis, X=a, X = b, and X =c.

This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com
Move the sliders to change $a,b,c$, and $N$. You can also change the function $P(x)$ by entering a new function in the box.

Mean Value Theorem for Integrals:
If $p$ is a continuous function on $[a,b]$ then there is an number $c_*$ between $a$ and $b$ where
$$\int _a^b P(x) dx = P(c_*)*(b-a).$$

This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com

The Fundamental Theorem of Calculus
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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 $f'(x) = P(x)$.

5.3 Euler's Method visualized with mapping diagram and graph, showing the connection for the Fundamental Theorem of Calculus between the mapping diagram and
$$\int_a^b P(x)dx = f(b) - f(a)$$
where $f'(x) = P(x)$.

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.

7.1 GeoGebra Applet for visualizing  sequences with Mapping Diagrams and Graphs for a sequence defined by $a_n = f(n)$

This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com

8.1 GeoGebra Applet for constructing MacLaurin Polynomials and Remainders $P_n(x,f(x))$ and $R_n(x,f(x))$

Thanks for participating
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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