Making Sense of Calculus with Mapping Diagrams Part II

5.2 The Additive Property for Integral
Theorem: [The Additive Property.] Suppose $y = P(x)$ is a continuous function on $[a,b]$ and $ a<c<b$ then
$$\int_a^c P(x)dx + \int_c^b P(x)dx = \int_a^b P(x)dx$$
In General: $\int_a^b P(x) dx = \int_a^c P(x) dx + \int_c^b P(x) dx$ for any $a, b,$ and $c$.

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.
Traditional 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.

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

Move the sliders to change $a,b$, and $c$. You can also change the function $P(x) = f'(x)$ by entering a new function in the box.¤

The additive property of the definite integral is visualized with mapping diagram and graph below, showing the connection between the integral estimates with 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.

5.4.1 Euler's Method visualized with mapping diagram and graph, showing the connection for the Fundamental Theorem of Calculus between the mapping diagram and estimating

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

x	f(x)	f'(x)	df=f'(x)dx
0.	0	1.0	0.2
0.2	0.2	1.4	0.28
0.4	0.48	1.8	0.36
0.6	0.84	2.2	0.44
0.8	1.28	2.6	0.52
1.0	1.80	3.0	0.60
1.2	2.40	3.4	0.68
1.4	3.08	3.8	0.76
1.6	3.84	4.2	0.84
1.8	4.68	4.6	0.92
2.0	5.60

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 $a,b$, and $N$. You can also change the function $P(x) = f'(x)$ by entering a new function in the box.

In context of the Sensible Calculus Text: IV.F Euler's Method Meets The Position, Cost, and Area Problem

5.4.2 Euler's Method visualized with mapping diagram and graph, showing the connection for the Fundamental Theorem of Calculus between the mapping diagram and
$$\int_a^b P(x)dx $$ where $f'(x) = P(x)$ evaluated by GeoGebra's CAS. ¤

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 $a,b$, and $N$. You can also change the function $P(x) = f'(x)$ by entering a new function in the box.

6 & 7.Sequences and Series

8. Multi-dimensional Calculus

Visualizing $z=f(x,y)$ and Understanding Partial Derivatives and the Differential in 3D with GeoGebra 5.0
A glimpse of multi-dimensional calculus with mapping diagrams.
Mapping Diagrams add much to visualizing multi-variable functions.

8.1 : Linear Functions of 2 variables: $z =f(x,y) = Ax + By + C$. As with functions of one variable, understanding linear (affine) functions is a key to the calculus of multi-variable functions, Mapping diagrams provide a valuable tool to understanding visually linear functions with 2 or more variables.¤

A mapping diagram for a linear function of 2 variables exhibits a focus point in a 3-D image.
This point completely determines the linear function values.
Using the plane determined in space by the three points: x on the x - axis, y on the y-axis, and the focus point,
the value of the function is determined by where that plane intersects the z axis.

8.2: Functions of 2 variables: $z =f(x,y)$ and partial derivatives. As with functions of one variable, functions of many variables are understood with linear functions (a key to the calculus of multi-variable functions), partial derivatives and the differential. Mapping diagrams provide a valuable tool to understanding visually these concepts for functions with 2 or more variables.¤

Mapping diagram for $f(x,y) = -x^2 -2y$ with evaluation for two pairs of data

Mapping diagram for $f(x,y) = -x^2 -2y$
illustrating estimates for partial derivatives $f_x(0.2,-0.4)$ and $f_y(0.2,-0.4)$.

Thanks for participating¤.

Questions?

The End!

References:¤
http://flashman.neocities.org/Presentations/JMM2016/JMM.MD.LINKS.html

M. Flashman:

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 (with Tami Matsumoto)

GeoGebra Book: Making Sense of Calculus with Mapping Diagrams[2016 MAA] http://ggbtu.be/bOB534MoK

AMATYC Webinar Martin Flashman - Using Mapping Diagrams to Understand Functions (YouTube)