MAA Minicourse #7 ¤

January 7 and 9, 2016

**Making Sense of Calculus with Mapping Diagrams**

Part I

1.Mapping Diagrams. ¤

What is a mapping diagram?

Introduction and simple examples from the past: Napiers Logarithm

See:
http://www.maa.org/press/periodicals/convergence/logarithms-the-early-history-of-a-familiar-function-john-napier-introduces-logarithms

*Logarithms: The Early History of a Familiar Function - John Napier Introduces Logarithms* by Kathleen M. Clark (The Florida State University) and Clemency Montelle (University of Canterbury)
**Figure 3.** The relation between the two lines and the logs and sines

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**

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Visualizing linear functions using mapping diagrams and graphs.**
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Notice how points on the graph pair with arrows and points on the mapping diagram.**
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**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.

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****Mapping diagrams and graphs visualize how the definition of a limit works for real functions.**

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

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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 be visualized using a tangent line on a graph or a focus point and derivative "vector" on a mapping diagram.**

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The derivative can also be understood as the
magnification factor of the best linear approximating function.****¤****
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The derivative can be visualized using focus points and derivative "vectors" on a mapping diagram.**
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**3.3 Mapping Diagrams for Composite (Linear) Functions ¤**

This is the fundamental concept for the chain rule.

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Visualizing the composition of linear functions using mapping diagrams. **

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**3.4. Intermediate Value theorem and Newton's method. **

3.4.1. Continuity can be understood by connecting it to the Intermediate
Value Theorem (IVT) and solving equations of the form $f(x) = 0$.

IVT: If $f$ is a continuous function on the interval $[a,b]$ and $f(a)
\cdot f(b) \gt 0$ then there is a number $c \in (a,b)$ where $f(c) = 0$.

Mapping diagrams provide an alternative visualization for the IVT. They
can also be used to visualize a proof of the result using the "bisection
method."

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Insert GEOGEBRA for Bisection and IVT.

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