### Section CCD Other Algebraic Functions

**Calculus I (Continuity and Differentiability)**

*Work in Progress! (3/20/2018)*Work in Progress! (3/20/2018)

**Introduction: **The concept of a function developed
after the relation between variables used for coordinates in geometry
was connected in the work on the calculus by G. Leibniz . As the
function concept evolved, its connection to geometry lessened until in
the 20th century the theory of functions and visualization with mapping
diagrams was considered a part of the theory of sets while the calculus
continued to be visualized primarily with graphs. In the mid and later
half of the 20th century some calculus texts (see for example M. Spivak
and S. Stein) used mapping diagrams to visualize important differential
calculus concepts (limits) and tools (the chain rule). These treated
some material that was not easy or convenient to visualize with graphs.

The methods for studying functions with calculus can become quite
abstract sometimes when visualization is restricted to the cartesian
graph, due in part to the limitations of visualizing the relationship
between two distinct variables with a single point. These subtle
concepts can sometimes be best understood by visually separating the information in a mapping diagram.

Mapping diagrams provide tools for visualizing functions beyond the
constraints of cartesian
geometry to investigate the calculus concepts of
continuity, differentiability, and integrability as well as some of the
computations related to the calculus.
In this chapter the emphasis will be on using mapping diagrams to
visualize many of the key concepts of continuity and differential
calculus leaving other calculus concepts to be visualized in Chapter 10
Calculus II (Differential Equations and Integration) and Chapter 11
Multi-variable Functions and Calculus.

Much of the work here appears in other formats as part of The Sensible Calculus

Since the study of calculus relies heavily on modelling elementary functions with the

Linear Functions(LF), reviewing the visualization of linear functions is most useful at this time.

9.1 Limits and Continuity

9.1.1 Definitions

9.1.15 Limit Theory

9.1.2 The Intermediate Value Theorem

9.1.3 The Extreme Value Theorem

9.2 The derivative

9.2.1 Definitions

9.2.2 Core Functions

9.2.3 Calculating Rules

9.2.3.1 Algebra Rules

9.2.3.5 The Chain rule

9.2.3.7 Implicit Differentiation

9.3. Numerical Applications

9.3.1 The Differential and Linear Estimation

9.3,2 Newtonâ€™s Method

9.4 The Mean Value Theorem

9.4.1 Finding Extremes with Calculus

9.4.2 The Second Derivative: Acceleration and Concavity

##### Subsection CCD.DLC Definitions of Limits and Continuity

##### Subsection CCD.LCT *Limits and Continuity Theory