### SectionCCDOther Algebraic Functions

Calculus I (Continuity and Differentiability)
*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         Newton’s Method
9.4        The Mean Value Theorem
9.4.1         Finding Extremes with Calculus
9.4.2        The Second Derivative: Acceleration and Concavity