Section CIS Calculus II Differential Equations, Integration, and Series

Calculus II (Differential equations, Integration and Series)
*Work in Progress! (5/2018)

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. This is due in part to the limitations of visualizing the relationship between two distinct variables with a single point. These subtle concepts can sometimes be better 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 the previous chapter, Calculus I (Continuity and Differentiability), the emphasis was on using mapping diagrams to visualize many of the key concepts and results of continuity and differential calculus. In this chapter the emphasis is on calculus concepts and results related to differential equations, integration and series. Chapter 11  remains to discuss multi-variable functions and their calculus.
Much of the work here appears in other formats as part of The Sensible Calculus

Since the study of differential equations, integration and series calculus relies heavily on continuity and differentiable functions reviewing the visualization of Calculus I (Continuity and Differentiability) is most useful at this time.

   10    Calculus II Differential equations, Integration and Series
    10.1         Euler’s method
    10.2         Definite Integration
    10.3         The Fundamental Theorem of Calculus
    10.4        Taylor and MacLaurin Theory and Practice
    10.5        Sequences and Series Tests
    10.6        Power Series

Subsection CIS.EM Differential Equations with Initial Conditions- Euler's Method

Subsection CIS.DI Definite Integration

Subsection CIS.FTC The Fundamental Theorem of Calculus

Subsection CIS.TM  Taylor and MacLaurin Theory and Practice

Subsection CIS.SST Sequences and Series Tests

Subsection CIS.PS  Power Series

Exercises (Not Done Yet)