### Section CIS Calculus II Differential Equations, Integration, and Series

**Calculus II (Differential
equations, Integration and Series)**

*Work in Progress! (5/2018)

**Calculus II (Differential equations, Integration and Series)**

*Work in Progress! (5/2018)

*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 Eulerâ€™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