Description

This subject is a continuation of the study of Calculus. It includes further exploration of derivatives of exponential, logarithmic, inverse, and hyperbolic functions. It extends the abilities to integrate more various forms of expressions and investigates techniques of Integration. It also explores indeterminate forms and improper integrals leading into Infinite Sequences and Sires. It finishes with Analytic Geometry in the Plane.

Sections of this topic include:

1. Inverse Functions

   We begin this course investigating the notion of inverse functions. During our investigation, we discover how to find, graph, and differentiate inverse functions. Groups within this section include: determining graphically if a function is one-to-one; finding the inverse function; determining algebraically if a function is one-to-one.

2. The Natural Logarithmic Function

   Here, we expand our knowledge of differentiation and integration by including the idea of how to simplify, differentiate and integrate the Natural Logarithmic Function. Groups within this section include: finding domains of the natural logarithmic function; determining if two functions are the same using logarithm properties; finding derivatives of the natural logarithmic function; using implicit differentiation; and applications involving lines.

3. The Exponential Function

   In this section, we explore one of the most useful tools in calculus, the Exponential Function. Groups within this section include: Finding derivatives of the exponential function; using implicit differentiation to find derivatives; finding higher derivatives; and applications involving lines.

4. Integrals Involving Logarithmic and Exponential Functions

   Now that we are proficient in differentiation of the Natural Logarithmic Function and the Exponential Function, we explore integrals involving these two functions. The group within this section is called: Evaluating Integrals.

5. Exponential and Logarithmic Functions to Other Bases

   Since we can now differentiate exponential and logarithmic functions with a base of e, we turn our attention on finding derivatives and integrals of exponential and logarithmic functions to other bases. Groups within this section include: Finding derivatives of exponential and logarithmic functions to other bases; finding higher order derivatives; using implicit differentiation; evaluating integrals; logarithmic differentiation; applications of logarithmic differentiation.

6. The Natural Logarithmic Function – A Historical Approach

   This section investigates the historical aspect of where the natural logarithmic function and exponential functions arise from.

7. The Hyperbolic Functions

   We now turn our attention to defining trigonometric functions based on a hyperbola as well as based on the exponential function. These functions are called Hyperbolic Functions. Groups in this section include: finding remaining hyperbolic function values given one value; derivatives of hyperbolic functions; finding equations of tangent lines; evaluating integrals; evaluating hyperbolic functions; proving hyperbolic function derivatives; proving hyperbolic function identities; and proving an interesting fact.

8. Applications and First-Order Differential Equations; Separate First-Order Equations; Linear First-Order Equations

   We apply our notion of exponential functions and logarithmic functions in solving differential equations. The group within this section is: solving differential equations using integrating factors.