Unit 9 - Sections 4.5 to 4.8

Module Overview

A lot of videos in this unit, but three of the four sections have very little in terms of testable material. Only L'Hopital's Rule requires a lot of practice. We are, however, setting the groundwork for some important mathematics. In these sections, we are introducing the idea of numerical methods, setting the stage for integration, and taking the first step toward Taylor series.

Learning Outcomes

Section 4.5: This section introduces us to numerical methods - ways of using mathematics to find approximations for values that we cannot find precisely. Linear approximations, or differentials, use the fact that the tangent line to a curve lies very close to the curve itself for a little while. The method itself has some very clear flaws, but we will return to it in APS 163 and build from linear approximations to much better ones, called Taylor polynomials. From these, we will go to Taylor series, which will allow us to write functions in terms of infinite polynomials.
Section 4.6: The Mean Value Theorem has few applications that can appear on an assignment or a test, but it does lay the foundation for the idea of integrals. As we prepare to discuss antiderivatives next week, this section is necessary for understanding what happens there.
Section 4.7: The greatest of the student-friendly shortcuts in mathematics, L'Hopital's Rule allows us to easily take derivatives that, earlier in the term, were very difficult. It also allows us to introduce a few more classes of undetermined forms to evaluate.
Section 4.8: One of the earliest numerical methods (and still one of the best), Newton's method gives a formula that can be repeated as many times as we want (or need) to approximate the roots of a function.

Type: Video Links
Title	Author	Description
Linear Approximation - Introduction \| Linear Approximation - Introduction	Shai Cohen
Linear Approximation - How It Works \| Linear Approximation - How It Works	Shai Cohen
Linear Approximation - The Method \| Linear Approximation - The Method	Shai Cohen
Differentials Example - Polynomials \| Differentials Example - Polynomials	Shai Cohen
Differentials Example - Tangent \| Differentials Example - Tangent	Shai Cohen
The Mean Value Theorem - Mean Value Theorem \| The Mean Value Theorem - Mean Value Theorem	Shai Cohen
Mean Value Theorem - Implications I \| Mean Value Theorem - Implications I	Shai Cohen
Mean Value Theorem - Implications II \| Mean Value Theorem - Implications II	Shai Cohen
Mean Value Theorem - Examples \| Mean Value Theorem - Examples	Shai Cohen
L'Hopital's Rule - Introduction \| L'Hopital's Rule - Introduction	Shai Cohen
L'Hopital's Rule - Repeated Uses \| L'Hopital's Rule - Repeated Uses	Shai Cohen
L'Hopital's Rule - An Example and a Warning \| L'Hopital's Rule - An Example and a Warning	Shai Cohen
L'Hopital's Rule - A New Limit \| L'Hopital's Rule - A New Limit	Shai Cohen
L'Hopital's Rule Example - Exponents \| L'Hopital's Rule Example - Exponents	Shai Cohen
L'Hopital's Rule Example - Shotgun \| L'Hopital's Rule Example - Shotgun	Shai Cohen
Newton's Method - Introduction \| Newton's Method - Introduction	Shai Cohen
Newton's Method - Newton's Method in Action \| Newton's Method - Newton's Method in Action	Shai Cohen
Newton's Method - Pitfalls \| Newton's Method - Pitfalls	Shai Cohen
Example - The Cube Root of Two \| Example - The Cube Root of Two	Shai Cohen