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.
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