This unit’s material is a collection of applications of integrals. The most important section, by far, is 6.7. In that section, we apply integrals to physical problems and start to get a sense of how useful this operation is in interpreting real-world processes.
Calculus for Engineers II
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Unit 2 – Sections 6.10 to 7.3
In the rest of this unit’s material we begin the study of methods of integration. There is a lot of material in this part of the course and it is important to be able to figure out when to use different methods, so make sure that you are careful in your studies of Chapter 7.
Unit 3 – Sections 7.4 to 7.6
This unit, we look at two major methods of integration. Like all of the methods other than IBP, these are actually just subclasses of substitutions, but they are far from obvious and deserve separate treatment. There are a lot of things happening here and plenty of example videos in which to see these details in action. Go through the material carefully and remember that you want to be able to understand when to use each method.
Unit 4 – Sections 7.7 – 8.1
This unit’s material includes three very separate topics. We begin with numerical integration – another of the most important numerical methods we glance at in first year (together with Newton’s method, linear approximation, and – soon – Taylor polynomials). The next section covers an interesting detail, looking at integrals that misbehave, either by having an unbounded integrand or an unbounded interval. Either way, the solution is to use limits. Finally, we begin to study differential equations, which are the basis for a tremendous amount of engineering.
Unit 5 – Sections 8.2 to 8.4
We begin in earnest our study of differential equations. These tremendously useful tools for science and engineering end up being rather pesky in terms of the ways we go about solving them. In this unit, we will be introduced to two types of DEs, as well as a numerical method for approximating solutions.
Unit 6 – Section 8.5, extra material on DEs, and Section 9.1
In this unit, we finish our study of differential equations. This includes a small section on second-order equations. The Briggs textbook includes an electronic-only chapter on these and we include it here. We will only be looking at the first two sections of that chapter (and, even then, only the aspects covered in the lecture videos).
Unit 7 – Sequences and Series
Through the three sections in this unit and the two sections in the next, we look at a few definitions and then study when series converge and diverge. We discover a number of tests that will help us figure out this information, including some very unusual-looking ones. The goal of this chapter is to allow us to use the information here when we try to express functions using certain infinite polynomials (we will call them power series). Knowing when a series converges or diverges will let us find out when our power series exist.
Unit 8 – Sections 9.5 and 9.6 – Convergence Tests Galore
In the sections covered in this unit, we do two things. First, we examine more convergence tests on positive (or non-negative) series. The tests range from ugly to beautiful, but are all useful. In the second section, we open up to the idea of allowing negative terms in our series and find that this leads to some very odd complications.
Unit 9 – Taylor Polynomials and Power Series – Sections 10.1 and 10.2
We begin our study of Taylor Series by looking at each half of the phrase individually – first we study (finite) Taylor polynomials. Then, before we extend them forever, we look at how series with a variable behave.
Unit 11 – Chapter 11
In this chapter, we start examining the effect of having an independent variable determine two (and, in the next chapter, three) dependent coordinates. The most common applications are those in which we use the parameter of time to describe physical motion through the plane and through space.
Unit 12 – Sections 12.5 to 12.7
The beginning of the study of vector-valued functions mirrors precisely the study of parametric equations. Most of the material in this week is nearly identical to what we had in the previous chapter.
Unit 13 – The End of the Journey
The material here takes us to the end of the course. I have included an extra section on multivariable calculus – simply an idea of what a multivariable function is and how to take partial derivatives.
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