The final exam is Tues, May 14, 2:00 – 3:50 pm in our classroom. There will be a calculator and a non-calculator part, so please bring a calculator. I have posted one more homework assignment with the solutions available (no credit, just for practice) involving optimization and double and triple integrals.
Good things to study include the tests, your solutions to problems we did from the notes, as well as the online homework problems.
Here's an overview of the objectives we studied from each chapter
In this chapter we generalize the idea of a functions $y=f(x)$ to a parametric curve $\vec r(t)=(x,y)$.
This chapter generalized our understanding of functions.
This chapter explored one particular plane transformation more in depth.
In this chapter, we generalized the Calculus 1 concept of limit and derivative to multiple variables. We use a matrix for the multivariable generalization of the derivative, even though the textbook does everything without mentioning the matrix underlying the calculations. Using a matrix (the total derivative) both simplifies things and is more powerful and general.
We skipped this chapter.
In this chapter, we generalize the idea of integrals. We also set up the framework for average value type of problems. When we integrate over vector fields, we get the concepts of work, flow, and flux. Finally, we generalize the fundamental theorem of calculus to work with line integrals.
In this chapter, we deal with functions like $f\colon \mathbb{R}^n\to\mathbb{R}$. We generalize the idea of $f_x$ and $f_y$ to the idea of a directional derivative. We discuss the importance of the gradient vector. We also generalize the second derivative test from calculus 1 and also use the gradient vector to get a way to optimize within some constraints.
Note that the book does not use eigenvalues and the matrix derivative in the second derivative test. I will expect you to be able to do the test using the matrix derivative and eigenvalues, like we discussed in class.
In chapters 10 and 12, we generalize integration to double and triple integrals. We also generalize the calculus 1 method of substitution to other transformations like polar, cylindrical, and spherical. For these problems, the focus is setting up an integration problem. I will expect you to be able to evaluate simple double or triple integrals involving trig functions, polynomials, and $u$-substitution. Note that we skipped chapter 11.