Hi all,
Remind you: please check all your grades on Sakai and let me know if there is any error. I will be on campus this Sunday, 12.30-3.30 and come to ask any questions you have!
Here you can find the solution to problem 8 on the practice final: 152final15_prac
Hi all,
Here is the practice final exam: 115final15_prac
This is basically a study guide, so the way to use it is to expand it into several types of problems. For example, when you do the linear equation problems, make sure to do some problems in section 2.3; when you solve quadratic equations, check your homework from section 1.3.
When doing these problems, make your sheet on the side, write down the methods you use, write down what was hard for you. We will go over these problems in class tomorrow, and bring any questions you have for me!
Important date: Our final exam is on Monday, Dec 14!!
Here is the practice exam for final: 152final15_prac
Now before you dive in on the problems, please take a few minutes to read the following study guide:
This final exam will *not* be just the same exam with numbers changed. The practice exam is intended for you to get a preview of the format, the content of the real exam. For example, you are expected to know the many ways of computing an integral, both indefinite and definite; you are expected to know how to apply integrations, such as use the definite integral to compute the volume and indefinite integral to solve a differential equation, etc.
That said, how do you use this practice test? You should treat each problem as an example of problems of that kind, review how to solve that kind of problems, find some example of that kind yourself, do several of those problems. Don’t only focus on doing that one problem 100% and ignore all other exercises you did of the same kind.
When you use this practice test, you should be also making your sheet on the side, doing problems of the same kind on a stack of paper and writing down all your questions for me (or whoever you normally get help from).
If you wish to check your answers, show me your work and I will be happy to check them for you. The point of doing this practice test is to review the content of the course, not obsessing with an extra factor of 1/2 in your final answer of problem 3.
Alright, let me know if you have any questions, and and and, do take advantage of the office hours!!
Hi all,
We have been looking at the power series for the whole week last week! The idea is that if the function is “nice” enough, we can represent it as the infinite sum of powers of x. Some functions are hard to study (for example, there is no way to integrate e^{-x^2} with FTC), but we know how to integrate, differentiate polynomials very easily. A lot of properties can be obtained just by looking at the powers of x, for example, we have seen that from the powers, we can tell sine is an odd function, but cosine is an even function.
Important fact: power series expansions in general only work locally, so it is very important that whenever we write down a power series, we immediately check on which interval the series converges. Such an interval is always centered at the center of the series, in the form of (a-R,a+R), here we do not study the convergence at the boundary. Make sure to test for the values of x where the series converge!!
We have looked at two kinds of power series: one for functions based on \sum^{\infty}_{n=0}x^n=1/(1-x) for |x|<1, and the more general Taylor series. For both kinds, remember they are just power series, so all the facts we learned about power series apply to these series, including the differentiation and integration rules. We mostly focus on how to write the functions in power series expansions and when they converge, but we will also look at some applications of these power series.
Next week, we will wrap up the Taylor series and review for the rest of the week.
Quiz corrections are due Friday, Dec 11, before 10am. Let me know if you have any questions!
Hi all,
One week left! Next week, we will finish up with trig functions and review for the final exam.
Last week, we talked about trig functions, in terms of ratios of length in a right triangle. Make sure to have the definitions of those trig functions and their relations written down on your sheet!
On Thursday, we looked at some applications of trig functions in right triangles. We can classify the information given to us as two categories: either angle relations or length relations. If you know one acute angle, you can find the other one by a+b=90 degrees; if you know two edges, you can solve for the last one by the Pythagorean theorem. The things that connect these two kinds of information are the trig functions, and you would often use trig functions to solve angle from edge and vice verse.
There are two types of information given in general: one angle and one edge, or two edges. Notice you will always need at least one edge in order to solve a triangle (similar triangles are of different size, but have the same angles!)
Next week, we will finish up the introduction of trig functions in the coordinate plane. It will be similar to trig functions in right triangles, but now it is easier to describe the vertices, etc. On Thursday, we will review for the whole semester.
Test 3 correction is due Tuesday and no later than that!
Math taught and learned 