Solution to # 8 on Practice final

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

 

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Practice final

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!!

 

Review of week 13 and preview of last week

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!

Review of week 11/12, preview of week 13

Hi all,

Hope you all had a fun and full Thanksgiving! For the past one and half weeks, we have started looking at series, whether they diverge or converge, how to compute their sums and such. For the most part, we are interested in power series and when they can be treated as functions. Next week we will continue with power series in reverse, in the sense that we look at normal functions, and see how to represent them with power series.

Series’ are sums of infinitely many items. Like how we always deal with infinities, we define series to be the limit of the partial sum. If the limit exists, we say the series converges; otherwise diverges. In this regard, we can treat the partial sums as a sequence, and all the results we had for sequences apply here. This method (looking at partial sums) normally require us to know a compact form of the partial sum. An example is the geometric series, with the compact form 1+r+r^2+…+r^n=(1-r^n)/(1-r), for all |r|>=1, the series would diverge and only for |r|<1 will the series converge.

In general, it is hard to tell if the series converges without a compact form of the partial sum, but we can tell if they diverge by looking at the limit of the term: the series will diverge for sure if the terms don’t converge to zero. You can use this standard to weed out a lot of divergent series.

In power series, we are adding polynomials of the shape c_n(x-a)^n. Note here x is the variable which is independent of the index n. As long as the series converge, we can look at its sum as a function. The values of x at which the series converge is naturally the domain of the function, therefore we focus on where the series converge.

If we are dealing with a geometric power series, the series converges if |r(x)|<1 (as in the power series). If we are dealing with a more general series, we can use the root test or the ratio test. Both tests are inconclusive at the boundary, so we omit the boundary cases(see sections 11.3 and 11.5).

For next week, we will write general functions in power series. The idea behind this is super cool: functions might be of different shapes and forms, but you can “always” write them in terms of the sum of powers of x. Therefore, if you are studying a “ugly” function, your life will be easier if you instead study the power expansion.

Alright, that is it for now. There are so many cool facts about power expansions and I can’t wait to share them with you guys! See you soon!

Homework (Due Dec 3/4)

Reading: 11.8, 11.9

Problems:

11.8: 6,7,9,10,12,15,18,20,29,30(a,b)

11.9: 4,5,6,8,11,12,13,14,15,18

Remark: For interval of convergence, we only look at the open interval without the endpoints. If you are done with all the homework, you can read section 11.3 and 11.5 to check the endpoints case by case. Turn them in to me and I will give you some extra credit.

Review of week ten and preview of week eleven (Homework due Thursday/Friday, Nov 19/20)

Hi all,

This week we had our test 2 and started looking into sequences. Next week, we will continue with properties of sequences and start with series.

Test 2 focused on general techniques of integration, application of integrations and improper integrals. From the questions I got from you guys, the hardest step is to start a problem and choose which method to use. I suggest you do the problems in section 7.5, as those problems use all the techniques we ever discussed this semester and you need to make a judgement.

For sequences, we care about convergence and divergence as n approaches to infinity. There are some general ideas to help you determine the convergence of a sequence:

  1. Look at corresponding continuous functions. As long as the function has a horizontal asymptote as x approaches to infinity, we can conclude the convergence of sequence. Note the divergence of function *does not* conclude the divergence of the sequence! The reason of looking at continuous functions is that we can use properties of functions that discrete sequences don’t have, such as L’Hospital rule.
  2. Use the properties of limit laws. If we have two convergent sequences, we can conclude the limits of their sums, differences, products, quotients, etc. We can use these properties to chop problems into small pieces and deal with each piece individually.
  3. Compare with known sequences (Squeeze Theorem). Same for all comparison rules, try some values and get a general direction before you charge ahead to find bounding sequences.
  4. In general, if you don’t know how to deal with infinities, try to convert them into zeros.

Next week, we will start looking at “sums” of the sequences, namely, series. You will use series in all sorts of applications, and they have interesting properties themselves as well!

Homework:

Reading: 11.1, 11.2

Problems:

11.1: 5,8,9,11,13,14,17,19,20,21,26,41,54,60,62,64,70

11.2: 9,10,15,16,17,21,22,25,34

Review of week nine and preview of week ten

Hi all,

Last week, we wrapped up integration with improper integrals and reviewed for the test this week on Tuesday and Wednesday. After the test, we will start with infinite sequences.

Improper integrals are all definite integrals that have singularities: one of the integration limits being infinity, the function is not defined at some point in the interval of integration or both. Improper integrals are defined as the limit of a normal definite integral, where the limit is taken for the integration interval to approach to the original interval with singularities. By definition, to evaluate an improper integral, you should follow two steps: evaluate a normal definite integral with parameters, then take the limit.

Sometimes all we care about is whether the improper integral converges and not the specific value of it, we can use the comparison rule. Given two non-negative functions f(x) and g(x), if we know f(x)>=g(x), then the improper integral of f converges implies the improper integral of g converges; the improper integral of g diverges implies that the improper integral of f diverges. A lot of the time, we use x^{-\alpha} as the comparison function,  and \int_1^{\infity} x^{-\alpha}dx converges if and only if \alpha>=1. Similarly,  \int_0^{1} x^{-\alpha}dx converges if and only if \alpha<1.

We will talk about infinite sequence on Thursday/Friday, and please review limits from Calc 1. We have already seen sequences when we studied Riemann sums, but we will study more general sequences in Chapter 11 and their properties.

Alright, that’s it for now. Sorry the sum up came later this week!