#2. In the comment section of the relevant week, submit a question or something you learned every week. These should be mildly relevant to the class. If you posted something along these lines on the Facebook page, you may simply copy and paste what you posted. Other comments such as interesting links, even those not related to math are welcome, but do not count toward the required participation. No duplicates, so the earlier you submit a question or comment, the less you'll have to read of other posts.
Read Appendix A.
Memorize/Learn Reference Page 1: Algebra.
[20120202:] If you don't want to sign with your name, that's fine. Next week in class, I will figure out which user names belong to which people.
Would it be possible that we could get a copy of last semesters final and go over it in section?
ReplyDelete-Ryan Schneider
I'll talk about this with Dr. Brown. I think it's a good idea. We can poll which problems students would like to see. And we can discuss low demand problems individually.
DeleteHi guys,
DeleteI would have no problem with that. Let's follow Tim's suggestion here and look for the problems you guys would like to see. Perhaps also we can make it a group discussion, since different people may have different insight as to how to attack the problems. And maybe either this forum or FB or on some other public place we can post the problems and seek input.
I also think this would be really helpful because it would give us a sense of what test questions might look like. It would also give us some problems other than homework questions to work on.
DeleteJulia Deutsch
This is just kind of a random thought...
ReplyDeleteSay we have to integrate an expression that is a product of three functions (For example x*e^x*cosx). Would it be correct to assign f(x)=x and g'(x)=x*cosx and then do integration by parts again for the integration of g'(x)? It would get messy, but I feel like it would work. I'll try to work it out on my own, but I figured it would be something to consider.
Chris Skoff
Awesome! See this link for a partial response: Useful Mnemonic: DETAIL.
DeleteDiscussion among students is welcome. Participation to the discussion will count towards your one week minimum.
One option to submit math is to write it on paper, then scan it or take a picture and email it to me. Another is to write in TeX. I learned TeX passively with LyX. As an example, I'd write g^{\prime}(x)=x \cos(x) enclosed with dollar signs instead of g'(x)=x*cos(x). The result would be $g^{\prime}(x)=x \cos(x)$.
Problem 7.1.32 in the book asks for the definite integral from 1 to 2 of ((lnx)^2) / (x^3).
ReplyDeleteThere is no way to simplify the (lnx)^2 part of this equation, so I'm thinking that I should use substitution. I'm running into problems when I try to use integration by parts, since I'm not sure whether to put u in terms of x or t. When I try to use t, it becomes more complicated because of exponents. Does anyone have a suggestion?
-Ashleigh Thomas
In this situation, it might be easier to clarify your question in person.
DeleteI can sidestep your question and suggest you read Useful Mnemonic: DETAIL and attack the problem without using substitution.
For fun, let's also do the problem with u-substitution. $$\int \frac{(\ln (x))^2 }{x^3} dx$$
Let $u=\ln(x)$. Then $du=\frac{1}{x} dx$ and $x=e^u$.
As an intermediate step, we have $$\int \frac{(\ln(x)^2}{x^2}\cdot \frac{1}{x}dx$$
Making the proper substitutions we can get $$\int u^2 e^{-2u} du$$. From here we can use integration by parts.
On one of the homework problems 7.2 problem 10 there came a point where cos(x)^3 had to be integrated. I used the reduction formula to solve this. Afterwards I also checked my answer with an internet source and it had simplified what I had [1/3sin(2x))+(1/6sin(2x)cos(2x)^2)] into (1/24)[9sin(2x)+sin(6x)]. I know that I dont necessarily need to simplify to this, but I am very curious on how they transformed the first part into the second. I tried using some trig identities, but I do not know how to make anything into a sin(6x).
ReplyDelete-Sophia Ottleben
Here are two (technically one) identities you probably know:
Delete$$\sin (v+w)=\sin(v)\cos(w)+\cos(v)\sin(w)$$
$$\sin (v-w)=\sin(v)\cos(w)-\cos(v)\sin(w)$$
Let's add them together!
$$\sin(v+w)+\sin(v-w)=2\sin(v)\cos(w)$$
I will let you try and figure out how to use this identity.
I encourage you, and all others, to come up with similar identities using identities for $\cos(s+t)$ and $\cos(s-t)$. Feel free to post your results.
Are there any good external sources that have solved examples of questions about integration by parts? I feel like I understand the general concept, but I need to understand the applications of the concepts to specific types of questions.
ReplyDeleteThe book provides some useful examples. But if you're looking for more, I Google'd "examples integration by parts" and glanced at the first two links. There are examples there, which you can try before looking at the solution.
Delete(1) http://www.math.ucdavis.edu/~kouba/CalcTwoDIRECTORY/intbypartsdirectory/IntByParts.html
(2) http://www.sosmath.com/calculus/integration/byparts/byparts.html
Is there any particular website that has quizzes and other types of problems for the material that was taught or will be taught in the semester. I think this is helpful to review the main concepts of last semester and will help when the midterm comes around.
ReplyDeleteI personally don't know of any particular websites. However, you have a great resource under your nose, your course book. There are plenty of exercises to do and if you run into trouble, just post a comment asking for help or head to the math help room for an immediate answer. The end of every chapter has a review section. Again, if you run into any problems, you have plenty of help available to you. At the minimum there's Dr. Brown, myself, the math help room, and your peers.
DeleteI have posted some problems. Depending on requests, I might post more.
DeleteTest Your Knowledge 20120209
If we want to come see you for help on homework or to get help on particular topics, what days and times are you in 201?
ReplyDeleteThis information has been updated on the side panel.
DeleteMy Office: Krieger 201
My Office Hours: Thursday 1:00-2:00 pm
My Help Room Hours: Thursday 11:00-1:00 pm
Email: tran at math dot jhu dot edu
The math help room is in Krieger 209. It's open M-Th 9 am to 9 pm and F 9 am to 6pm.
This website is always here.
On the first homework problem from the set due at the end of this week (7.1 #8), it first seemed like I was going in circles when I applied Integration by Parts. I picked the wrong thing for my initial g'(x), I chose t^2. I then realized that then the integral became more comlicated because then g(x) became t^3/3. By picking sin(beta*t), however, I realized that helped make the integral simpler - but it required the application of Integration by Parts for a second time. That took me a while to figure out.
ReplyDelete-Jackson Berger
I'm glad you shared this. As a side note, DETAIL would have definitely helped in this situation.
DeleteFor section 7.3 where trigonometric substitutions are used to sold non-trigonometric functions, is there a reference place where to find the trig identities. I know the sin^2(x)+cos^2(x) = 1 identity but have trouble with the others.
ReplyDeleteI believe Appendix D (starting page A28) in the textbook has most, if not all of the trig identities.
DeleteThis is where we have our one comment a week right? Anyway, I don't really have a question, but I thought it was cool in class when Prof. Brown went through the derivation of integration by parts.
ReplyDeleteI always like knowing this stuff in case I forget what the actual rule is on tests.
-Nathan Patterson
I am still having problems with the reduction formula. Is it necessary to use it on 7.2.10? Also regarding that same matter, can we go over 7.1.48a?
ReplyDeleteThanks,
Marcelo
I believe I addressed your question in class. But for 7.1.48a, I hope you can figure out the answer by reading Example 6 on page 467. For 7.2.10, see Example 4 on page 472, that will answer your question.
DeleteI lost my notebook with my calc notes in it, so I'm having issues with the homework. For 7.1, question 26, when we're using a definite integral, do we use the anti-product rule as normal, and then just plug in the two values for y?
ReplyDeleteThanks,
Jack
I think you can use the formula on page 467 (red box numbered 6) which evaluates definite integrals by parts. It's also similar to Example 5 in the book.
Delete-Tiffany
I was having trouble understanding when integrating by parts the role that the constant c plays and when to include the constant in our calculations.
ReplyDeletePerhaps reading section 5.4 on page 397 for a refresher on the topic of indefinite integrals will clarify the role that C plays in general.
DeleteThe idea is we include C when we need a family of antiderivatives, but we can omit it when we only need a particular antiderivative.
Thus, when applying the formula for integration by parts, we only need to come up with one antiderivative.
Example:
Compute $\int x e^x dx$
Let $u=x$ and $dv=e^x dx$
Then $du=dx$ and $v=e^x$
It would be excessive to choose $v=e^x+C$.
If this doesn't help your understanding, feel free to let me know why.
I learned integration by parts in high school but I always relied on instinct when picking which function to use for "u" and which to use form"dv." DETAIL is proving to make that process less of a guessing game and more of an actual strategy! Thank you for introducing it!
ReplyDelete- Erica Zehnder
How would you solve this problem:
ReplyDeleteintegral of (dx)/((x^2+4x+8)^3) with trig functions in denominator
General Idea:
DeleteComplete the square.
Make a substitution that would allow you to use a trig identity.
Major Hints:
$(x+2)^2+4$
Let $x+2=2 \tan(\theta)$
Remark:
Don't forget about taking the cube of the expression.
I was wondering what could be done if we were to solve an integral with the integrand sinmxcosnxdx if m and n equal the same number?
ReplyDeleteUse a useful trig identity. =)
DeleteIf $m=n$ then $\int \sin(mx)\cos(mx)dx=\frac{1}{2}\int \sin(2mx) dx$
Delete$=-\frac{1}{4m}\cos(2mx)$
Here I used a special case of the product identities listed on page 476.
The special cases are:
$$ \sin(\theta) \cos (\theta)=\frac{1}{2}\sin(2\theta)$$
$$ \sin^2(\theta)=\frac{1-\cos(2\theta)}{2}$$
$$ \cos^2(\theta)=\frac{1+\cos(2\theta)}{2}$$
The first special case is more often written
$$ \sin(2\theta) = 2\sin(\theta) \cos (\theta)$$