There's actually no due date. Just post.
I'll do my best to address finding bounds, but ultimately you just need to practice a lot. Also, I know there's a lot of posts, but it might help to read through questions that other students have come up with, and my replies to them.
I know it's not easy to do some of the above, since in general I find it hard to follow my own advice. But I feel there's no harm in giving you ideals which I strive to follow.
Reiterate Some Goals with the Commenting System:
(1) I want students to comment consistently and abundantly.
(2) Students wait until its too late to ask questions.
(3) It's good to ask questions, but sometimes we need to ask the right questions. Something I'm still learning to do myself.
(4) Realize that it's okay to ask the TA and professor questions.
(5) Realize that it helps to reflect on what you've learned.
(6) Helping each other.
I'm understanding how to recognize and calculate basic improper integrals, but I was wondering what are some applications of improper integrals?
ReplyDeleteI'm not sure about real world applications, but they can come up a bunch in math (depending on what field of math you study).
DeleteActually, I'd like to go back to Dr. Brown's example near the beginning of the semester:
$$\int_{-\infty}^\infty e^{-x^2} dx$$
This integral and it's integrand are relevant to the field of statistics. In particular, if you were to graph $e^{-x^2}$, you'd get a bell curve.
If you take statistics, then you'd find that you can compute probabilities by taking integrals. Sometimes you're interested in the area under the right half of the bell curve. That's $\int _0 ^\infty e^{-x^2}dx$.
In introductory statistics, this is pushed out of sight, because the integral is computed for you and put into a bunch of tables that confuse a lot of students. Also, three standard deviations from the mean (the tip of the curve) make up for 99.7% of the area under the curve.
Thus while numerically you'd work with calculators or computers, theoretically you know that the area under the curve is finite. I think that's often practical. To use theory to understand the behavior of a system, but when doing calculations, you can approximate the system.
As an example of theory versus computation. Consider when looking at slope fields of a differential equation, you can see singularities or places where a slight change in value may result in significantly different solutions. Knowing this, you'd do a better job approximating the solution further away from the singularities.
I haven't taken a course, but from what I know, this is all relevant in numerical analysis.
Thank you Tim! I appreciate your thorough responses!
Delete-Erica
Could you do a problem similar to 7.8.56, where you have to use trig substitutions in order to see if the function converges or diverges? I'm confused on the bounds.
ReplyDeleteThanks.
Could you discuss problem 7.8.52 because I'm having problems discussing the Comparison Theorem. Thank you
ReplyDeleteNahyr
I never really understood squeezing first semester so I still don't get it now in terms of using it as a method for sequences. Can we do an example of sequences that involves this method?
ReplyDeleteSqueezing is typically applied to sequences which fluctuate. Since their behavior is more erratic, we control them from above by a sequence that decreases and from below by a sequence that increases.
DeleteAt first, do you understand the example in the book?
Also, I did not understand how you can tell if a recursive sequence converges or not. Can you explain this?
ReplyDeleteOne way is to figure out a formula for the sequence.
DeleteAnother way is to figure out if its monotonic (increasing or decreasing) and bounded. This method will typically require induction. Do you understand the method of induction?
Another way would be to use the squeeze it between two sequences which you know converge.
Ultimately, a recursive sequence is not much different than a regular sequence.
There is another method I know, but I didn't see it in the book, and it might be difficult to use, so I will refrain from explaining it for now.
Are we going to have to go back to the epsilon delta definition of limits when being tested on this chapter? Also when talking about convergence, a lot of it seems so obvious that we should just be able to state what it converges to. What exactly constitutes a good explanation of how we know what we know?
ReplyDeleteMy guess is no, you won't have to use the $\epsilon$-$\delta$ definition of limits, but you can ask for an official answer from Dr. Brown.
DeleteIndeed some limits are clear, such as $\lim_{x\to \infty} \frac{1}{x^2}=0$. But more complicated expressions might clearly require l'Hospital's rule or multiplying by a conjugate.
Other than those two extremes, "good" is subjective, so try to put yourself in the grader's position and what the grader will expect.
A professor once explained that the person giving the examination knows the solutions. So it's not important to him/her that you know the solution, but to show you know how to get there. And since he/she can't read your mind, then it's up to you to present your understanding on paper to the extent that he/she grasps your understanding.
Hypothetical example: You're a genius and can do crazy computations all in your head. Unfortunately, that isn't going to help you on a written exam.
Can we go over in class how we should show a sequence converges on an exam to get full credit? I think I'm explaining with too many words and not using enough mathematical notation. Thank you!
ReplyDelete-Erica Zehnder
Yesterday in class Dr. Brown showed that we had to make valid assumptions in order to test the sequences for convergence. Could we practice a few in order to get a hold of how to choose the right assumptions.
ReplyDeleteValid assumptions? For example?
DeleteThis question actually pertains to last assignment that was due but for questions like the type of #38 where the integration is extremely difficult, is the only way to prove convergence by the method of theoretical analysis and use of theorems?
ReplyDeleteThis is really late, but I'm still a bit confused about how to do problems 71 and 78 in section 7.8. Could you explain how to do these? Thanks!
ReplyDelete