Practice Exam 2.
Made on 20120420. Partially distributed on 20120420. Made available online 20120423.
No calculators!
#1) Evaluate the integral.
\[
\int\frac{s^{3}}{\sqrt{s^{2}+4}}ds
\]
#2) Find the exact length of the polar curve.
\[
r=3\cos\theta,\quad0\leq\theta\leq\frac{\pi}{2}
\]
#3) Determine whether the integral is convergent or divergent. If
it is convergent, evaluate it.
\[
\int_{0}^{\infty}\frac{x\arctan x}{(1+x^{2})^{2}}dx
\]
#4) Determine whether the following are convergent or divergent.
State the type of test you use. You may use each type of test only
once.
a)
\[
\sum_{n=5}^{\infty}(-1)^{n}\frac{e^{-n}}{n!}
\]
b)
\[
\sum_{n=4}^{\infty}\frac{n!}{n^{n}}
\]
c)
\[
\sum_{n=3}^{\infty}\frac{\sin\left(n+1\right)}{n^{2}}
\]
d)
\[
\sum_{n=2}^{\infty}\frac{2}{3}\frac{7^{n}}{5^{n-1}}
\]
e)
\[
\sum_{n=1}^{\infty}\frac{2-2n}{3n^{4}}
\]
#5) There should be three forms/strategies from section 11.7 which
you didn't use. State/describe them.
Reply here to discuss problem 1.
ReplyDeleteReply here to discuss problem 2.
ReplyDeleteI don't know if i did it right but I did L= integral from 0 to pi/2 sqrt(r^2+ (dr/dtheta)^2) dtheta -which is the formula stated in the book. I then plugged in the r= 3costheta and dr/dtheta which i found out to be -3sintheta. I was able to remove the constant 9 out of the square root nicely, and by trig identity, I figured that cos^2 theta + sin^2 theta is one, hence my length became 3. I think my arithmetics were right but I'm a bit confused as to why I didn't have to use my limits at all. If my answer is wrong it would be nice to know where I made a mistake.
DeleteI also want to ask Tim if we'll have to know how to draw the polar equations/graph them because I find some of them simply too challenging to draw.
Thanks! -Sun Joo LEE
To answer your first "question," think about the following and how it relates to your answer:
Delete$\int_a ^b 1 dx = \left. x \vphantom{\int}\right|_a^b=b-a$
For your second question, you *might* not be asked to, however, I find being able to draw them useful in determining bounds. I don't write the test, so I don't know if you will/won't have to. My suggestion is to be able to show the main properties, such as x-intercept, y-intercept, number of loops, r versus $\theta$ graph. Math professors aren't concerned, in general, if you're a great artist; they want you to show them you learned math and/or developed thinking skills.
Reply here to discuss problem 3.
ReplyDeleteAt first, do you think the integral converges or diverges?
DeleteI'm having trouble seeing how to integrate this! The extra "x" in the numerator is throwing me off!
Delete-Erica
If we think it converges, then we should evaluate it. And if we want to evaluate it, we might consider integration by parts or a u-substitution. Maybe both. What have you tried?
DeleteOverall, this problem is a great review of many concepts depending on how you approach it. It might not be relevant for the midterm, but being able to do this problem is definitely great review for the final.
Reply here to discuss problem 4a.
ReplyDeleteReply here to discuss problem 4b.
ReplyDeleteThis problem has a factorial and so I decide to try the ratio test. Problem 4a also had a factorial, but there was another important aspect to 4a that is more important than having a factorial.
DeleteReply here to discuss problem 4c.
ReplyDeleteCan the divergence test be applied here?
DeleteIf we take the limit as $n$ goes to $\infty$, we get $0$. Thus, the divergence test is inconclusive on the convergence or divergence of this series.
DeleteCan the squeeze theorem be applied in this case?
DeleteWhich test can be used?
Delete@ericazehnder
DeleteShort Answer:
No
Long Answer:
In general, the squeeze theorem is hard to apply to series. Recall from class, I emphasize that a series is a limit of the partial sums. So if we were to apply the squeeze theorem, we'd have to apply it to the partial sums, not to the individual terms.
Understanding the difference between the limit of the terms and the limit of the partial sums is not easy, but useful. This difference is particularly useful to understanding what the divergence test does and/or does not tell you.
@Megan Kelly
I suspect the series converges absolutely, so we can look at the absolute value of the terms.
From here can
1) use the fact that $\left| \sin (\theta) \right|\leq 1$ and then use the p-series test
2) use the limit comparison test and then use the p-series test.
Note that when using the limit comparison test, we are taking advantage of the fact that $\sin(\theta)$ is a bounded function.
You might have other options, but those seem the simplest to me. Also, I designed my practice tests to train you into learning the different tests. That's why I added the rule that you can't use any two tests twice (as the primary step).
The example to keep in mind: the terms of the harmonic series go to zero, but the series itself is divergent.
Delete$$\sum_{n=1}^\infty \frac{1}{n}$$
Reply here to discuss problem 4d.
ReplyDeleteReply here to discuss problem 4e.
ReplyDeleteTim, I am having trouble with this question. It seems like a simple limit comparison test but the tests gives a -1.
DeleteWe can adapt the series in order to use the limit comparison test as it is stated in the book:
DeleteFactor out the minus sign and consider the series
$$\sum_{n=1}^\infty \frac{2n-2}{3n^4}$$
If we go back to the definition of an infinite series as a limit of partial sums, we would understand that the convergence of the above series is the same as the convergence of our original series.
This new series (only different by a sign) has positive terms for $n \geq 2$. So we can apply the limit comparison test.
Note that on the test, you should mention how the limit comparison test generalizes as long as the terms are all positive for large enough $n$.
In any case, the same argument basically generalizes to saying that any finite non-zero limit means the two series have the same convergence and/or divergence.
What are the implications of when the limit equals 0 or $\infty$?
Reply here to discuss problem 5.
ReplyDelete