Practice Test 4
Made available 20120428 11:59 PM.
1) Determine the tangent line of the polar function $r=6\cos\theta$
at the point $(x,y)=(3,3)\in\mathbb{R}^{2}$.
2) Find the area enclosed by the graph of $r=\cos(3\theta)$? Hint:
I may or may not be trying to trick you.
3) Evaluate the following (improper) integral.\[
\int_{-1}^{5}\frac{1}{(x+1)^{1/6}}dx\]
4) Convergent or divergent. State the test you use. You may not use
the same test twice. Extra: What about absolutely convergent?
a)\[
\sum_{n=1}^{\infty}\frac{(\cos n)^{n}}{n^{2n}}\]
b)\[
\sum_{n=2}^{\infty}\sin n\]
c)\[
\sum_{n=1}^{\infty}2\frac{3^{n}}{5^{n-1}}\]
Hint: This one converges. Why? Find the value.
d)\[
\sum_{n=2}^{\infty}\frac{\cos(n\pi)}{n^{2}\ln n}\]
e)\[
\sum_{n=1}^{\infty}\frac{n!}{(2n)!}\]
5) State the tests you didn't use.
6) Determine the interval of convergence.
a)\[
\sum_{n=1}^{\infty}\frac{2^{n-1}}{n^{n}}x^{n}\]
b)\[
\sum_{n=1}^{\infty}\frac{1}{n^{2}}(x+1)^{n}\]
7) Find the power series for the following functions. What is the
interval of convergence of the series?
a)\[
\frac{1}{1-x^{2}}\]
b)
\[
\tan^{-1}x\]
8) Compute the Taylor series for the following functions around the
given $a$.
a)\[
x^{2}+2x+1\]
around \[
a=1\]
b)\[
x^{2}+2x+1\]
around \[
a=3\]
c)
\[
e^{x}\]
around \[
a=0\]
(Note: Actually make the computation, don't just state the result)
9) Conceptual Check: What does $\sum_{n=1}^{\infty}a_{n}$ mean?
10) (SUPERCALIFRAGILISTICEXPIALADOCIOUS BONUS) Does the following
converge or diverge? Hint: Don't worry if you don't get the answer.
\[
\sum_{n=1}^{\infty}\frac{\sin n}{n}\]
Update 20120501: If you tried working on this problem, I applaud your efforts. While it has a solution, the solution I found requires a theorem you have not learned. Note that I said I found the solution. This means I tried to figure out the answer (with two friends in the department), but couldn't. This doesn't mean you can't figure out the solution, but that it is a difficult problem.
10') Does the following converge or diverge?
\[
\sum_{n=1}^{\infty}\frac{\sin (1/n)}{(1/n)}\]
Was browsing through Professor Brown's notes and found the answer to 10!
ReplyDeleteIn lecture Dr. Brown used the squeeze theorem saying that -1/n<sin(n)/n<1/n
Thus it converges to 0!
Ah, this is the limit of the terms. I will repeat a comment I made on Practice Exam 2 Problem 4c.
Delete"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."
The example to keep in mind: the terms of the harmonic series go to zero, but the series itself is divergent.
$$\sum_{n=1}^\infty \frac{1}{n}$$
Oh ok I see. Thanks for the feedback!
Deletein regards to 4a, does cos(n) converge or diverge as n approaches infinity?
ReplyDelete-Erica
The limit of $\cos(n)$ as $n$ approaches infinity does not exist. But we should take advantage of the fact that $\cos(x)$ is a bounded function.
DeleteHello Tim,
ReplyDeleteI was wondering if you will have our graded tests available after class this Friday, or sometime that day?
Thank you,
Sophia
Hey this question doesn't apply to the practice test but I wasnt sure where else to post it.
ReplyDeleteIn terms of trigonometric substitutions, what do we do when the power of tangent is even and the power of secant is odd?
I believe that is treated on a case-to-case basis. Such problems require you to think on your feet. Try using various trigonometric identities and integration by parts (possibly twice). Remember to use your time wisely and not get bogged down by any one problem.
DeleteHow do we approach a problem like
ReplyDelete"For what values of r does the function y=e^(rx) satisfy the differential equation 2y" + y' - y = 0 ?
In general, let's say you have $ay^{\prime \prime}+by^\prime+cy=0$. After you plug in $e^{rx}$ and factor out $e^{rx}$ you will obtain $ar^2+br+c=0$. Now just solve the quadratic equation for $r$.
DeleteWhat attributes can we infer about a solution to a differential equation just by looking at the differential equation?
ReplyDeleteI suppose that depends on the differential equation. If you have a differential equation of the form $$\frac{dy}{dx}=f(y)$$, then you can determine the equilibrium solutions and various other properties.
DeleteIn section 9.3 (separable equations), is it ok if we choose to separate C with K like they do in one of the early examples?
ReplyDeleteWill we have to translate a word problem into a mathematical equation (section 9.3 - separable equations).
ReplyDeleteI don't make the tests, so I do not know. However, as I have said repeatedly, Dr. Brown thus far has emphasized that his test problems come from homework problems or in-class examples. During the semester, he probably dropped key phrases like "this might be on the test" or "this is good to know." Hopefully you were able to pick up on those moments.
Delete