<p>Integration (0 and 2pi) [sin(x)^4] dx</p>
<p>I might have made a mistake somewhere but I got 3(pi) / 4. Apply the half-angle formula twice.</p>
<p>A) Bump! Don't let this thread die!</p>
<p>B) Along with Q and A's, how about everyone pool together exam tips... you know, what you seem to find more often on the exams, test-taking tips for Calc BC, etc.?</p>
<p>What half angle formula, snipez?</p>
<p>and Diamondbacker, look at my previous post. I need help with finding convergence/divergence on the third summation.</p>
<p>Thanks</p>
<p>So we know that 1 CONVERGES and 2 DIVERGES.</p>
<p>3 DIVERGES because of the integral test. Becaue we have the series:</p>
<p>III. Sigma[n = 2, +inf] 1/(n*ln(n)) </p>
<p>We can look at the similar integral,</p>
<p>(Integral from 2 to +inf) 1/(x*ln(x))</p>
<p>If this integral converges to a finite sum, then the series will converge to a finite sum. If it diverges, then the series diverges.</p>
<p>When we integrate we get ln(lnx), from 2 to +inf. If we plug in the values, we get:</p>
<p>ln(ln(+inf)) - ln(ln(2)). the first term goes to infinity, so the sum goes to infinity - It is divergent...</p>
<p>The only one that converges is I.</p>
<p>hmm yeah I tried everything else before I tried the integral test, mainly because I did not know the integral test. anyways here's what I do when I have to test for convergence/divergence:</p>
<ol>
<li>check to see if it is an alternating series (obvious)</li>
<li>check to see if geometric (obvious as well)</li>
<li>test for divergence </li>
<li>reduce to p-series form, direct compare with p-series</li>
<li>ratio test</li>
<li>limit compare with p-series</li>
<li>root test, integral test</li>
</ol>
<p>usually I don't reach step 7, but if I can see a nice integrand I'll apply integral test.</p>
<p>RESMonkey, cos(2x) = cos^2(x) - sin^2(x) = 2cos^2(x) - 1 => cos^2(x) = (1+cos(2x))/2. Similarly (applying pythagorean identity), cos(2x) = 1-2sin^2(x) => sin^2(x) = (1-cos(2x))/2. I doubt you will encounter a non-calculator problem involving trigonometric powers though.</p>
<p>Some key points to remember:
1. always keep in mind the chain rule. it's the most important in differential calculus
2. f'(a) doesn't have to exist for f to have an extremum at x = a, similarly f''(a) doesn't have to exist for f to have an inflection point there. Just consider sign changes.
3. The infinite series question on the FR only requires knowledge of finding the interval of convergence (which is just ratio test, which usually results in the endpts tested by alternating series test or comparison to p-series), the taylor series formula (easy to remember), and understanding of the basic function to series expansions. Typically the series is a slight modification of the basic functions so that should point you in the right direction.</p>
<p>Another series question and two improper integrals:</p>
<p>Which of the following series converge?
I. Sigma[n=1, inf] n/(n+2)
II. Sigma[n=1, inf] cos(n*pi)/n
III. Sigma[n=1, inf] 1/n</p>
<p>Evaluate integral<a href="x%5E2">0, inf</a>(e^(-x^3)) dx</p>
<p>Evaluate integral[1, inf] x/(1+x^2)^2 dx</p>
<p>1) Evaluate integral<a href="x%5E2">0, inf</a>(e^(-x^3)) dx
lim[b--->inf]Integral(0 to b) x^2 / e^(-x^3); use u-substitution, and get (-1/3)lim[b--->inf.]e^(-b^3) - 1 = (-1/3) * -1 = **1/3*</p>
<p>2) Evaluate integral[1, inf] x/(1+x^2)^2 dx
lim[b--->inf.]Integral(1 to b) x / ((1 + x^2)^2) dx; use u-substitution, and get (1/2)lim[b--->inf. (-1)/1+b^2 + 1/2] = 1/2 * /12 = **1/4*</p>
<p>I. DIVERGES by nth term test
II. Diverges by nth term test? (if you use L'Hopital, the limit is oscillating and doesn't exist.
III. DIVERGES - harmonic series</p>
<p>I think the second series converges by the alternating series test. You can rewrite cos( n * pi ) as -1 ^ n.</p>
<p>Ah, yes, good point. I had seen a similar problem before too. You're correct, Begoner.</p>
<p>good job guys, all the reasoning is correct, vader, both integrals are evaluated correctly</p>
<p>Another review Question:</p>
<p>If the first 3 terms of series:</p>
<p><a href="-1%5En">sigma, n=0 to n=+inf</a>/(1+n^2)</p>
<p>to approximate the series, then which of the following statements is (are) true?</p>
<p>I. The estimate is 0.7
II. The estimate is too low
III. The estimate is off by less than 0.1</p>
<p>@ Siddharth:</p>
<p>You're correct.</p>
<p>Is the answer to your question B, 2.017?</p>
<p>man you really need to use a calculator for that. But im too lazy togo get mine :)</p>
<p>Actually you barely need the calculator for it. Most of the calculator required questions can be worked out by hand (in fact, you MUST), until you can finally use the calculator. There reallly aren't many questions you can do wholly on the calculator.</p>
<p>Find the antiderivative of f'(x)=sqrt(1-x^2)</p>
<p>(I just did it and thought it was a bit tricky...)</p>
<p>On the multiple choice, in a calculator active question like that one, you can do siddharthdhami's problem entirely with the calculator:</p>
<p>Y1 = 100 + 20sin(piX/2) + 10cos(piX/6)
Y2 = nDeriv(Y1,X,X)</p>
<p>Set the table for the Independent Variable to "Ask".</p>
<p>Type in the five values for X.</p>
<p>Check to see which number is lowest.</p>
<p>Move on to the next question. :)</p>
<p>That took me too long...</p>
<p>You use trig substitution, then regular substitution to realize that the integral can be wrriten as cos(t)^2 dt, then you integrate that and plug t back into the initial equation
I believe it is .5(sqrt(1-x^2) x+.5arcsin(.5x)</p>
<p>Next question</p>
<p>What are all values for which the series <a href="(x+3)%5En">sigma, n=1 to n=+inf</a>/(sqrt(n) converges</p>
<p>a)[-4,-2)
b)[,2,00
c)(-2,0)
d)(-4,-2)
e)(-4,-2]</p>
<p>TheMathProf's way will work, but I believe s/he meant to take the double derivative.
g'(x) = rate of change of the function, so the question is asking for when g'(x) is most negative so you have to take the derivative of it again to find the abs. extrema
g''(x) will equal zero when g(x) is relatively increasing or decreasing the fastest.
so you must use the second derivative test to figure out which one (increasing or decreasing). so:
g'''(x) > 0 when the function is at the minimum</p>
<p>@ raller</p>
<p>Is the answer (a), [-4, 2) ?</p>