<p>For series, you can use any of the following tests for convergence:</p>
<ul>
<li>Direct Comparison Test</li>
<li>Integral Test</li>
<li>Ratio Test</li>
</ul>
<p>Also, there are specific rules for geometric series, p-series, and alternating series that determine their respective convergence/divergence.</p>
<p>COCOFORCOLLEGE</p>
<p>6b. The base is 1 - the x intercept of line l. that is because triangle T is a right triangle with the vertical leg extending from (1,1) to the x axis and line l is the hypotenuse. y=x^n intercepts line l at (1,1) so we want to find the equation of line l in terms of n. y’= nx^(n-1) (power rule), so at x = 1, y’ = n(1^(n-1)) = n . so the slope of line l is n and we have the point (1,1) so line l is y-1 = n(x-1). The x intercept is where y = 0 so we have
0-1= n (x-1) -> -1=nx - n -> n-1 = nx -> (n-1)/n = x. Now that we have the x intercept, the base is 1 - the intercept so we have 1- (n-1)/n -> (n-(n-1))/n -> (n-n +1)/ n = 1/n. So the base is (1/n) the height is 1, and the area is (1/2)(1/n)1 = 1/(2n) QED.</p>
<p>6c. The area of S is simply the area under the curve x^n minus the area of triangle T. We take the integral of x^n from 0 to 1. The antiderivative is (x^(n+1))/(n+1) + C and evaluated from x=0 to x=1 gives us (1^(n+1))/(n+1) = 1/(n+1) (the 0 cancels out). So our desired area of S is 1/(n+1) - 1/(2n). To find the value of n that maximizes the area, we take the derivative and set it equal to 0 find the maximum of n (think maxima minima). so the derivative is -1/(n+1)^2 + 2/(2n)^2 = 0 -> 2/(2n)^2 = 1/(n+1)^2 -> take the radical of both sides -> (2^1/2)/(2n)= (n+1) cross multiply n(2^.5) + 2^.5= 2n -> 2n-n(2^.5)= 2^.5 -> n(2-2^.5)=2^.5 -> n= 2^.5/(2-2^.5) And after rationalizing the denominator you get 1+2^.5</p>
<p>does anybody know how many series/taylor and mclauren questions are typically on the test?</p>
<p>-random post to have this thread in my participated column so I can access as I study-</p>
<p>studentabcd: there’s normally one FRQ on taylor/maclauren and 3-5 multiple choice questions.</p>
<p>@studentabcd: there’s always taylor series as one of the last questions in FRQ. In MCQs, I’m not so sure. Probably around 3~5 I’m assuming.</p>
<p>^ LOL cocoforcollege, I actually did not copy what you wrote. I didn’t refresh my page and just commented and didn’t know you had commented before hahaha.</p>
<p>Good luck, everyone! Hoping for some nice FRQs to compensate for the mass of points I’m almost definitely going to lose on MC, aka really hoping for little-to-no limits. Does anyone know the likelihood that we’ll have to know a standard volume formula (i.e. cylinder, cone) for related rates?</p>
<p>And a reminder-to-self/maybe helpful reminder: express your integral boundaries in terms of whatever the equation is in! i.e. when converting an integral with respect to x to an integral with respect to u, you also have to re-express the integral boundaries in terms of u (i.e. u=2x+1; integral boundary changes from 2 to 2(2)+1=5).</p>
<p>Does anyone have last minute tips for improving speed on the first multiple choice section? I can never finish in time… :(</p>
<p>Should we know any other Maclaurin series other than e^x, cosx, sinx, and 1/1-x?</p>
<p>@Ronaldo: hmm…write faster, know how to use your calculator efficiently…sorry I’m betting that didn’t really help…you want to spend 2 minutes per question.</p>
<p>@starchy: my teacher told me to memorize ln(x).</p>
<p>I think it’s supposed to be ln(1 + x)?</p>
<p>our class was practically done in april 21st and i haven’t really reviewed since then -__- got a 3 then 5 then 5 on three practice exams but who knows what could happen tomorrow. i haven’t really done math for 3 weeks. my goshh.</p>
<p>Good luck everyone!</p>
<p>Just took the international version of the test! I don’t know how similar the topics are going to be for the domestic version, but if there is any resemblance, then you guys had better review your Taylor/Maclaurin series. You should even know anything and everything there is to know about series (all the tests for convergence, etc.). </p>
<p>Personally, the exam went very well. I’m predicting a raw score in the high 80’s (out of 108), and if things go particularly well, it could even be in the low 90’s. Too bad I blanked out on the last FRQ. :(</p>
<p>Just a reminder:</p>
<p>lim as n -> infinity of (1 + 1/n)^n = e</p>
<p>This may show up in series.</p>
<p>EDIT: calgirl15, I believe arithmetic, binomial, and Fourier series are not on the exam.</p>
<p>OK, so that was a bit of a hyperbole… 
What I meant was anything and everything to do with series as they pertain to the Calculus BC syllabus. Better? :D</p>
<p>Is Lagrange error bound important? It’s the only thing I don’t understand. :(</p>
<p>It probably won’t cover more than one part of an FRQ, if it appears at all.</p>
<p>Thanks! What about exponential growth?</p>