AP Calc BC Exam UPDATED ACTUAL Exam Questions and CORRECT Answers

AP Calc BC Exam UPDATED ACTUAL
Exam Questions and CORRECT Answers
nth term test given sigma(x) - CORRECT ANSWER - if lim(n-->inf.) x does not equal 0, then
sigma(x) DIVERGES. if lim(n-->inf.)x =0, then the test is INCONCLUSIVE


justification of nth term test - CORRECT ANSWER - "diverges by nth term test" ... NEVER
"converges by nth term test"


geometric series given sigma(x) w common ration "R" - CORRECT ANSWER - if |R| < 1 -->
converges; if |R| > 1 --> diverges


justification of geo series test - CORRECT ANSWER - "diverges bc of geometric, R=___,
|R|>1" OR "converges bc of geometric, R=___, |R|<1"


telescoping test/partial fraction decomp ONLY works if... - CORRECT ANSWER - the
denominator is factorable; if values cancel out, the leftover values are the convergence value; if
nothing cancels out, the test is INCONCLUSIVE


justification of telescoping test - CORRECT ANSWER - "converges by telescoping"



P-Series Test given sigma(1/n^p) - CORRECT ANSWER - p>1 --> converges; p</= 1 -->
diverges; it can be ANY number over n^P


justification of P-series test - CORRECT ANSWER - "diverges by p-series test, p=__, p</= 1"
OR "converges by p-series test, p=___, p>1"


requirements of integral test given sigma(x) - CORRECT ANSWER - x must be continuous,
positive, and factorable

,integral test given sigma(x) - CORRECT ANSWER - x must be continuous, positive, and
factorable; find the value of the improper integral: lim(n-->inf.) [(1 to n) x; if lim = #, then the
series converges; if lim = inf., then the series diverges


justification of the integral test - CORRECT ANSWER - "the series converges/diverges by the
integral test"


when to use the integral test - CORRECT ANSWER - use this test if you can easily
antidifferentiate; cannot use this if it doesn't meet the requirements (x must be continuous,
positive, and factorable) or if it is not antidifferentiable


limit comparison test given a positive, unknown series sigma(a); sigma(b) is a positive, known
series (p-series/geo) - CORRECT ANSWER - lim (b/a) or (a/b) = L ; if L is a positive, real
number, then either both series converge or both diverge (they behave the same); limit cannot
equal infinity or a negative #


justification of limit comparison test - CORRECT ANSWER - sigma(a) conv/div. by the limit
comparison test since sigma(b) conv/div. by _______ (p-series, geo...)


direct comparison test with sigma(a) - unknown and sigma(b) - known - CORRECT
ANSWER - CONVERGES: b>a, sigma(b) converges; DIVERGES: b<a, sigma(b) diverges;
go-to comparison series: geo + p-series; this test helps w series that would have messy limits and
integrals


benefit of direct comparison test - CORRECT ANSWER - helps w series that would have
messy limits and integrals


justification of direct comparison test - CORRECT ANSWER - since sigma(b) conv/div
(geo/p-series) and b<>a, hence it shall be known that sigma(a) conv/div by Direct Comparison
Test


series units reminders and tips - CORRECT ANSWER - pull out constants --> sigma(1/2n) =
1/2sigma(1/n) and it becomes a p-series!;;; factorials grow the fastest;;; trial and error;;; write out

, justifications;;; don't forget ER+ (positive real #) for LCT;;; try simple methods first;;; don't
forget about LPT, IBP, etc.;;; if you see a factorial, think ratio test; for finding interval of
convergence, test endpoints - for finding the radius, DON'T!;;; abs. value in ratio test and error
bounds; "write the taylor series" - include sigma, "write the general term" - no sigma;;; 3^n / -
3^n = (3/-3)^n = (-1)^n;;; |Taylor(x) - actual(x)| < error;;; can only use taylor polynomial
shortcuts when they are centered at 0


what function grows the fastest - CORRECT ANSWER - factorials



fastest growth to slowest growth - CORRECT ANSWER - n! --> 5^x --> 2^x --> x^7 --> x^2



lim(n-->inf.) 2^n/(2^n - 1) = - CORRECT ANSWER -1



if you see a factorial in a series, think: - CORRECT ANSWER - ratio test



power series (x-3)^n / 2n - CORRECT ANSWER - deriv: n(x-3)^n-n; antideriv: (x-
3)^n+n(n+1)


if you see "write the taylor series" vs "write the general term" - CORRECT ANSWER - taylor
series - include sigma; general term - don't include sigma


3^n / -3^n = (3/-3)^n = - CORRECT ANSWER - (-1)^n



taylor series -- "find f''(0)" - CORRECT ANSWER - f''(0)/2! = ______ (whatever the problem
gives you for the 2nd degree term)


you can only use taylor polynomial shortcuts when they are centered at - CORRECT
ANSWER - 0



alternating series test requirements given sigma (-1)^n x - CORRECT ANSWER - x > 0, x>
x+1 (take deriv), x --> 0 (lim=0)