Will send the work where its readable, calculus 2
Show all work and leave all answers as exact answers (do NOT estimate with a calculator).
Findthegreatestlowerboundandleastupperboundofthefollowingsets(provided they exist).
(a) (−2,5)∪{5.1,5.01,5.001,5.0001,...} (b) {x∈R|sinx=cosxandx≥0}
(c) {x∈R||x−5|>2}
Determine if the sequence is monotone and justify your answer. State what type of
specific behavior it exhibits (increasing, nondecreasing, etc.). (a) an = n
3n
(b) an =(−1)n 3 n
√
3. Determine if the following sequences converge or diverge. If a sequence converges, find its limit.
(a) an = sin2 n. n
(b) an = 2n n2
√
(c)an= n5n
2n/2
(d) an = 1 + n
4. Use L’Hopital’s Rule to find the following limits (of sequences or functions).
(a) lim (b) lim
n+3 n→∞ n2 +5n+6
n+2n n→∞ n−1
(c) limln(x)−ln(sinx). x→0+
(d) limxsinx x→0+
5. Find lim sin(aθ) for real numbers a, b > 0. θ→0 bθ
6. (Exploration,nottobegraded)Writethefollowingsequenceasarecursivesequence (the n-th term gn in terms of the (n-1)-th term gn−1).
1,2=1+1,1+1,1+ 1 ,1+ 1 ,... 121+11+1
2 1+1 2
(a) Write the first 5 terms g1, ..., g5 as rational numbers.
(b) Use a calculator, MS excel (this is most efficient), or some other program to
compute the first 25 values (to at least 10 decimals) of {gn}.
(c) Is the sequence monotone?
(d) Multiply the 25th term by 2, subtract 1, and then square it. What integer is this near?
(e) What (irrational number) do you now expect the sequence to converge to? The limit is a very famous number called the golden ratio (google it!)
(f) The Fibonacci sequence is given by the recursive formula f1 = 1, f2 = 1, and
fn = fn−1 + fn−2 for n ≥ 2. Compare the sequences gn and an = fn+1 . fn
Subject | Mathematics |
Due By (Pacific Time) | 03/01/2013 02:00 am |
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