CALCULUS FINAL EXAM QUESTIONS WITH CORRECT ANSWERS

CALCULUS FINAL EXAM QUESTIONS
WITH CORRECT ANSWERS
For: ∫sin(mx)cos(nx)dx use, - ANSWER-sinAcosB=½[sin(A-B)+sin(A+B)]

For: ∫sin(mx)sin(nx)dx use, - ANSWER-sinAsinB=½[cos(A-B)-cos(A+B)]

For: ∫cos(mx)cos(nx)dx use, - ANSWER-cosAcosB=½[cos(A-B)+cos(A+B)]

√(a²-x²) - ANSWER-x=a sin∅ and 1-sin²∅=cos²∅

√(a²+x²) - ANSWER-x=a tan∅ and 1+tan²∅=sec²∅

∫(1/(x²+a²))dx → - ANSWER-(1/a)tan⁻¹(x/a)+c

Squeeze Theorem - ANSWER-a_n ≤ b_n ≤ c_n and lim (as n→∞) a_n = lim (as n→∞)
c_n = L, then lim (as n→∞) b_n = L

Geometric Series - ANSWER-n=1∑∞ arⁿ⁻¹ = a + ar + ar² + ... = a/(1-r) |r| < 1 means it is
convergent

Test for Divergence - ANSWER-If lim (as n→∞) a_n DNE, of if lim (as n→∞)≠0, it'd
divergent

Integral Test - ANSWER-I. f is continuous, II. positive, III. decreasing on [1,∞) and a_n =
f(n).
(n=1)∑∞ a_n converges iff, 1∫∞ f(x)dx is convergent

P-series - ANSWER-(x=1)∑∞ 1/(xⁿ) if n>1, it converges if n≤1, it diverges

Comparison Tests - ANSWER-If ∑b_n is convergent and a_n≤b_n for all n, ∑a_n is
convergent
If ∑b_n is divergent and a_n≥b_n for all n, ∑a_n is divergent.
lim (as n→∞) (a_n/b_n) = c if c>0 both either converge or diverge.

Alternating Series - ANSWER-(n=1)∑∞ (-1)ⁿ⁻¹b_n
(i) b_n+1_ ≤ b_n for all n
(ii) lim (as n→∞) b_n = 0, then it converges

Absolute Convergence - ANSWER-If ∑|a_n| converges

Conditional Convergence - ANSWER-If ∑a_n converges, but ∑|a_n| does not

Ratio Test - ANSWER-lim (as n→∞) |(a_n_+1)/(a_n)| = L