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The past exam paper of Calculus, key points are: Good Order, Second Order, Taylor Polynomial, Estimate, Largest Possible Error, Occurred, Previous Estimate, Comparison, Good Upper Bound, Converges or Diverges
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Math 106: Review for Final Exam, Part II
(a)
1
7 + 5 sin x x^2
dx
(b)
1
1 + 3x^2 + 2x^3 √ (^310) x (^12) + 17x 10 dx
(a) ak = 3 +
10 k
(b) ak = (−1)k
(c) ak =
5 ek 7 ek^ + ln(k + 1)
Strategy. The following is a good order in which to consider the various series convergence tests.
(a) Do the individual terms approach 0? If they don’t approach 0, the nth Term Test tells you the series must diverge. If they do approach 0, try another test. (b) Is the series geometric? (That is, do you multiply by the same constant r to get from each term to the next?) If so, the series converges if |r| < 1 and diverges otherwise. (c) Does the series contain something such as (−1)k^ or (−1)k+1^ or cos (kπ) that makes its terms alternate? If so, try the Alternating Series Test. (d) Does the series contain a factorial (k!) or exponential (such as 2k^ or ek)? If so, try the Ratio Test. (e) If the series has positive terms, does it remind you of a simpler series (especially a p-series: powers of k such as 1/k or 1/k^2 )? If so, try the Comparison Test. (f) Is the formula something you can integrate easily? If so, try the Integral Test.
(a) 3.1 + 3.01 + 3.001 + 3.0001 + · · ·
(b) 1 + 1/2 + 1/3 + 1/4 + · · ·
(c) 5 − 5 /3 + 5/ 9 − 5 /27 + · · ·
k=
(x + 3)k k · 5 k^
(a) 1 − 1 +
(b)
(c) 1 − π^2 2
π^4 24
π^6 720
0
e−x
2 dx and show that it converges.
(b) If f(x) = e−x
2 , what is f(400)(0)? What is f(401)(0)?
