Fall 2010 Math 152
2
Week in Review VII
Key Concepts:
Section 10.1
(covering 9.4, 10.1)
1. Limit of a Sequence: if an = f (n), recall
all techniques for finding limits of real-valued
functions as x → ∞
an+1
2. an increasing: show f ′ (x) > 0,
> 1
an
(provided an > 0 for all n), or an+1 − an > 0
(reverse signs for decreasing)
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3. Induction (Appendix E in text)
courtesy: David J. Manuel
Section 9.4
Examples:
Key Concepts:
1. about x-axis: SA =
ˆ
b
´b
2πx ds
1. Define the nth term of the sequence
3 4 5
, · · · and find the limit.
2, , ,
4 9 16
2πy ds
2. Find the limits of the following sequences:
a
2. about y-axis: SA =
a
n2 + 2n − 5
2n2 + 1
1
(b) an = n sin
n
√
2
(c) an = n + 4n − n
2n
3
(d) an = 1 +
n
(a) an =
3. ds can be done with respect to x, with respect
to y, or with respect to t (changing the other
variable as needed)
Examples:
(e) an =
4n + (−1)n
n
3. Determine whether the following sequences
are increasing or decreasing and find their
limits:
1. Find the area of the surface obtained by rotating the following curves about the given
axis:
ln n
n
2n
(b) an =
n!
(a) an =
1
(a) f (x) = (ex + e−x ), x ∈ [0, ln 2] about
2
the x-axis
√
(b) y = 2 x, x ∈ [0, 2] about the x-axis
4. A sequence is defined recursively by a1 =
1
2, an+1 = 2 + an . Show an is increasing
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and less than 3 for all n. Explain why the
sequence has a limit, then find the limit.
(c) y = 2x − 1, x ∈ [1, 3] about the y-axis
(d) x = 3t2 , y = t3 , t ∈ [−2, 2] about the
y-axis
(e) hx = i2t + sin(2t), y = 1 − cos(2t), t ∈
π
0,
about the y-axis
4
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