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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) 1 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 4 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 1