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Unformatted text preview: Unoﬃcial Math 31B Practice Final by TA Ning Khamsemanan Disclaimer: this practice exam is my attempt to help you study for the ﬁnal. Things in here might not be in the real exam and vice versa. Don’t use this as your main study. There might be some typos and/or mistakes. 1. Find the radius of convergence and Interval of convergence of the following (a) (b)
∞ 3n (x−2)n n=1 (n+1)! ∞ (x−1)n n=1 n2n 2. Find the power series representation of the following (a) ln (b)
1+x 1− x x (1−2x)2 3. Determine whether the following series if they are conditionally convergent, absolutely convergent or divergent. (a) (b)
∞ (−1)n (n+1)3n n=1 22n+1 ∞ (−1)n+1 √ 5n n=1 4. Test the series for convergence of divergence (a) (b) (c)
∞ n=1 n+sin2 n2
1 n +1 ∞ (−1)n 3n n=1 1+8n ∞ √1 n=2 n (ln n)3 5. Find the Maclaurin series for the following functions: (a) x2 cos x (b)
e−x x 6. Let A be the area between y = x3 , y = 2x − x2 and x ≥ 0. (a) Find A 1 (b) Find the volume generates by rotating A around x-axis by slicing method. (c) Find the volume generates by rotating A around y -axis by shell method. 7. Evaluate the following limits. (a) limx→1+
x x− 1 − (b) limx→π/2 (tan x)
√ 1 ln x cos x 8. Evaluate the following integrals. (a) (b) (c)
x2 − 1 dx x sec6 x dx tan2 x x2 +1 dx x2 − 1 2 ...
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This note was uploaded on 02/22/2010 for the course MATH Math 31B taught by Professor Houdayer during the Spring '09 term at UCLA.
- Spring '09