Review-Exam2

# Review-Exam2 - x 2 and(b 1 7-x 4 For what values of x is...

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Math136 Review for the 2nd exam Fall 2010 1. Find the solution of the initial value problem dy dx + 2 y x = ln x x , y (1) = 3 2. Determine whether the following improper integrals converge or diverge, and ﬁnd the values of the ones that converge. (a) Z 1 2 x x 2 + 1 dx (b) Z 0 x 2 e - x dx (c) Z π/ 2 0 tan xdx 3. Find the limits of the following sequences (if they exist): (a) n p n 2 + 2 n - n o (b) ±² 2 n + 7 2 n + 1 ³ n ´ (c) ± 3 n - 4 n 2 100 n 2 - n - 1 ´ . 4. Use the integral test to show that the series X n =2 1 n (ln n ) 2 converges. Make sure to check the hypotheses. 5. For each of the following inﬁnite positive series, determine whether it converges or diverges. Identify the test used, and justify the use of it. (a) X n =1 2 n 2 + 1 (b) X n =1 n n ( n + 2)! 3 n (c) X k =1 2 + cos(3 k ) 9 k 2 + k (d) X k =1 k sin ² 1 k ³ 6. For the power series X n =0 ( x - 4) n ( n + 1)5 n ﬁnd the radius of convergence R , and the interval of convergence I . For which values on I is the series absolutely convergent? For which values on I (if any) is the series conditionally convergent? 7. Use the geometric power series for 1 / (1 - x ) to ﬁnd the Maclaurin series for (a) 1 (1 + 2
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Unformatted text preview: x ) 2 and (b) 1 7-x 4 . For what values of x is each valid for? 8. (a) Find the 3rd degree Taylor polynomial T 3 ( x ) of the function f ( x ) = ln(1 + 2 x ) centered at c = 1. (b) Use Taylor’s inequality to estimate the accuracy of the approximation f ( x ) ≈ T 3 ( x ) when x satisﬁes 0 . 5 ≤ x ≤ 1 . 5. 9. Find the third Taylor polynomial for the function f ( x ) = x 3-2 x 2 + x-5 centered at c = 1. What is the error? 10. Given the alternating series ∞ X k =1 (-1) k k 3 k , (a) Prove it is convergent. (b) Estimate the AST error when using the ﬁrst 4 terms. (c) How many terms must be added in order to get an approximation to within 0.001? 11. Given the series ∞ X k =1 2 3 √ k + 5 k , (a) Prove it is convergent, by direct comparison with a geometric series. (b) Find a number N so that the partial sum up to N approximates the value of the series to within 10-4 ....
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