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35 Pages

### Day40-1432classa

Course: MATH 1432 11278, Spring 2012
School: U. Houston
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Word Count: 786

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1432 Have Math you registered for Test 4? Have you registered for the final? Dont forget online quizzes. THINK: There is one number less than 1000 that gives a remainder of 1 when it is divided by 2, 3, 4, 5, 6, 7 or 8. What is that number? Review Test 4: Series: Convergence, Divergence, Absolute Convergence, Conditional Convergence, Sum (geometric, telescoping) LHopital recognize and apply Improper Integrals...

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1432 Have Math you registered for Test 4? Have you registered for the final? Dont forget online quizzes. THINK: There is one number less than 1000 that gives a remainder of 1 when it is divided by 2, 3, 4, 5, 6, 7 or 8. What is that number? Review Test 4: Series: Convergence, Divergence, Absolute Convergence, Conditional Convergence, Sum (geometric, telescoping) LHopital recognize and apply Improper Integrals recognize type of improper integral, compute improper integrals using CORRECT notation Series use known tests (alternating series, root, ratio, p-series, limit comparison, integral, basic comparison, geometric, basic divergence) to determine convergence Taylor Polynomials and Series Give Taylor polynomials using given information (values, functions, etc); be able to find the error, or n for a given error Find a Taylor series for cos (x2) centered at x = 0: Find a Taylor series for e2x centered at x = 0: Find the Taylor polynomial P5(x) for f (x) = xcosx2. Find the nth Taylor polynomial Pn for the function f (x) = ex Find the nth Taylor polynomial Pn for the function f (x) = sinh x Find the nth Taylor polynomial Pn for the function f (x) = ln (1 x) Give the 5th degree Taylor polynomial for f (x) = sin(x) centered at 0. Give the 5th degree Taylor polynomial for f (x) = ex centered at 0. Give the 5th degree Taylor polynomial for f (x) = ln(x+1) centered at 0. Give the 5th degree Taylor polynomial for f (x) = cos(x) centered at 0. Find a power series, centered at 0, for f ( x ) = 5 . 3x 1 f ( 2 ) = 1, f ' ( 2 ) = 2, f '' ( 2 ) = 1 Give the 2nd degree Taylor polynomial for f centered at 2. Rewrite f (x) = 3x3 +2x2 x + 1 in powers of (x 2). Create the 3rd degree Taylor Polynomial for f (x) = arctan(x) centered at x = 0. n +1 ( c ) n+1 Rn ( x ) = x ( n + 1) ! f Use the Lagrange formula to find the smallest value of n so that the Taylor polynomial of degree n for f (x) = cos (x) centered at x = 0 can be used to approximate f (x) within 10 4 at x = 1. Use the Lagrange formula to find the smallest value of n so that the nth degree Taylor Polynomial for f (x) = ln (1 + x) centered at x = 0 approximates ln (2) with an error of no more than 0.01. Assume that f is a function such that f (n ) ( x ) 22 for all n and x. 1 1 Estimate the error if P6 is used to approximate f . 3 3 Which is term truncated if we want to approximate the sum of with an error of less than 1 ? 1000 ( 1 ) 2n n =1 n +1 3 1 k =0 ( 1) ( x + 1) k k k k2 + 1 Give the 9th derivative of f at x = 1. Use a power series to give the 14th derivative of f at x = 0 for f ( x ) = e x 2 Use a power series to give the 12th derivative of f at x = 0 for () f ( x ) = cos 2x 3 Give a power series for f centered at x = 0 for f ( x ) = ln (1 + x ) Give a power series for f centered at x = 1 for f ( x ) = ln ( x ) Give a power series for f centered at x = 0 for f ( x ) = arctan ( x ) 1. Determine if LHopital applies. If so, state the indeterminate form and compute the following limits: ln (n + 5 ) a. lim n+8 n b. lim ( 5n ) n 3 n 3 c. lim 1 + n n 2n cos ( 2x ) d. lim 3x x0 e. lim x 1 5 2x ln x x 1 1 f. lim + x x 0 x g. lim x 0 x x e e sin ( 3x ) 2 h. lim x x ln x i. j. lim ( cos 2x ) x 0 ( 5 lim x + 2 x ) 3 x 1 ln x 2. For each of the following, find the exact value, if possible. a. 2 n=0 b. n=0 c. n=2 1 n n +1 3 n 5 1 n ( n + 1) d. cos ( n ) n 3 n=2 e. n 72 n 5 n=0 f. n=0 n+2 2 n 5 3. Evaluate each improper integral, and explain why it is improper. Use correct notation. a. b. 2 1 1 1 x 06 2 dx 1 dx 1 x c. d. 1 14 ( x + 6) 64 0 x 2 / 3 2 dx dx e. f. 7 x 0 7x 1 0 9+x 2 2 dx dx = g. 5 2 dx = x 2 4. Converge or diverge? a. n=2 b. n =1 2 4n + 5n 2 5 n 3n 1 2 5n + 3n 2 6 2n + n 10 c. 2 3 n =1 d. n =1 n 1 . n (n + 1) e. 3 2 n =1 f. n =1 n n 3 n + 2n g. 7n n=0 h. n =1 2 ( 1 ) n 2 n i. 1 1 n n +1 n =1 j. n =1 5 2n 1 k. n =1 l. n =1 n 2n 5n 1 m. 2n 3 n! n =1 n 1 2 ( 1 ) n 3 3n + 1 5 3 2 n. n=0 5 n 6 o. n =1 n n p. 1 + e n n =1 q. 1 n n n =1 5 3 1 r. n ( ln n ) n=2 s. n ( ln n) n=2 1 2 t. ne n =1 u. n =1 v. n =1 w. 3 n n n +1 n 1 n( n + 1 )( n + 2 ) en n =1 n! x. ( 1) n n n n=2 y. n=2 z. n=2 25n ! (n + 3) ! ( 1) n ! n ( n + 1) ! aa. n! n ( 1) (n 1) 5n n=2 2 n + 2n 1 bb. n=2 cc. n=2 dd. n=2 cos ( n ) n+7 ( 1 ) n n 2 2 +1 n (2) n n
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