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# test110 - 8 ≤ x f for all x 2 1 ≤ ≤ x a Find the 7-th...

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Name 11 February 2010 Teaching Assistant Hour Test 1 Math 1502 E Andrew Instructions: 1. Closed book. 2. Show your work and explain your answers and reasoning. 3. Calculators may be used, but are by no means necessary. Pay particular attention to instruction 2. To receive credit, you must show your work. Unexplained answers, and answers not supported by the work you show, will not receive credit. 4. Express your answers in simplified form. 1. (25) a. Find the general solution of the differential equation x e y dx dy 3 3 - = + b. Solve the initial value problem 1 ) 0 ( 2 2 = = y y x dx dy 2. (25) a. Evaluate the limit ) sin( ) cos( 2 lim 0 x x e x e x x x - + - . b. Determine whether the integral converges, and if so, evaluate the integral. - 0 2 dx e x x

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Name Page 2 of 3 Math 1502 E Andrew Hour Test 1 Teaching Assistant 11 February 2010 3. (25) A mysterious function named f has these properties A. Whenever k is odd, the k-th derivative at 0 is ( 29 ( 0 0 = k f . B. ( 29 ( 29 ( 29 6 ) 0 ( , 4 ) 0 ( , 2 ) 0 ( ' ' , 0 0 6 4 = - = = = f f f f C. ( 29 ( 29
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Unformatted text preview: 8 ≤ x f for all x , 2 1 ≤ ≤ x . a. Find the 7-th Taylor polynomial at 0 for the function f described above. b. Estimate the accuracy with which this 7-th Taylor polynomial approximates 2 1 f 4. (25) a. Determine which series converge and which diverge. Be sure to tell which convergence test you are using. i. ∑ ∞ = 1 2 ! k k k ii. ∑ ∞ = + 1 2 3 ) cos( 2 k k k iii. ∑ ∞ = + 1 2 2 ) 1 ( k k k k b. Calculate the radius of convergence of ( 29 ∑ ∞ =--1 2 3 ) 1 ( k k k k x k Name Page 3 of 3 Math 1502 E Andrew Hour Test 1 Teaching Assistant 11 February 2010 ANSWERS 1. a. x x e C e x y 3 3--+ = b. 1 1 2--= x y 2. a. 0 b. Converges to 2 1 3. a. ( 29 6 4 2 7 ! 5 1 ! 3 1 x x x x P +-= b. ( 29 ( 29 8 2 ! 7 1 4. a. Diverges by the Basic divergence test, terms do not tend to 0. b. Converges by the Ratio Test c. 3 = R...
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test110 - 8 ≤ x f for all x 2 1 ≤ ≤ x a Find the 7-th...

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