Math 1B Final Examination N Reshetikhin May 18 2004 Student s Name TA s name Student s i d number 1 10 pnts Evaluate the integral Z 2 x3ex dx 1 2 15 pnts Evaluate the integral Z t2 1 dx 1 t 1 2 3 15 pnts Indicate which of the following statements are true and which are false Do not show your work Z Z 1 sin2 x dx converges by comparison test with dx 1 3 x x3 1 1 2 3 4 5 Z 1 2 Z 1 Z Z Z 1 dx dx is a convergent improper integral x 1 2 0 sin x1 dx diverges by comparison test with x 1 dx is a divergent improper integral x2 ln x dx is a convergent improper integral x 1 3 2 3 1 dx x 4 15 pnts Find the radius and the interval of convergence of the power series X n 1 x 1 n n 2 n 3 2 n 1 4 5 15 pnts State whether the following series is absolutely convergent conditionally convergent or divergent Do not show your work 1 X cos n X 1 n n 1 2 n 1 3 X n 1 27 n3 27 4n4 1 1 n n 1 4 X 3n 1 n n n 1 5 X n 2 cos n 1 n2 ln n 5 6 15 pnts For each statement indicate whether it is true or false Do not show your work 1 If X c2n converges then n 1 X 1 n c2n also converges n 1 2 If f x 0 is monotonically increasing and then X Z f x dx is convergent 1000 f n converges n 1 3 If the sequence an converges and the sequence bn diverges then an bn diverges 4 If the sequence an converges and and the sequence bn diverges then anbn diverges 5 If X n 0 an5n converges and X an 6 n diverges then n 0 X n 0 6 an 8n diverges 7 15 pnts For each statement indicate whether it is true or false Do not show your work 1 X ncnxn converges absolutely inside of the interval of convergence of the n 1 power series X c n xn n 1 2 X cnxn has radius of convergence R then X cnxn converges for x R then R is the radius of convergence of this X cnxn diverges for x a 0 then a R where R is the radius of n 1 n 1 cn Rn converges condition n 1 ally 3 X power series 4 n 1 convergence of this power series 5 The radius of convergence of X xn X xn is 1 n n 1 n100 n 1 7 8 15 pnts Solve the initial value problem x2 1 2yy0 2y0 x y 0 0 8 9 15 pnts Find the general solution to the differential equation dy 1 x2 y x2 y dx 9 10 10 pnts Find the general solution to the differential equation y00 y ex y 0 0 y0 0 5 2 10 11 0 0 5 1 1 5 y 2 2 5 3 0 5 1 1 5 Flow x 2 2 5 3 11 15 pnts In a box near each picture of directional fields indicate which differential equation it is representing Flow 3 2 5 2 12 y 1 5 1 0 5 0 0 5 1 1 5 2 x 2 5 3 Flow 3 2 5 2 13 y 1 5 1 0 5 0 0 5 1 1 5 2 x 2 5 3 Flow 3 2 5 2 14 y 1 5 1 0 5 0 0 5 1 1 5 2 x 2 5 3 Flow 3 2 5 2 15 y 1 5 1 0 5 0 0 5 1 1 5 2 x 2 5 3 12 20 pnts Find the power series solution to the differential equation y00 xy0 y 0 y 0 1 y 0 0 0 16
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