Math 1B Final Examination N Reshetikhin May 13 2005 P roblem P oints Grade 1 2 3 4 5 6 7 8 9 10 11 12 T otal 10 15 15 15 15 15 15 15 15 10 15 20 175 Student s Name GSI s name Student s i d number 1 10 pnts Evaluate the integral Z e 1 x dx 2 15 pnts Evaluate the integral Z t2 1 dt 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 x 2 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 X n 1 n 2 2n 5 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 X 1 n n 1 2 n 1 3 X n 1 n2 13 n 13 4n2 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 continuos and convergent then X Z f x dx is 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 excluding boundary points of the n 1 interval of convergence of the power series X c n xn n 1 2 If X cnxn has radius of convergence R then X cn xn converges for x R then R is the radius of convergence of X cnxn diverges for x a 0 then a R where R is the radius of n 1 n 1 cn Rn converges condi n 1 tionally 3 If X this power series 4 If 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 y0 ex 2y 8 9 15 pnts Find the general solution to the differential equation 2 dy 2xy ln x ex dx 9 10 10 pnts Find the general solution to the differential equation y00 y cos x y 0 0 y 0 0 5 2 10 11 15 pnts Match pictures to differential equations 1 dy dy dy dy dy y2 2 y x 3 y1 2 4 y 2 5 xy dx dx dx dx dx Flow 3 2 5 2 y 1 5 1 0 5 0 0 5 1 1 5 2 2 5 3 x Figure 1 Equation number Flow 3 2 5 2 y 1 5 1 0 5 0 0 5 1 1 5 2 2 5 x Figure 2 Equation number 11 3 Flow 3 2 5 2 y 1 5 1 0 5 0 0 5 1 1 5 2 2 5 3 x Figure 3 Equation number Flow 2 y 2 1 0 1 1 2 x 1 2 Figure 4 Equation number Flow 3 2 5 2 y 1 5 1 0 5 0 0 5 1 1 5 2 2 5 x Figure 5 Equation number 12 3 12 20 pnts Find the power series solution to the differential equation y00 xy0 y 0 y 0 1 y 0 0 0 13
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