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MATH 137 CALCULUS 2 SPRING 2012 FINAL EXAM REVIEW :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: The examples I've listed aren't meant to be exhaustive, so don't just study these examples and expect to have covered all your bases. Instead, each example should just give you an idea of the type of problem to expect. Integration by parts EX 6 x cos(2 x)dx EX ln( x ) dx x2 EX x e EX arctan( 4 x)dx EX 3 x dx Improper integration EX 0 xe x dx (u-sub also) 2 1 0 2 x ln( x)dx (improper because of the 0; have to use parts also) EX 0 1 dx 2x 5 EX 2 4 dx (look like x(ln( x)) 3 something we've done recently?) Area between curves EX Find the area bounded by f ( x) cos( x) and g ( x) sin( x) between x 0 and x . EX Find the area of the region in the first quadrant bounded by y x 3 and y 2 x x 2 . Volumes of revolution EX Find the volume of the solid obtained by rotating the region bounded by y 40 x 5 x 2 in the first quadrant about the line x 10 . (have to use shells) EX Find the volume of the solid obtained by rotating the region bounded by y 40 x 5 x 2 and y x 2 8 x about the line y 2 . (use discs) EX Find the volume of the solid obtained by rotating the region bounded by y x and y x about the line x 3 . Now try it about the line y 3 . Separable first order differential equations EX Find a general solution to the differential equation dy xy cos(x) . dx EX Solve the initial value problem: dy 2x 2 , y (0) 3 . dx y x 2 y 2 EX Solve the initial value problem: dy y cos( x) , y (0) 0 . dx 1 y2 EX Find a general solution to the differential equation x cos( x) (2 y e 3 y ) y' . Differential equation (other) EX Sketch the slope field for y ' ( y 4)( y x) and sketch on it solutions which start at (-3, 2) and (-4, 6). EX Given the differential equation y' y 2 ( y 6)( y 5) , sketch a graph of possible solutions by finding the equilibrium solutions and then sketching a representative solution from each region they define. Finally, label each equilibrium solution as stable, semi-stable, or unstable. EX Find the values of r which would make y e rx a solution of y ' ' '2 y ' '24 y ' 0 Convergence of sequences EX Determine whether the following sequences converge or diverge. a) an n ln( n) b) an n 2 3n n! c) a n cos 2 (n) 2n d) a n arctan( n) 1 1n Convergence of series EX Determine whether the following series converge or diverge. a) 3n 7 n 6n c) b) n 3 n(ln( n)) d) n6 2 2 (1) n 2n 2 n 1 EX* Explain the difference between “if” and “iff”. Use convergence tests as examples. Power series EX Find the interval of convergence for the following power series. (1) n (2 x 1) n n EX Find the first five nonzero terms (not just the coefficients) of the Taylor expansion of a) f ( x) ln( x) about a 1 2 b) f ( x) 5 sin( x) about a . EX Write the following function as a power series. 2 a) f ( x) x arctan( x) sin( x4 ) b) f ( x) x c*) f ( x) 2 x 3 ln( 2 x 3) d*) f ( x) x 2 e x 4 EX Evaluate 2 x cos(3x)dx using MacLaurin series. Regions in R 3 EX Describe the region represented in R 3 by the following equations and inequalities. a) yx z0 b) x2 y 2 z 2 1 y 2 z 2 16 Vectors EX Let a 2,3,1 and b 3,1,2 . a) Find a 3b b) Find a b c) Find the angle between a and b d) Find a b Lines and planes in R 3 EX Find the equation of the line through the points (2,5,1) and (1,1,4) . EX Find an equation of the plane through the points (1,2,1) , (2,5,1) and (1,1,4) . x 1 t EX Find the equation of the plane containing the point (3,3,3) and the line y t z 1 2t .