MATH 201 Problems For Final, Fall 2011

Practice Final Exam Exercise 1 Discuss the convergence of the following series: a) +∞ � n=0 1 √ n! b) +∞ � n=0 n + 10 n ln3 n c) +∞ � n=0 (−1)n2 1 n2 + √n d) +∞ � n=1 1 − cos n nln n e) +∞ � n=1 n(e1/n − 1) Exercise 2 a) Estimate the value of the integral � 1 0 e−3x2 dx with an error of magnitude no less than 10−3 (hint: Maclaurin then alternating series and ...) b) Find f(2001)(0) if f(x) = x6 sin(x5) Exercise 3 If w = f(x, y) is differentiable and x = r + s, y = r − s, show that ∂w ∂r × ∂w ∂s = �∂f ∂x �2 − �∂f ∂y �2 Exercise 4 Find the absolute extremum of f(x, y) = 5 + 4x − 2x2 + 3y − y2 on the region R bounded by the lines y = 2, y = x, and y = −x. Exercise 5 Use Lagrange multipliers to find the maximum and minimum of f(x, y) = 4xy subject to x2 + y2 = 8. Exercise 6 Convert to polar coordinates, then evaluate the following integral � 2 0 � 0 −√ 1−(y−1)2 xy2dxdy Exercise 7 Here is the region of integration of the integral = j x 6 z=y2 y z (0,-1,1) ............... ............... (1,-1,1) (1,-1,0) � 1 0 � 0 −1 � y2 0 dz dy dx Rewrite the integral as an equivalent iterated integral in the other 5 orders, then evaluate one of them Exercise 8 Let V be the volume of the region D enclosed by the paraboloid z = x2 + y2, and the plane z = 2y a) Express V as an iterated triple integral in cartesian coordinates in the order dz dx dy (do not evaluate the integral) b) Express V as an iterated triple integral in cylindrical coordinates, then evaluate the resulting integral (you may use the result: � sin4 x dx = −sin3 x cos x 4 − 3 cos x sin x 8 + 3x 8 ) Exercise 9 Let V be the volume of the region D that is bounded below by the xy-plane, above by the sphere x2 + y2 + z2 = 4, and on the sides by the cylinder x2 + y2 = 1 a) Express V as an iterated triple integral in spherical coordinates in the order dρ dφ dθ (do not evaluate the integral) b) Express V as an iterated triple integral in spherical coordinates in the order dφ dρ dθ (do not evaluate the integral) Exercise 10 Let V be the volume of the region that is bounded form below by the sphere x2 + y2 + (z − 1)2 = 1 and from above by the cone z = � x2 + y2. Express V as an iterated triple integral in spherical coordinates, then evaluate the resulting integral (sketch the region of integration) Exercise 11 Set up an integral in rectangular coordinates equivalent to the integral � π/2 0 � √ 3 1 � √ 4−r2 1 r3(sin θ cos θ)z2dzdrdθ (do not evaluate the integral) Exercise 12 a) Find the work done by the force −→ F = x−→i +y2−→j +(y −z)−→k along the straight line from (0, 0, 0) to (1, 1, 1) b) Evaluate � (1,π/2,e) (0,0,1) (ln z + ex sin y)dx + ex cos ydy + (x/z − z)dz c) Find the outward flux of the field −→ F = (y − 2x)−→i + (x + y)−→j across the curve C in the first quadrant, bounded by the lines y = 0, y = x and x + y = 1. i) by direct calculation ii) by Green’s theorem Exercise 13 a) The Curl vector of a vector field in space F = M(x, y, z)−→i + N(x, y, z)−→j + P(x, y, z)−→k is defined by: Curl(F) = ∇ × F = ������� −→i −→j −→k ∂ ∂x ∂ ∂y ∂ ∂z M N P ������� = �∂P ∂y − ∂N ∂z � −→i + �∂M ∂z − ∂P ∂x � −→j + �∂N ∂x − ∂M ∂y � −→k Show that div(Curl(F)) = 0 b) Show that Curl(grad f) = 0 if f(x, y, z) is a differentiable function Exercise 14 Use the transformation u = y −2x and v = x+2y to find the value of the integral � � R x + 2y y − 2x dA where R is the region in the second quarter of the plane, and enclosed by the lines y − 2x = 2, and y − 2x = 4