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Exam 2 Math 16C, Summer Session I 2020 Due August 1, 2020 at 6:00pm PST (GMT -7) ”Bears are hype in some instances, but not so much in others.” - Raymond Chou 1 Those Lagrange Multipliers Sure Can Multi- ply All Right Use the method of Lagrange Multipliers for the following optimization prob- lems. (20 points total) 1.1 (9 points): People like it when things are optimal. Optimize the func- tion f(x, y, z) = x2y2z2 according to the constraint 2x + 3y + 5z = 9. 1.2 (9 points): Boxes are pretty cool. Consider a lidless box with dimensions x, y, and z. If this box must have surface area of 60 sq. units, find the dimensions that give the maximum volume. 1.3 (2 points): A rectangular prism in four dimensions with side lengths x, y, z, w has volume V = wxyz. If this rectangular prism is subject to the con- straints xyz+wxy+wxz+wyz = 10 and xy−yz+zw−wx = 20, write, but do not solve a Lagrange multiplier function for this scenario. (Find L(x, y, z, w); I think I once used the notation F(x, y, z, w) for this as well) 2 Dubious Double Integrals So I heard you like double integrals, so I put some double integrals on the exam. (10 points total) 2.1 (3 points): ∫ 1 0 ∫ x x2 xy dy dx 2.2 (3 points): ∫ 2π π ∫ π 0 (sin(x) + cos(y)) dy dx 2.3 (4 points): ∫ 4 1 ∫√x 0 3 2e y/ √ x dy dx 1 3 Impossible Integrals and Rambunctious Re- versals 3.1 (10 points): The following integral, as is, is impossible to compute:∫ 1 0 ∫ 1 x ey 2 dy dx I don’t like giving impossible problems, and as it turns out, there is a way to solve this problem. Reverse the order of integration and solve this integral. (You may use a graphing utility to visualize the region if necessary) 3.2 (3 points Extra Credit): Solve the following integral by any means.∫ 8 0 ∫ 2 3 √ x 1 y4 + 1 dy dx 4 Abhorrent Areas and Vile Volumes? So now we’re going to compute some volumes over some areas! (10 points) 4.1 (5 points): Compute the volume bounded between the xy-plane and the surface z = 1(x+1)2(y+2)2 in the region x ≥ 0, y ≥ 0. 4.2 (5 points): Compute the volume of the region bounded below by the xy- plane and above by z = 2y1+x2 , whose shadow on the xy-plane is bounded by the curves y = √ x, y = 0, x = 25. 5 Convergence Conundrums Determine whether the following series converge or diverge using the tests we learned. Please explicitly state which test you use (20 points total) 4.1 (4 points): ∑∞ n=0 1 7n 4.2 (4 points): ∑∞ n=1 n n+1 4.3 (4 points): ∑∞ n=0 1 n! 4.4 (4 points): ∑∞ n=0 n( 4 5 ) n 4.5 (4 points): ∑∞ n=1 1 n2−1 4.6 (2 points Extra Credit): ∑∞ n=2 ln(n) n1.5 2 6 Turmoil and Taylor Series So now comes the part where we get to be tortured by Taylor series. Please show your work. (Make a chart!) (20 points) 6.1 (10 points): Compute the Taylor polynomial for f(x) = sin(x) with a = 0. Write the first five terms, then express concisely in sigma notation. 6.2 (10 points): Compute the Taylor polynomial for ln(x) with a = 1. Write the first five terms, then express concisely in sigma notation. 7 Miscellaneous Mischief I needed to fill 10 more points worth of problems. (10 points) 7.1 (3 points): Does the sum ∑∞ n=0 3( 99 100 ) n converge? If so, can you com- pute its sum? If you can, compute it. 7.2 (2 points): Does the sum ∑∞ n=0 n0.3 n+1 converge? If so, can you compute its sum? If you can, compute it. 7.3 (5 points): Use the ratio test to determine whether the sum ∞∑ n=0 (−1)nn!3n (2n)! converges or not. 3