Math 211-02 Multivariable Calculus 2016 Final Exam

Math 211-02 Multivariable Calculus, Spring 2016 Final Exam May 10 1. (10 points) (a) (5 points) Find the distance between the skew lines x = y = z and x+1 = y/2 = z/3. (b) (5 points) Find the equation of the tangent plane and the equation of the normal line to the ellipsoid x2 + 2y 2 + 4z 2 = 7 at (−1, −1, −1). √ 2. (10 points) We are given the curve ~r(t) = h2t, 4 t, ln ti where 1 ≤ t ≤ e. (a) (5 points) Find the arc length of this curve. (b) (5 points) Find the curvature of this curve when t = 1. 3. (10 points) Consider the function   2x3 + y 3 f (x, y) = 2x2 + y 2  0 if (x, y) 6= (0, 0) if (x, y) = (0, 0). (a) (5 points) Prove that both fx (0, 0) and fy (0, 0) are 1. (b) (5 points) Prove that f (x, y) is not differentiable at (0, 0). You can use the fact that if g(x, y) is differentiable at (x0 , y0 ), then g(x, y) − g(x0 , y0 ) − ∇g(x0 , y0 ) · hx − x0 , y − y0 i p = 0. (x,y)→(x0 ,y0 ) (x − x0 )2 + (y − y0 )2 lim 4. (10 points) (a) (5 points) Suppose f is a differentiable function of x and y, and g(r, s) = f (2r − s, s2 − 4r). Calculus gs (1, 2) using the following information. (Note that you do not need to use all of the information given below.) f (0, 0) = 3 g(0, 0) = 6 fx (0, 0) = 4 fy (0, 0) = 8 f (1, 2) = 6 g(1, 2) = 3 fx (1, 2) = 2 fy (1, 2) = 5 (b) (5 points) Let ~u be the unit vector in the direction of h1, 1i. Let f (x, y) = x2 +xy+y 2 . Compute D~u2 f (x, y). 5. (10 points) Find the points at which the absolute maximum and minimum of the function f (x, y) = xy − 1 on the disk x2 + y 2 ≤ 2 occur. State all points where the extrema occur as well as the maximum and minimum values. 6. (10 points) (a) (5 points) Evaluate the double integral Z 8Z 2 4 ex dxdy. y 1/3 0 (b) (5 points) Rewrite the following triple integral in the order dxdydz. Z 1 Z 1 Z 1−y f (x, y, z)dzdydx. √ 0 x 0 7. (10 points) (a) (5 points) Prove that the limit does not exist y 2 sin2 x . (x,y)→(0,0) x4 + y 4 lim RRR (b) (5 points) Compute zdV where E is the region between the spheres x2 +y 2 +z 2 = E p 4, x2 + y 2 + z 2 = 1 and above the cone z = x2 + y 2 . 8. (10 points) R 3 (a) (5 points) Use Green’s theorem to compute C sin(x2 )dx + (x + yey )dy where C is the ellipse x2 + 4y 2 = 4, in the counterclockwise direction. R (b) (5 points) Evaluate the line integral F~ · d~r where F~ is the gradient of C f (x, y, z) = esin(x) cos(y/2) tan(z) and C is the straight line segment from (0, 0, 0) to (1, π, 1). 9. (10 points) (a) (5 points) Compute the area of the part of the surface y = xz that lies within the cylinder x2 + z 2 = 1. RR ~ where F~ = hx, y, zi and S is the part of the cylinder (b) (5 points) Compute S F~ · dS 2 2 x + y = 1 bounded between z = −1 and z = 1, oriented outward. 10. (10 points) RR (a) (5 points) Rewrite the surface integral of a scalar function S x2 + y + 4zdS, where S is the sphere x2 + y 2 + z 2 = 4, as a surface integral of a vector field, and then use the Divergence Theorem to compute its value. RR ~ where F~ (x, y, z) = hexy , exz , x2 zi (b) (5 points) Use Stokes’ theorem to evaluate S curlF~ ·dS 2 2 2 and S is the half of the ellipsoid 4x + y + 4z = 4, y ≥ 0, oriented in the direction of the negative of y−axis.