1. Suppose f (x, y) has values and partial derivatives as in the table at right.(a) Find all the critical points you can from the given data and classify them into local mins, local maxes,and saddles. (3 points)(x, y) f fxfyfxxfxyfyy(1,0) 2 01 102(0,1) 3 00 −31−2(2,0) 0 −11 102(1,2) 5 00 −132Local mins (if any):Local maxes (if any):Saddles (if any):(b) Let L(x, y) denote the linear approximation to f (x, y) at the point (0,1). Is L(0.1, 1.1) likely to be largerthan, equal to, or smaller than f (0.1, 1.1)? Circle your answer below (1 point)L(0.1, 1.1) > f (0.1, 1. 1) L(0.1, 1.1) = f (0.1, 1. 1) L(0.1, 1.1) < f (0.1, 1.1 )2. Find an equation of the tangent plane to the surface defined by the equation12x2+xy+3yz2= 7 at thepoint (2, 1,1). (3 points)Equation: x+ y+ z =13. A tiny spaceship is orbiting on the path given by x2+y2=4. The solar radiation at a point (x, y) in the planeis f (x, y) =(x +2)y = xy+2y.(a) Use the method of Lagrange multipliers to find the maximum value M and minimum value m of solarradiation experienced by the tiny spaceship in its orbit. (5 points)M = m =(b) The orbit of the spaceship is shown below. On the same axes, graph the level set f (x, y) =m. (2 points)-4 -3 -2 -1 1 2 3 4-3-2-1123Tiny rocket from xkcd.com24. Let C be the curve in R3parameterized by r(t) =〈sin t,2t, cos t〉for 0 ≤ t ≤π/2.(a) Compute the length of C. (3 points)Length =(b) Evaluate the integral�Cx2z ds . (4 points)�Cx2z ds =(c) What is the average value of f (x, y,z) =x2z on the curve C? (1 point)Average =35. Let C be the curve in R2parameterized by r(t) =�−t2, t�for −1 ≤ t ≤0.(a) Mark the picture of C from among the choices below. (1 point)xy(−1,1)xy(−1,1)xy(−1,−1)xy(−1,−1)(b) For the vector field F =�y, x +3�directly calculate�CF·dr using the given parameterization. (4 points)�CF ·dr =(c) The vector field F is conservative. Find f : R2→R with ∇f =F. (2 points)f (x, y) =(d) Use your answer in part (c) to check your answer from part (b). (2 points)46. (a) Mark the picture of the curve in R3parameterized by r(t) =〈t cos t , t sint , t〉for 0 ≤ t ≤4π. (2 points)xyzxyzxyzxyz(b) Consider the three vector fields E, F, and G on R2shown below.E FG(i) One of these vector fields is yi +xj. Circle its name here: EFG (1 points)(ii) Exactly one of these vector fields is not conservative. Circle it here: EFG (1 points)(iii) Exactly one of the following is a flowline (also called a streamline or integral curve) for G parameter-ized by time for 0 ≤t ≤1. Circle it. (1 point)r(t ) =〈t,1−t〉r(t ) =�e−t,e−t�r(t ) =�t,�t�r(t ) =〈1 −t,1−t〉(c) Consider the curve C and vector field H shown at right.Is the integral�CH ·dr:positive negative zero (1 points)57. Find a parameterization r(t) of the curve of intersection of the cylinder (x −1)2+ y2= 1 and the planex −y +z =1; specify the range of the parameter t. (4 points)r(t ) =�,,�for ≤ t ≤8. Consider the function f (x, y) on the rectangle D = {0 ≤ x ≤ 3 and 0 ≤ y ≤ 2} whose contours are shownbelow right. For each part, circle the best answer. (1 point each)(a) The value of Duf (P) is:negative zero positive(b) The number of critical pointsof f in D is:0123(c) The integral�Cf ds is:negative zero positive(d) The integral�C∇f ·dr is:−4 −20240.51.01.52.02.53.00.51.01.52.012312f =−3f =−2f =−1f =0f =1f =2f =3f =4f =5f =0f =−1f =−2f =−1PuC(e) Mark the plot below of the gradient vector field ∇f
View Full Document
Unlocking...