1. Consider the vectors: a = h1,0,°1i b = h1,1,1i c = h°1,1,0i.(a) Compute a £ b. (3 points)a £ b =D,,E(b) Compute the volume of the parallelepiped determined by the vectors a,b, and c. (2 points)Volume =2. Given that u and v in the picture at left have length 1, compute u · v, u · w, and projvw. (1 point each)uvwu · v = u · w = projvw =3. A particle moves with constant velocity h3,1,°1i starting from the point (3,2,4) at time t = 0. When andwhere will it cross the xy-plane? (3 points)When: t = Where:≥,,¥14. Let A be the plane given by x ° z = 1 and B the plane given by x + y + z = 2.(a) Find a normal vector n for the plane A. (1 points)n =D,,E(b) Find the angle between the two planes. (2 points)µ =(c) Find the equation of a plane C which is perpendicular to both A and B. (3 points)Equation: x+ y+ z =5. Exactly one of the following two limits exists. Circle the one that exists and justify your answer. (5 points)lim(x,y)!(0,0)√x2° y2px2+ y2!lim(x,y)!(0,0)√xy°x2+ y2¢2!26. For each function label its graph from among the options below: (a) (x + y)2(b) x +cos(y)(3 points each)7. Circle the equation for the quadratic surface shown at ri ght. (3 points)(a) x2+ y2+ z2= 1(b) x2° y2° z2= °1(c) x2+ y2° z2= °1(d) x2° y2° z2= 1(e) x ° y2° z2= 1xyz38. Is the function f : R2! R given at right continuous at (0,0)? Justify your answer. (2 points)f (x, y) =(x2+ y + 1 if (x, y) 6= (0,0),0if(x, y) = (0,0).9. Let f (x, y) be a function with values and derivatives in the table. Use linear approximation to estimatef (2.1,0.9). (3 points)(x, y) f ( x, y)@ f@x(x, y)@ f@y(x, y)(°1,3) 0 4 4(2,1) 2 -1 3(2,4) 3 7 7(3,6) 1 -3 -5f (2.1,0.9) º10. Suppose f (x, y) has the contour plot below right, with points labelled. Circle the best answer to each of thefollowing questions: (1 point each)(a) f ( a) is: positive negative 0(b)@ f@x(b) is: positive negative 0(c)@2f@2y(c) is: positive negative 0(d) Circle one:@ f@x(a) >@ f@x(b)@ f@x(a) <@ f@x(b)°4°3°2°1012354cabxy411. An exceptionally tiny spaceship positioned as shown is travelling so that its x-coordinate increases at a rateof 1/2 m/s and y-coordinate increases at a rate of 1/3 m/s. Use the Chain Rule to calculate the rate at whichthe distance between the spaceship and the point (0, 0) is increasing. (6 points)xy12341 2 3 4 5Distances in metersRocket courtesy of xkcd.comrate = m/s12. Let f : R2! R be the function whose graph is shown at right.(a) Find the equation of the tangent plane to the graph at(0,0,0). (2 points)(b) The partial derivative@2f@x@y(0,0) is (circle your answer):positive negative 0 (1 point)xyz513. Extra Credit Problem. Suppose f : R2! R is continuous at (0, 0) with f (0,0) = 2 and partial derivativesfx(0,0) = 1 and fy(0,0) = °1. If in additionlimt!0f (t , t ) ° 2t= 1can f be differentiable at (0, 0)? Carefully justify your answer. (3 points)Scratch work may go below and on the back of thi s
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