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1 Name: Class: Date: ID: A Final Exam Review Short Answer 1. Find the distance between the sphere (x 1) + (y + 1) + z = 1 4 and the sphere (x 3) + (y + ) + (z + ) = 1. Find, a a + b, a b, a, and 3a + 4b given a = 3,, 1 and b =,6,7. 3. Given a = 1,1, b = 4,, and c = 5,, find s and t such that c = sa + tb. 4. Let =1 a and b = and the angle between a and b be 15. Find (i) a b (ii) b (3a + b) (iii) (a + b) (a b) (iv) a b 5. Let a = 1,, and suppose that b is a vector parallel to a. Find proj b a. 6. Find a b, where a and b are given in the figure. 7. Find the volume of the parallelepiped below given P = (1, 3, ), Q = (3, 1, 3), R = (, 1, 4), and S = ( 1,, 1). 8. Compute a b if =3, a b = 7, and a b =. 1

2 Name: ID: A 9. Find symmetric equations of the line passing through (, 3, 4) and parallel to the vector AB, where A and B are the points (, 1, 1) and (,, 3). 1. Determine whether the lines L 1 : x = 1 + 7t, y = 3 + t, z = 5 3t and L : x = 4 t, y = 4, z = 7 + t are parallel, intersecting or skew. If they intersect, find the point of intersection. 11. Find equations of all planes containing the x axis. 1. Find equations of all planes containing the y axis. 13. Find the angle between the lines x 1 = 1 y 3 = z 3 1 and x = y = z Find an equation of the plane containing P( 1,, 1) and the line x + 1 = y 5 = z Do the two lines x 1 ()=1,1,3 t + t 1,, and x = 1,1,4 + s,,1 intersect? If so, find the point of intersection. 16. Sketch the curve of intersection of the two surfaces z = y x and x + y = Let f(x, y) = 5 (a) Evaluate f 1, 1ˆ. (b) Find the domain of f. (c) Find the range of f. 18. If Q = 1, π,3 ˆ in cylindrical coordinates, find rectangular coordinates of Q. 19. Find cylindrical and spherical equations for the surface whose equation in rectangular coordinates is x =. Describe the surface.

3 Name: ID: A. Find the set of intersection of the surfaces whose equations in spherical coordinates are ρ = 4 and θ = π Find the set of intersection of the surfaces whose equations in spherical coordinates are ρ = 4 and θ = π 6.. Describe in words the solid represented in spherical coordinates by the inequality ρ Sketch the solid given in cylindrical coordinates by θ π,1 r 3, r z A curve is given by the vector equation r (t) = ( + cos t) i + (1 + sin t) j. Find a relation between x and y which has the same graph. 5. Consider the curve in the xy-plane defined parametrically by x = t 3 3t, y = t, z =. Sketch a rough graph of the curve. 6. Show that the curve with vector equation r (t) = cos t i + sin (t) j + sin t k is the curve of intersection of the surfaces (x 1) + y = 1 and x + y + z = 4 Use this fact to sketch the curve. 3

4 Name: ID: A 7. Let u()= t ti + sint j cos t k and v()= t i + t j tk. Find d dt È ÎÍ u() t v() t. 8. If u()= t t sint,t,t 3 and v()= t t sint,cos t, t 1 3, compute d dt u() t v() t ˆ and d dt u() t u() t ˆ. 9. A helix has radius 5 and height 6, and makes 4 revolutions. Find parametric equations of this helix. What is the arc length of the helix? 3. Find the center of the osculating circle of the parabola y = x at the origin. 31. Find the center of the osculating circle of the curve described by x = 4sint, y = 3t, z = 4costat,,4 ˆ. 3. Consider y = sinx, π < x < π. Determine graphically where the curvature is maximal and minimal. 33. Find a parametric representation for the surface consisting of that part of the elliptic paraboloid x + y + z = 4 that lies in front of the plane x =. 34. Find a parametric representation for the surface consisting of that part of the hyperboloid Ï x y + z = 1thatliesbelowthedisk Ô x,y ˆ Ì x + y ÓÔ 4 Ô Ô. 35. Are the two planes r 1 s,t ˆ = 1 + s + t,s t,1+ s and r s,t ˆ = + s + t,3+ t,s + 3t parallel? Justify your answer. 36. A picture of a circular cylinder with radius a and height h is given below. Find a parametric representation of the cylinder. 37. For the function z = x + y 1, sketch the level curves z = k for k =, 1,, and Describe the difference between the horizontal trace in z = k for the function z = f(x,y) and the contour curve f(x,y) = k 4

5 Name: ID: A 39. The graph of level curves of f(x,y) is given. Find a possible formula for f(x,y) and sketch the surface z = f(x,y). 4. Let f(x,y) = Ô Ì Ï x 4 + y 4 if (x,y) (,) (x + y ) ÓÔ if(x,y) = (,) If so, prove it. If not, demonstrate why not. Does this function have a limit at the origin? Ï x + xy 41. Determine if f(x,y) = Ô Ì x + y if (x,y) (,) ÓÔ 1 if(x,y) = (,) discontinuity. is everywhere continuous, and if not, locate the point(s) of 4. Let f(x,y,z) = (xyz). Find all second-order partial derivatives of f. 43. If z = x siny + ye x, find δz δx, δz δy, δ z δx, δ z δxδy,and δ z δy 44. Find δz δx for x 3 y z + sin(xyz) = 45. Let f(x,y,z) = (x y)(y z)(z x). Compute δf δx + δf δy + δf δz. 46. Show that f(x,y) = e y cos x xy satisfies Laplace Equation f xx + f yy =. 5

6 Name: ID: A Ï x 4 + y 47. If g(x,y) = Ô Ì x + y 4 if (x,y) (,) ÓÔ if(x,y) = (,) find g g (1,) and x y (1,). 48. If f(x,y) = 4 x, find f x (1,1) and f y (1,1) and interpret these numbers as slopes. Illustrate with sketches. 49. Consider a function of three variables P = f(a,r,n), where P is the monthly mortgage payment in dollars, A is the amount borrowed in dollars, r is the annual interest rate, and N is the number of years before the mortgage is paid off. (a) Suppose f(18,,6,3) = 18. What does this tell you in financial terms? (b) Suppose δf (18,,6,3) = What does this tell you in financial terms? δr (c) Suppose δf δn (18,,6,3) = What does this tell you in financial terms? 5. If f(x,y) = ye xy, find the values x for which f(x,5) = 5, and then find an equation of the plane tangent to the graph of f at (x,5,5). 51. The dimensions of a closed rectangular box are measured to be 6 cm, 4 cm, and 3 cm. The ruler that is used has a possible error in measurement of at most.1 cm. Use differentials to estimate the maximum error in the calculated volume of the box. 5. Suppose that z = x 3 y, where both x and y are changing with time. At a certain instant when x = 1 and y =, x is decreasing at the rate of units/s and y is increasing at the rate of 3 units/s. How fast is z changing at this instant? Is z increasing or decreasing? 53. Suppose that z = u + uv + v 3, and that u = x + 3xy and v = x 3y +. Find δz at (x,y) = (1,). δx 54. Let z = ln x + y, x = r cos θ, and y = r sinθ. Use the chain rule to find δz δz and δr δθ. 55. Let w = x 3 + y 3 + z 3, x = s + t, y = s t and z = st. Use the chain rule to show that s( δw δw ) + t( δs δt ) = 3x 3 + 6y 3 + 6z Let z = x y + xy and x = u + v, and y = u 3v. Show that δ z δuδv = 16x 6y. 6

7 Name: ID: A 57. Let f(x,y,z) = x y + y 3 z + xz 3 and let P(,1, 1). (a) Find the directional derivative at P in the direction of 1,,3. (b) In what direction does f increase most rapidly? (c) What is the maximum rate of change of f at the point P? 58. Find the directional derivative of f(x,y,z) = x + y z at the point (1,3,5) in the direction of a = i j + 4k 59. Let f(x,y,z) = x + y + xz. Find the directional derivative of f at (1,,) in the direction of the vector v = 1, 1,1. 6. Let the temperature in a flat plate be given by the function T(x,y) = 3x + xy. What is the value of the directional derivative of this function at the point (3. 6) in the direction v = 4i 3j? In what direction is the plate cooling most rapidly at (3, 6)? 61. Suppose that the equation F(x,y,z) = defines z implicitly as a function of x and y. Let (a,b,c) be a point such that F(a,b,c) = and F(a,b,c) =, 3,4. Find δz δz (a,b) and δx δy (a,b). ln((x + h) y) ln(x y) 6. Find lim h h 63. Find an equation of the tangent plane to the surface 4x y + 3z = 1 at the point (, 3,1). 64. Find an equation of the tangent plane to the surface z = f(x,y) = x 3 y 4 at the point ( 1,, 16). 65. Find the points on the hyperboloid of one sheet x + y z = 1 where the tangent plane is parallel to the plane x y + z = Let f(x,y) = x e y and P(,). (a) Find the rate of change in the direction of f(p). (b) Calculate D u f(p) where u is a unit vector making an angle θ = 135 with f(p). 67. Find the local maximum and minimum values and saddle points of the function f(x,y) = x 3 3xy y 3. 7

8 Name: ID: A 68. Use the level curves of f(x,y) shown below to estimate the critical points of f. Indicate whether f has a saddle point or a local maximum or minimum at each of those points. 69. Find the critical points (if any) for f(x,y) = 1 x + 1 y + xy and determine if each is a local extreme value or a saddle point. 7. Compare the minimum value of z and sketch a portion of the graph of z = 3x + 6x + y 8y near its lowest point. 71. Find the absolute maximum and minimum value of f(x,y) = x 3y x + 6y on the square region D with vertices (,), (,), (,), and (,). 7. Give an example of a non-constant function f(x, y) such that the average value of f over R = {(x,y) 1 x 1, 1 y 1} is. 73. Compute xy e xy 3 dxdy Evaluate the iterated integral x sinxy dy dx. 1 π / x 75. Use a double integral to find the volume of the solid bounded by the planes x + 4y + 3z = 1, x =, y =, and z =. 8

9 Name: ID: A 76. Find the volume of the solid under the surface z = x + y and lying above the region {(x,y) x 1,x y x }. 77. Write f(x,y) da as an iterated integral in polar coordinates, where R is the region shown below. R 78. Find the mass and center of mass of the lamina that occupies the region D = {(x,y) x, y 3} and has density function p(x,y) = y. 79. Find the moments of inertia I x, I y, and I for the lamina that occupies the region given by D = {(x,y) x, y 3}. 8. Find the y-coordinate of the centroid of the semiannular plane region given by 1 x + y 4, y. Sketch the plane region and plot the centroid in the graph. 81. Compute the area of that part of the graph of 3z = 5 + x 3/ + 4y 3/ which lies above the rectangular region in the first quadrant of the xy-plane bounded by the lines x =, x = 3, y =, and y = Find the area of that part of the plane x + 3y z + 1 = that lies above the rectangle [1, 4] [, 4]. 83. Find the area of that part of the surface z = x + y that lies above the triangle with vertices (, ), (1, 1), and (, 1). 84. Find the area of the surface with vector equation r (s, t) = s cos t, s sint, s, 1 s 5, t π. 85. Find the area of the surface with vector equation r (u,v) = u cosv, u sinv, u, u, v π. 86. Find the volume, using triple integrals, of the region in the first octant beneath the plane x + y + 3z = Use the method of iterated integration in order to evaluate the triple integral xdv N cut o from the first octant by the plane defined by x + y + z = 3. where N is the region 9

10 Name: ID: A 88. Evaluate e (x + y + z 3/ ) dv, where E is the solid bounded by the sphere x + y + z = 1. E 89. A sphere of radius k has a volume of 4 3 πk 3. Set up the iterated integrals in rectangular, cylindrical, and spherical coordinates to show this. π 5π /6 9. Sketch the region E whose volume is given by the integral ρ sinφ dρ dφ dθ. π /6 1/sinφ 91. Give a geometric description of the solid S whose volume in spherical coordinates is given by π π / V = ρ sinφ dρ dφ dθ. π /4 9. Use the change of variables x = au, y = bv, z = cw to evaluate ydv E ellipsoid x a + y b + z c = 1., where E is the solid enclosed by the 93. Compute the Jacobian of the transformation T given by x = 1 (u v), y = 1 (u + v), and find the image of S = {(u,v) u 1, v 1} under T. 94. Evaluate (x + y) e x y da,where R is the rectangular region bounded by the lines x + y =, x + y = 1, R x y =, and x y = 1. 1

11 Name: ID: A 95. Sketch the vector field F where F x,y ˆ = xi yj. 96. Sketch the vector field F where F x,y,z ˆ = zk. 97. Determine the points x,y,z ˆ where the gradient field f x,y,z ˆ for f x,y,z ˆ = xy + xz + yz has z-component. 98. Find a function of f x,y ˆ such that f = F x,y ˆ = x x + y ˆ 3, y x + y.. 3 ˆ 11

12 Name: ID: A 99. Evaluate the line integral xdy, where the curve C is given in the figure below. C 1. Determine whether F x, y, z ˆ = xy + 1n z, x + z cos yz ˆ, a potential function. x z + y cos yz ˆ ˆ is conservative and if so, find 11. Determine whether F x, y, z ˆ = x y, y, z is conservative and if so, find a potential function. 1. Evaluate e x ˆ 1 + cos x + y dx + c 1 + y + ye y ˆ + x if C is the path starting at, dy ˆ and going along the line segment from, ˆ to 1,1ˆ, and then along the line segment from 1,1ˆ to, ˆ. È 13. Evaluate 3x y + cos È x + yˆ C ÎÍ dx + x 3 y + cos x + yˆ ÎÍ dy if C is the closed path starting at,ˆ and moving clockwise around the square with vertices, ˆ, 1,ˆ, 1,1ˆ, and,1 ˆ. 14. Let C be the closed path from, ˆ to,4ˆ along y = x and back again from,4ˆ to, ˆ along y = x. Evaluate x 3 ˆ + y dx + x y ˆ C dy directly, and then using Green's Theorem. 15. Use Green's Theorem to evaluate the line integral along the given positively oriented curve: y tan 1 ˆ x dx + 3x + sin y C ˆ dy, where C is the boundary of the region enclosed by the parabola y = x and the line y = Use Green's Theorem to find the area of the region formed by the intersection of x + y and y 1. 1

13 Name: ID: A Evaluate 4 y 4 1 x ˆ ÿ 3 cos y 3 dx + xy C 3 + x x ˆ 4 y sin y 3 dy, if C is the path shown below. 18. Evaluate ÿ dx + 4xdy, if C is the path shown below, starting and ending at 1,1 C ˆ. 19. Find (a) the curl and (b) the divergence of the vector field F x, y, z ˆ = x yi + yz j + zx k. 11. Find (a) the curl and (b) the divergence of the vector field F x, y, z ˆ = sin xi + cos xj + z k Evaluate the surface integral F ds S for the vector field F x, y, z ˆ = x yi 3xy j + 4y 3 k where S is the part of the elliptic paraboloid z = x + y 9 that lies below the square x 1, y 1 and has downward orientation. 11. Evaluate the flux of the vector field F = i + j + 3k through the plane region with the given orientation as shown below. 13

14 Name: ID: A 113. Find the flux of the vector field F x, y, z ˆ = xi + yj + k across the paraboloid given by x = u cos v, y = u sin v, z = 1 u with 1 u, v π and upward orientation: 114. Use Stokes' Theorem to evaluate xy dx + yz dy + zx dz, where C is the triangle with vertices C 1,, ˆ,,1,ˆ, and,,1 ˆ, oriented counterclockwise as viewed from above Evaluate the flux integral xi yj + 3zk ˆ n ds over the boundary of the ball x + y + z 9. S 116. Let F x, y, z ˆ = x y i x zj + z yk and let S be the surface of the rectangular box bounded by the planes x =, x = 3, y =, y =, z =, and z = 1. Evaluate the surface integral F ds Let F x, y, z ˆ = zi + x j + y k x + y + z 3 ˆ and let S be the boundary surface of the solid Ô Ï E = Ô x, y, z ˆ Ì 1 x + y + z ÓÔ 4. Evaluate the surface integral Ô F S ds Find the flux of F x, y, z ˆ = x 3 i + xz j + 3y z k across the surface of the solid bounded by the paraboloid z = 4 x y and the xy plane Evaluate F S ds, where F x, y, z ˆ = is the sphere x + y + z = 9. x x + y + z ˆ 3 i + S y x + y + z ˆ 3 j + z x + y + z 3 ˆ k and S 14

15 Final Exam Review Answer Section SHORT ANSWER 1. ANS: The closest distance between the two spheres is 1:5. (Hint: The centers of two spheres are (1, 1,) and (3,, ) and distance between centers is 3. Sketch the two spheres.). ANS: = a 14, a + b = 3,8,6, a b = 3, 4, 8, a = 6,4,, and 3a + 4b = 9,3,5 3. ANS: s = 3, t = 1 4. ANS: (i) 3, (ii) 4 3 3, (iii) 3, (iv) ANS: proj b a = 1,, 6. ANS: 6,4,6 7. ANS: ANS: 1 9. ANS: x = y = z 4 1

16 1. ANS: Skew 11. ANS: by + cz = where b and c are not both zero. 1. ANS: ax + cz = where a and c are not both zero. 13. ANS: cos ANS: 3x y z + 6 = 15. ANS: 1 3 ˆ Yes;,1, ANS: 45.3 or.794 radians 17. ANS: (a) 5 (b) x and y can be any real numbers. (c) {} ANS: (, 1, 3)

17 19. ANS: Cylindrical: r cosθ = ; spherical: ρ sinφ cosθ =. ANS: A half circle radius 4 on the plane y = x 1. ANS: A circle radius 4 on the plane z = 3. ANS: solid sphere of radius 5 centered at origin, with hollow ball inside of radius 3. ANS: 4. ANS: (x- ) + (y 1) = 1 3

18 5. ANS: 6. ANS: 7. ANS: d È u() t v() t dt ÎÍ = tcost sint t ˆ sint i + ( 4t + sint )j + 6t ˆ cost k 8. ANS: d dt u() t v() t ˆ =, d dt u() t u() t ˆ = sin t + t sint cos t + t t 3 4

19 9. ANS: r()= t 5cosπ t,5sinπ 3 t, t 4; L = π ( ) ANS:, 1 ˆ 31. ANS:,,.5 ˆ 3. ANS: Minimal at x =, maximal at x =± π 33. ANS: x = 4 y z, y = y, z = z where y + z 4 since x. 34. ANS: x = r cos θ, y = r sinθ, z = 1 + r, θ π, r 35. ANS: Yes 36. ANS: r s,t ˆ = a coss,a sins,t, s π, t h. Other answers are possible. 5

20 37. ANS: 38. ANS: The contour curve f(x,y) = k is the horizontal trace of z = f(x,y) in z = k projected down to the xy-plane. 39. ANS: z = y. There are other possible answers. 4. ANS: Approaching the origin along y =, the limit equals 1; approaching the origin along y = x, the limit equals 1. Thus, this function does not have a limit at the origin. 41. ANS: f is continuous everywhere its denominator does not equal zero. The limit does not exist at (,) and thus f is discontinuous at (,). 6

21 4. ANS: f xy = 4xyz, f yx = 4xyz, f zx = 4xy z, f xz = 4xy z, f yz = 4x yz, f zy = 4x yz, f xx = y z, f yy = x z, f zz = x y 43. ANS: δz δx = x siny + yex, δz δy = x cos y + e x, δ z δx 44. ANS: δz δx = 3x yz cos(xyz) y + xy cos(xyz) 45. ANS: δf = (y z)(z x) (x y)(y z) δx δf δy δf δz = (y z)(z x) + (x y)(z x) = (x y)(y z) (x y)(z x) δf δx + δf δy + δf δz = 46. ANS: = siny + yex, δ z δxδy = x cosy + e x,and δ z δy f x = e y sinx y, f y = e y cosx x f xx = e y cosx, f yy = e y cos x f xx + f yy = 47. ANS: δg δg (1,) =, δx δy (1,) = = x siny 7

22 48. ANS: f x (1,1) =, f y (1,1) = 49. ANS: (a) If you borrow $18, with annual interest rate 6% and 3 year amortization, then your monthly payment is $18. (b) If you borrow $18, with annual interest rate 6% and 3 year amortization, then your monthly payment increases by $ per 1% increase in interest rate. (c) If you borrow $18, with annual interest rate 6% and 3 year amortization, then your monthly payment decreases by $1.86 for each additional year of amortization. 5. ANS: x = ; z = 5x + y 51. ANS: 54 cm 3 5. ANS: dz = 1, z is decreasing dt 53. ANS: ANS: δz δr = x cosθ x + y + y sinθ x + y = 1 r, δz δθ = x sinθ + y cosθ =. x + y x + y 8

23 55. ANS: Answers may vary 56. ANS: Answers may vary 57. ANS: 6 (a) 14 (b) 3,1,7 (c) ANS: ANS: ANS: 6 ; the plate is cooling most rapidly in the direction i j ANS: δz δx (a,b) = 1, δz δy (a,b) = ANS: x 63. ANS: 16x + 6y + 6z = 9

24 64. ANS: 48x 3y z + 96 = 65. ANS: 1, 1, 1 ˆ, 1, 1, 1 ˆ 66. ANS: (a) 4 (b) D u f(p) = f(p) cos 135 =4 ˆ = ANS: (,) is a saddle point; f( 1,1) = 1 is a local maximum 68. ANS: f has a saddle point at (,) and maximum points at ( 1, 1) and (1,1). 69. ANS: f(1,1) = 3 is a local minimum. 7. ANS: z( 1,) = 11 is the absolute minimum value. 1

25 71. ANS: Absolute maximum value is, and absolute minimum is. 7. ANS: Possible functions include f (x, y) = x and f (x, y) = y. Any linear function of x and y with no constant term will work, as will many other functions. 73. ANS: 1 (e ) ANS: ANS: ANS: ANS: π 4sinθ f(r cos θ,r sinθ) rdrdθ 78. ANS: m = 9, (x, y) = (1, ) 79. ANS: I x = 18, I y = 8,I = 6 11

26 8. ANS:, 8 ˆ 9π 81. ANS: 4 ( ) ANS: ANS: ANS: 4 π 85. ANS: π( ) ANS: ANS: 7 8 1

27 88. ANS: 4π(e 1) ANS: Rectangular coordinates: V = k k k x k x k x y k x y dzdydx ; cylindrical coordinates: k r V = rdzdrdθ ; spherical coordinates: V = ρ sinφ dρ dφ dθ π k k r π π k 9. ANS: A sphere of radius with a hole of radius 1 drilled through the center. 91. ANS: A hemisphere with a cone cut out. 13

28 9. ANS: 93. ANS: J = 1, image of S = {(x,y) 1 x 1, x y x } 94. ANS: e 95. ANS: 96. ANS: 97. ANS: All points of the form a, a,bˆ 14

29 98. ANS: f x,y ˆ = 99. ANS: 9 1 x + y ˆ + C 1/ 1. ANS: f (x, y, z) = x y + sin (yz) + x ln z 11. ANS: F is not conservative. 1. ANS: e + sin ANS: 14. ANS: ANS: ANS: π ANS: 15

30 18. ANS: ANS: (a) yzi + xzj + x ˆ k (b) xy + z + x 11. ANS: (a) sin x k (b) cos x + z 111. ANS: ANS: ANS: 18π 114. ANS: ANS: 144π 116. ANS: ANS: 16

31 118. ANS: 3π 119. ANS: 4π 17

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