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MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the linearization of the function at the given point. 1) f(x, y) = 2x - 7y - 2 at (-9, -10) A) L(x, y) = -9x - 10y - 2 B) L(x, y) = -18x + 70y -.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Apply Green's Theorem to evaluate the integral. 1) (8y + x) dx + (y + 4x) dy C: The circle (x - 1)2 + (y - 3)2 = 4 A) -16 B) -60 C) -16π D) 16π 2) (y2 + 2) dx +.
MULTIPLECHOICE. Choosetheonealternativethat best completes thestatement or answers the question. Find (δf/δx) and (δf/δy). 1) f(x, y) = 4x - 6y2 - 9 A) (∂f/∂x) = -12y; (∂f/∂y) = 4 B) (∂f/∂x) = 4; (∂f/∂y) = -12y C) (∂f/∂x) = 4x; (∂f/∂y) = -12y D) (∂f/∂x) = -5; (∂f/∂y) = -12y - 9 2) f(x, y) = (6x5y3.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. 1) Find the centroid of the region cut from the first quadrant by the line x + y = 9. A) = 3, = 3 B) = 6, = 6 C) = (9/2), = (9/2) D) = (9/4),.
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Find two paths of approach from which one can conclude that the function has no limit as (x, y) approaches (0, 0). 1) f(x, y) = (9y/√(10x2 + 9y2)) 2) f(x, y) = (x5 - y/x5 +.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Set up an integral for the area of the surface generated by revolving the given curve about the indicated axis. 1) y = sinx, 0 ≤ x ≤ π/4; x-axis A) dx B) 2πdx C) 2πdx D) dx 2) y = x5,.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Evaluate the integral. 1) A) 64π2 B) (256/3)π2 C) (512/3)π2 D) (512/9)π2 2) A) (243/8)π2 B) (243/8)π C) (81/2)π D) (81/2)π2 3) A) (854,296,875/16)π B) (43,046,721/32)π C) (6561/16)π D) 69,984π 4) A) (7/4)π B) (7/2)π C) (7/3)π D) (7/6)π 5) A) (1/32)π(15π - 16) B) (1/64)π(15π - 8) C) (1/64)π(15π - 16) D) (1/32)π(15π - 8) 6).
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Evaluate the cylindrical coordinate integral. 1) A) 30π B) 270π C) 540π D) 90π 2) A) 16π B) 64π C) 128π D) 8π 3) A) 1458π B) 2187π C) 7290π D) 2916π 4) A) 945,750 B) (2,994,875/2) C) (788,125/3) D) (788,125/2) 5) A) 40 B) 20π C) 60 D) 60π 6) A) 6π + ln2 B) 12 -.
Find the volume of the solid generated by revolving the region about the given axis. Use the shell or washer method. 15) The triangle with vertices (0, 0), (0, 2), and (2, 2) about the line x = 2 A) (20/3)π B) (16/3)π C) (4/3)π D) (8/3)π 16) The region bounded by x = 3√(y), x.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. 1) A solid of constant density is bounded below by the plane z = 0, above by the cone z = r, and on the sides by the cylinder r = 8. Find the.
Evaluate the integral by changing the order of integration in an appropriate way. 41) A) 1 - sin9 B) (1 + sin9/2) C) 1 + cos9 D) (1 - cos9/2) 42) A) 2sin10 B) 2(1 - cos10) C) 2cos10 D) 2(1 + sin10) 43) A) 6ln4lncos7 B) - 6ln4lncos7 C) 6ln2lncos7 D) - 6ln2lncos7 44) A) - 9ln2lncos7 B) (9/2)ln2lncos7 C) 9ln2lncos7 D) - (9/2)ln2lncos7 45) A).
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the Jacobian (∂(x, y)/∂(u, v)) or (∂(x, y, z)/∂(u, v, w)) (as appropriate) using the given equations. 1) x = -4u + 5v, y = 3u - 5v A) -5 B) -35 C) 35 D) 5 2) u = -5x -.
Reverse the order of integration and then evaluate the integral. 24) A) cos8 B) - cos8 C) 1 - cos8 D) 1 + cos8 25) A) 9cos10 B) 10sin9 C) 9sin10 D) 10cos9 26) A) (7/4)e9 B) (7/2)e9 C) (7/4)(e9 - 1) D) (7/2)(e9 - 1) 27) A) (33/2) B) (27/2) C) 10 D) 20 28) A) (1/3)(2e - 1) B) (2/3)(e - 1) C) (2/3)(2e - 1) D) (1/3)(e.
Solve the problem. 21) Find the average height of the paraboloid z = x2 + y2 above the disk x2 + y2 ≤ 81 in the xy-plane. A) 27 B) (243/2) C) (405/2) D) (81/2) 22) Find the average height of the paraboloid z = 8x2 + 9y2 above the disk x2 + y2 ≤ 100.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find all the local maxima, local minima, and saddle points of the function. 1) f(x, y) = x2 - 6x + y2 - 16y - 6 A) f(3, 8) = -79, local minimum B) f(3, -8) = 177, saddle.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Calculate the circulation of the field F around the closed curve C. 1) F = - (1/2)x2yi - (1/2)xy2j; curve C is r(t) = 4costi + 4sintj, 0 ≤ t ≤ 2π A) - 2 B) 0 C) - 1 D).
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Match the vector equation with the correct graph. 1) r(t) = 3i - 2j - tk; -1 ≤ t ≤ 1 A) Figure 8 B) Figure 6 C) Figure 3 D) Figure 5 2) r(t) = 2costi +sintk; 0 ≤ t ≤.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the gradient field of the function. 1) f(x, y, z) = x8y4 + (x6/z10) A) ∇f = (8x7 + 6x5)i + 4y3j - (10/z11)k B) ∇f = (8x7 + 6x5)i + 4y3j + (10/z11)k C) ∇f = 8x7y4i +.
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Use the limit definition of the partial derivative to compute the indicated partial derivative of the function at the specified point. 1) Find (∂f/∂x)at the point (-3, -8): f(x, y) = 2x2 + 5xy + 8y2.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the extreme values of the function subject to the given constraint. 1) f(x, y) = 6x2 + 4y2, x2 + y2 = 1 A) Minimum: 4 at (0, ±1); maximum: 0 at (0, 0) B) Minimum: 4 at.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the specific function value. 1) Find f(-2, -3) when f(x, y) = 5x + 2y - 8. A) -16 B) -24 C) -29 D) -26 2) Find f(3, 4) when f(x, y) = 5y2 - 8xy. A) -51 B) -19 C) -16 D) -47 3) Find.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. 1) Evaluate (dw/dt) at t = (1/2)π for the function w = x2 - y2 + 6x; x = cost, y = sint. A) 2 B) 8 C) 4 D) -6 2) Evaluate (dw/dt) at t = 10 for.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. 1) Find the equation for the tangent plane to the surface 6x + 2y - 2z = 0 at the point (1, -1, 2). A) 6x - 2y - 4z = 6 B) 6x - 2y.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Compute the gradient of the function at the given point. 1) f(x, y) = -7x - 9y, (-7, -10) A) ∇f = -35i - 90j B) ∇f = -35i + 90j C) ∇f = 5i - 10j D) ∇f = -7i.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Express the area of the region bounded by the given line(s) and/or curve(s) as an iterated double integral. 1) The coordinate axes and the line x + y = 7 A) B) C) D) 2) The coordinate axes.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Using Green's Theorem, compute the counterclockwise circulation of F around the closed curve C. 1) F = (x2 + y2)i + (x - y)j; C is the rectangle with vertices at (0, 0), (2, 0), (2, 4),.
Find the flux of the vector field F across the surface S in the indicated direction. 20) F = 7xi + 7yj + 2k; S is "nose" of the paraboloid z = 7x2 + 7y2 cut by the plane z = 2; direction is outward A) (12/7) B) (24/7)π C) (32/7)π D) - (32/7)π 21) F(x,.
Solve the problem. 54) What region in the xy-plane maximizes the value of ∫∫(4900 - 100x2 - 49y2) dA? R A) The ellipse (x2/49) + (y2/100) = 1 B) The ellipse 49x2 + 100y2 = 4900 C) The ellipse 100x2 + 49y2 = 1 D) The ellipse (x2/100) + (y2/49) = 1 55) What region in the.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the length of the curve. 1) y = 4x3/2 from x = 0 to x = (5/16) A) (335/432) B) (9/16) C) (335/3) D) (335/288) 2) y = (4 - x2/3)3/2 from x = 1 to x = 8 A) 9 B) 12 C).
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the divergence of the field F. 1) F = -21xz6i + 3yj + 3z7k A) -42z6 + 3 B) -21z6 + 3 C) -42z6 D) 3 2) F = -7x3i - 3xyj - 5xzk A) -21x2 - 8x B) -21x2 - 3y -.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Evaluate. The differential is exact. 1) A) 3 B) (1/3) C) 1 D) 0 2) A) 2 B) 1 C) -2 D) 0 3) A) 0 B) -21 C) -42 D) 29 4) dz)) A) e18 - 1 B) e18 - 3 C) 0 D) e9 + e4 + e5 - 1 5) A).
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the limit. 1) A) 1 B) -1 C) 2/3 D) No limit 2) A) Ö143 B) Ö143/143 C) 143 D) No limit 3) ln|x + y/xy| A) -ln 2 B) ln 2 C) 0 D) No limit 4) A) 0 B) ∞ C) 1 D) No limit 5) sin(x2 + y7/ x –.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Set up the iterated integral for evaluating ∫∫∫f(r, θ, z) dzrdrdθ over the given region D. D 1) D is the right circular cylinder whose base is the circle r = 5sinθ in the xy-plane and whose top.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. 1) Set up the triple integral for the volume of the sphere ρ = 10 in spherical coordinates. A) B) C) D) 2) Set up the triple integral for the volume of the sphere ρ.
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Parametrize the surface S. 1) S is the portion of the plane 4x + 7y + 8z = -1 that lies within the cylinder x2 + y2 = 4. 2) S is the portion of the cone.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Change the Cartesian integral to an equivalent polar integral, and then evaluate. 1) A) (27π/2) B) (π/2) C) (3π/2) D) (9π/2) 2) A) (625π/8) B) (125π/8) C) (125π/4) D) (25π/8) 3) A) 32π B) 16π C) 4π D) 64π 4) A) (π(3 - ln4)/2) B) (π(3 + ln4)/4) C) (π(3 -ln4)/4) D).
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Evaluate the integral. 1) A) 1 B) 34 C) 22 D) 80 2) A) - 36 B) - 4 C) - 2 D) - 30 3) A) - 3456 B) - 2304 C) - 1152 D) 576 4) A) 7 B) 280 C) 42 D) 1680 5) A) - 5 B) 34 C) - 3 D).
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Evaluate the surface integral of G over the surface S. 1) S is the parabolic cylinder y = 4x2, 0 ≤ x ≤ 5 and 0 ≤ z ≤ 2; G(x, y, z) = 5x A) (5/96)(1601√(1601) +.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Evaluate the integral. 1) A) 5 B) - (7/2) C) (9/2) D) - 3 2) A) (625/3) B) (625/4) C) (125/3) D) (125/4) 3) A) (4096/15) B) (4096/7) C) (8192/15) D) (8192/7) 4) A) (2/6) B) (2/5) C) (1/6) D) (1/5) 5) A) (364/3) B) (350/9) C) (350/3) D) (364/9) 6) A) 8 B) 1 C) 2 D) 4 7) A).
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Use Stokes' Theorem to calculate the circulation of the field F around the curve C in the indicated direction. 1) F = 2yi + yj + zk; C: the counterclockwise path around the boundary of the ellipse.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the surface area of the surface S. 21) S is the paraboloid x2 + y2 - z = 0 below the plane z = 12. A) (343/6)π B) (57/2)π C) 57π D) (171/2)π 22) S is the paraboloid x2 + y2.
Find the volume of the indicated region. 21) the tetrahedron cut off from the first octant by the plane (x/5) + (y/6) + (z/7) = 1 A) 35 B) (105/2) C) 105 D) 70 22) the region bounded by the paraboloid z = 64 - x2 - y2 and the xy-plane A) 256π B) 2048π C) (4096/3)π D) (512/3)π 23) the.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Use Taylor's formula to find the requested approximation of f(x, y) near the origin. 1) Quadratic approximation to f(x, y) = sin(8x + y) A) 8x + y + 4x2 + y2 B) 8x + y + 4x2.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. 1) Write an iterated triple integral in the order dzdydx for the volume of the rectangular solid in the first octant bounded by the planes x = 6, y = 4, and z =.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Sketch the surface z = f(x,y). 1) f(x, y) = x2 A) B) C) D) 2) f(x, y) = -x2 – y2 A) B) C) D) 3) f(x, y) = 3 – x2 A) B) C) D) 4) f(x, y).
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the center of mass of a thin plate of constant density covering the given region. 1) The region bounded by y = x2 and y = 3 A) = 0, = 27/5 B) = 0, = 18/5 C) =.
Solve the problem. 21) A swimming pool has a rectangular base 14 ft long and 28 ft wide. The sides are 5 ft high, and the pool is full of water. How much work will it take to lower the water level 2 feet by pumping the water out over the.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. 1) The spring of a spring balance is 6.0 in. long when there is no weight on the balance, and it is 8.3 in. long with 9.0 lb hung from the balance. How much.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Test the vector field F to determine if it is conservative. 1) F = 7x6y7z7i + 7x7y6z7j + 7x7y7z6k A) Conservative B) Not conservative 2) F = x7y7z7i + x7y7z7j + z7k A) Not conservative B) Conservative 3) F = 6x6y5i + (5x6y4.
Use the given transformation to evaluate the integral. 21) u = x + y, v = x - y; ∫∫(x + y)sin(x - y) dx dy, R where R is the parallelogram bounded by the lines y = x, y = x - 10, y = -x, y = -x + 5 A) (25/4)(1 -.
