1. If the line l has symmetric equations. = y 3 = z+2 find a vector equation for the line l that contains the point (2, 1, 3) and is parallel to l.

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1 . If the line l has symmetric equations MA 6 PRACTICE PROBLEMS x = y = z+ 7, find a vector equation for the line l that contains the point (,, ) and is parallel to l. r = ( + t) i t j + ( + 7t) k B. r = ( + t) i j + (7 t) k r = ( + t) i + ( t) j + ( + 7t) k r = ( + t) i + ( + t) j + (7 t) k r = ( + t) i + j + (7 t) k. Find parametric equations of the line containing the points (,, ) and (,, 5). x = t, y = + 4t, z = 5t B. x = t, y = t, z = x = t, y = + t, z = 5t x = t, y = t, z = 5t x = + t, y = t, z = 5. Find an equation of the plane that contains the point (,, ) and has normal vector i + j + k. x y z + 9 = B. x + 4y + 6z + 9 = x x y z = x + y + z = = y+ = z+ 4. Find an equation of the plane that contains the points (,, ), ( 5,, ), and (,, 4). 6x y + z = 5 B. 6x + y + z = 5 x 6y + z = r = 8 i j + k x 6y z = 5. Find parametric equations of the line tangent to the curve r(t) = t i + t j + t k at the point (, 4, 8) x = + t, y = 4 + 4t, z = 8 + t B. x = + t, y = 4 + 4t, z = + 8t x = t, y = 4t, z = 8t x = t, y = 4t, z = t x = + t, y = 4 + t, z = 8 + t 6. The position function of an object is r(t) = cos t i + sin t j t k Find the velocity, acceleration, and speed of the object when t = π. Velocity Acceleration Speed i π k j π k + π 4 B. i j + π k i k + 4π j π k i k 9 + 4π j π k i k 9 + 4π i k j π k 5

2 7. A smooth parametrization of the semicircle which passes through the points (,, 5), (,, 5) and (,, 5) is r(t) = sin t i + cos t j + 5 k, t π B. r(t) = cos t i + sin t j + 5 k, t π r(t) = cos t i + sin t j + 5 k, π t π r(t) = cos t i + sin t j + 5 k, t π r(t) = sin t + cos t j + 5 k, π t π 8. The length of the curve r(t) = ( + t) i + ( t) j + t k, t is B. 9. The level curves of the function f(x, y) = x y are circles B. lines parabolas hyperbolas ellipses. The level surface of the function f(x, y, z) = z x y that passes through the point (,, ) intersects the (x, z)-plane (y = ) along the curve z = x + 8 B. z = x 8 z = x + 5 z = x 8 does not intersect the (x, z)-plane. Match the graphs of the equations with their names: () x + y + z = 4 (a) paraboloid () x + z = 4 (b) sphere () x + y = z (c) cylinder (4) x + y = z (d) double cone (5) x + y + z = (e) ellipsoid b, c, d, 4a, 5e B. b, c, a, 4d, 5e e, c, d, 4a, 5b b, d, a, 4c, 5e d, a, b, 4e, 5c. Suppose that w = u /v where u = g (t) and v = g (t) are differentiable functions of t. If g () =, g () =, g () = 5 and g dw () = 4, find when t =. dt 6 B. / 4 4. If w = e uv and u = r + s, v = rs, find w r. e (r+s)rs (rs + r ) B. e (r+s)rs (rs + s ) e (r+s)rs (rs + r ) e (r+s)rs ( + s) e (r+s)rs (r + s ).

3 4. If f(x, y) = cos(xy), f x y = xy cos(xy) B. xy cos(xy) sin(xy) sin(xy) xy cos(xy) + sin(xy) cos(xy) 5. Assuming that the equation xy + z = cos(z ) defines z implicitly as a function of x and y, find z x. y sin(z ) B. y +sin(z ) y +z sin(z ) y +z sin(z ) y z sin(z ) 6. If f(x, y) = xy, then f(, ) = i + 9 j B. 8 i + 8 j 9 i + j. 7. Find the directional derivative of f(x, y) = 5 4x y at (x, y) towards the origin 8x B. 8x y 8x x +y 64x +9 8x + y 8x +y. x +y 8. For the function f(x, y) = x y, find a unit vector u for which the directional derivative D u f(, ) is zero. i + j B. i+ j i j i j i j. 9. Find a vector pointing in the direction in which f(x, y, z) = xy 9xz + y increases most rapidly at the point (,, ). i + 4 j B. i + j 4 i j i + k i + j.. Find a vector that is normal to the graph of the equation cos(πxy) = at the point ( 6, ). 6 i + j B. i j i + j j i j.. Find an equation of the tangent plane to the surface x + y + z = 6 at the point (,, ). x + y + z = B. x + 4y 6z = 6 x y + z = 4 x + 4y 6z = x + y z = 6.. Find an equation of the plane tangent to the graph of f(x, y) = π + sin(πx + y) when (x, y) = (, π). 4πx + y z = 9π B. 4x + πy z = π 4πx + πy + z = π 4x + πy z = 9π 4πx + y + z = 9π.

4 . The differential df of the function f(x, y, z) = xe y z is df = xe y z dx + xe y z dy + xe y z dz B. df = xe y z dx dy dz df = e y z dx xye y z dy + xze y z dz df = e y z dx + xye y z dy xze y z dz df = e y z ( + xy xz) 4. The function f(x, y) = x 6xy y has a relative minimum and a saddle point B. a relative maximum and a saddle point a relative minimum and a relative maximum two saddle points two relative minima. 5. Consider the problem of finding the minimum value of the function f(x, y) = 4x + y on the curve xy =. In using the method of Lagrange multipliers, the value of λ (even though it is not needed) will be B Evaluate the iterated integral x x dydx. 8 9 B. ln ln. 7. Consider the double integral, R f(x, y)da, where R is the portion of the disk x + y, in the upper half-plane, y. Express the integral as an iterated integral. x x f(x, y)dydx B. x f(x, y)dydx x f(x, y)dydx. x f(x, y)dydx x f(x, y)dydx x 8. Find a and b for the correct interchange of order of integration: x x f(x, y)dydx = 4 b a f(x, y)dxdy. a = y, b = y B. a = y, b = y a = y, b = y a = y, b = y cannot be done without explicit knowledge of f(x, y). 9. Evaluate the double integral R yda, where R is the region of the (x, y)-plane inside the triangle with vertices (, ), (, ) and (, ). B. 8.. The volume of the solid region in the first octant bounded above by the parabolic sheet z = x, below by the xy plane, and on the sides by the planes y = and y = x is given by the double integral x ( x )dydx B. x ( x )dydx x x dydx. x x dydx x x ( x )dydx 4

5 . The area of one leaf of the three-leaved rose bounded by the graph of r = 5 sin θ is 5π 6 B. 5π 5π 6 5π 5π.. Find the area of the portion of the plane x + y + z = 6 that lies in the first octant. B A solid region in the first octant is bounded by the surfaces z = y, y = x, y =, z = and x = 4. The volume of the region is 64 B An object occupies the region bounded above by the sphere x + y + z = and below by the upper nappe of the cone z = x + y. The mass density at any point of the object is equal to its distance from the xy plane. Set up a triple integral in rectangular coordinates for the total mass m of the object x 6 x x y x +y z dz dy dx B. 4 6 x 4 x y z dz dy dx 6 x x +y 4 x x y z dz dy dx 4 x x +y 4 6 x 4 x y xy dz dy dx. 6 x x +y 4 6 x x y x +y z dz dy dx 5. Do problem 4 in spherical coordinates. π π π π 4 ρ cos ϕ sin ϕ dρ dϕ dθ B. π 4 ρ sin ϕ dρ dϕ dθ π 4 ρ cos ϕ dρ dϕ dθ. π π π 4 ρ cos ϕ sin ϕ dρ dϕ dθ π ρ cos ϕ sin ϕ dρ dϕ dθ 6. The double integral x y (x + y ) dydx when converted to polar coordinates becomes π π r9 sin θ dr dθ B. r8 sin θ dr dθ 7. Which of the triple integrals converts 4 x x dz dy dx 4 x +y from rectangular to cylindrical coordinates? π π r9 sin θ dr dθ. r8 sin θ dr dθ π r8 sin θ dr dθ π r r dz dr dθ B. π r r dz dr dθ π r r dz dr dθ π r r dz dr dθ π r r dz dr dθ. 8. If D is the solid region above the xy-plane that is between z = 4 x y and z = x y, then D x + y + z dv = 4π B. 6π 5π 8π 5π. 5

6 9. Determine which of the vector fields below are conservative, i. e. F = grad f for some function f.. F (x, y) = (xy + x) i + (x y y ) j.. F (x, y) = x y i + y x j.. F (x, y, z) = ye z i + (xe z + e y ) j + (xy + )e z k. and B. and and only all three 4. Let F be any vector field whose components have continuous partial derivatives up to second order, let f be any real valued function with continuous partial derivatives up to second order, and let = i x + j y + k z. Find the incorrect statement. curl(grad f) = B. div(curl F ) = grad(div F ) = curl F = F div F = F 4. A wire lies on the xy-plane along the curve y = x, x. The mass density (per unit length) at any point (x, y) of the wire is equal to x. The mass of the wire is (7 7 )/ B. (7 7 )/8 7 7 ( 7 )/ ( 7 )/ 4. Evaluate C F d r where F (x, y) = y i + x j and C is composed of the line segments from (, ) to (, ) and from (, ) to (, ). B Evaluate the line integral C x dx + y dy + xy dz where C is parametrized by r(t) = cos t i + sin t j + cos t k for π t. B. 44. Are the following statements true or false?. The line integral C (x + xy)dx + (x y )dy is independent of path in the xy-plane.. C (x + xy)dx + (x y )dy = for every closed oriented curve C in the xy-plane.. There is a function f(x, y) defined in the xy-plane, such that grad f(x, y) = (x + xy) i + (x y ) j. all three are false B. and are false, is true and are true, is false is true, and are false all three are true 45. Evaluate C y dx + 6xy dy where C is the boundary curve of the region bounded by y = x, y = and x = 4, in the counterclockwise direction. B

7 46. If C goes along the x-axis from (, ) to (, ), then along y = x to (, ), and then back to (, ) along the y-axis, then C xy dy = x y dy dx B. x x dy dx x y dy dx x x dy dx 47. Evaluate C F d r, if F (x, y) = (xy ) i + (x y x) j and C is the circle of radius centered at (, ) and oriented counterclockwise. B. π π 48. Green s theorem yields the following formula for the area of a simple region R in terms of a line integral over the boundary C of R, oriented counterclockwise. Area of R = R da = C y dx B. C y dx C x dx C y dx x dy x dy 49. Evaluate the surface integral Σ x ds where Σ is the part of the plane x + y + z = 4 in the first octant B If Σ is the part of the paraboloid z = x + y with z 4, n is the unit normal vector on Σ directed upward, and F (x, y, z) = x i + y j + z k, then Σ F n ds = B. 8π 4π 4π 8π 5. If F (x, y, z) = cos z i + sin z j + xy k, Σ is the complete boundary of the rectangular solid region bounded by the planes x =, x =, y =, y =, z = and z = π, and n is the outward unit normal on Σ, then Σ F n ds = B. π 5. If F (x, y, z) = x i + y j + z k, Σ is the unit sphere x + y + z = and n is the outward unit normal on Σ, then Σ F n ds = 4π B. π 4π 4π 7

8 ANSWERS C, A, B, 4 B, 5 A, 6 D, 7 B, 8 D, 9 E, B, A, E, B, 4 B, 5 D, 6 C, 7 E 8 D, 9 A, C, E, A, D, 4 B, 5 E, 6 B, 7 C, 8 B, 9 E, A, B, D, B, 4 B, 5 A, 6 E, 7 B, 8 C, 9 B, 4 C, 4 A, 4 D, 4 D, 44 E, 45 D, 46 B, 47 D, 48 A, 49 B, 5 E, 5 A, 5 E 8