Calculus III-Final review Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the corresponding position vector. 1) Define the points P = (-, ) and Q = (-9, -). Find the position vector corresponding to PQ. A), 8 B) -11, C), - D) -, -8 1) Express the vector in the form ai + bj. ) P1P if P1 is the point (1, -1) and P is the point (, -) A) v = i - j B) v = -i + j C) v = -i + j D) v = -i - j ) Find the indicated vector. ) Let u = 9, 9, v = 1,. Find u - v. A) 5, - B) 1, 5 C) 18, - D) 1, -5 ) Express the vector in the form ai + bj + ck. ) P1P if P1 is the point (,, ) and P is the point (5,, ) A) v = i - j - k B) v = -i + j - k C) v = -i + j + k D) v = -i - j + k 5) u - 5v if u = 1, 1, and v =,, 1 A) v = -1i + j - 5k B) v = -1i + 8j - 5k C) v = i + j - 5k D) v = 18i + j - 5k ) 5) Find an equation for the sphere with the given center and radius. 6) Center (-, 5, ), radius = A) x + y + z + 1x + 1y = -58 B) x + y + z + 1x - 1y = -58 C) x + y + z - 1x - 1y = -58 D) x + y + z - 1x + 1y = -58 6) Find v u. ) v = 5i - j and u = -8i - j A) -i - 9j B) -5 C) -6 D) -i + 1j ) Find the angle between u and v in radians. 8) u = 1i + j + 5k, v = i + 6j + 9k A) 1.11 B) 1.9 C). D) 1.1 8) 9) u = j - 9k, v = i - 5j - k A) -. B) 1.6 C) 1.8 D) 1.5 9) Find projv u. 1) v = i + j + k, u = i + 1j + k A) 19 1 i + 19 1 j + 19 1 k B) 19 169 i + 19 169 j + 19 169 k C) 19 i + 19 j + 19 k D) i + j + k 1) 1
11) v = k, u = i + 6j + 9k A) 18 11 i + 5 11 j + 81 18 k B) 11 11 i + 5 11 j + 81 11 k C) 9k D) 9 11 k 11) 1) How much work does it take to slide a box meters along the ground by pulling it with a 196 N force at an angle of 11 from the horizontal? A) 19. joules B) 9 joules C). joules D) 91 joules 1) Find the magnitude and direction (when defined) of u v. 1) u = i, v = 8j A) 16; -k B) ; 8k C) 16; k D) 16; 16k 1) 1) u = i + j - k, v = -i + k 1) A) ; i + 1 j - k B) 9; 9 i + 1 9 j - 9 k C) 9; 9 i - 1 9 j + 9 k D) ; i - 1 j + k 15) Find the area of the parallelogram determined by the points P(, -5, 5), Q(-1, -, 8), R(6,, -) and S(-, 1, 1). A) 1,85 B) 1 91 C) 1 91 D) 1,85 15) Find the triple scalar product (u x v) w of the given vectors. 16) u = i + j + j; v = i + j + 5k; w = 8i + 6j + k A) -88 B) C) 56 D) - 16) 1) u = i + j - j; v = 1i + 5j - 5k; w = 1i + 6j - k A) 55 B) 95 C) -15 D) -5 1) Find parametric equations for the line described below. 18) The line through the points P(-1, -1, -) and Q(5,, -) A) x = t - 6, y = t -, z = -t - B) x = 6t + 1, y = t + 1, z = t + C) x = 6t - 1, y = t - 1, z = t - D) x = t + 6, y = t +, z = -t + 18) Find a parametrization for the line segment beginning at P1 and ending at P. 19) P1(6, -5, ) and P(, -5, -) A) x = 6t, y = -5, z = t -, t 1 B) x = 6t, y = -5t, z = t -, t 1 C) x = -6t + 6, y = -5t, z = -t +, t 1 D) x = -6t + 6, y = -5, z = -t +, t 1 19) ) Find symmetric equations for the line through the points P(-1, -1, -) and Q(,, ). A) x - 1 C) x + 1 = y + 1 = y - 1 = z + = z + B) x + 1 D) x - 1 = y + 1 = y - 1 = z + = z + )
1) Find the symmetric equations of the line through (,, 1) and perpendicular to the plane x + y - z =. A) x - 1 = y - = z - 1 - C) x - -1 = y - - = z - 1 B) x + 1 = y + = z + 1 - D) x + -1 = y + - = z + 1 1) ) Find the equation of the plane containing the line x = t, y = 6 - t, z = and the point 16 5, 5,. A) x - y - z = 1 B) x + y - z = 1 C) x + y + z = 1 D) x - y + z = 1 ) Find the equation of the plane through the point P(-, 1, ) and perpendicular to the line x = 1 + 8t, y = -5 + 8t, z = - t. A) 8x + 8y - z = 11 B) 8x + 8y + z = -11 C) 8x + 8y - z = 19 D) 8x + 8y - z = -11 ) ) Find the following. ) If v = -1, 6,, find v. A) B) C) D) 5 ) Find the distance between points P1 and P. 5) P1(5, -8, -1) and P(1,, 9) A) 15 B) 1 C) 1 D) 18 5) 6) Find the center and radius of the sphere with the equation x + y + z - x + y - z - 9 =. A) center = 1, - 1, 1 ; radius = 5 6 C) center = 1, - 1, 1, radius = 5 6 B) center = - 1, 1, - 1 D) center = 1, - 1, 1 ; radius = 5 8,; radius = 5 6) ) Find the equation of the sphere with center = (, -6, 1) and radius = 5. A) x + (y + 6) + (z - 1) = 5 B) x + (y + 6) + (z + 1) = 5 C) x + (y - 6) + (z - 1) = 5 D) x - (y + 6) - (z - 1) = 5 ) Identify the type of surface represented by the given equation. 8) x + y 8 = z 9 A) Hyperbolic paraboloid B) Paraboloid C) Elliptic cone D) Ellipsoid 8)
9) x + z = y 8 A) Hyperbolic paraboloid B) Elliptic cone C) Ellipsoid D) Paraboloid 9) ) The Cartesian coordinates of a point are (-1, -1, -). Find the cylindrical coordinates. A), 5π, - B), 5π, - C), 5π, - D), 5π, - ) 1) The cylindrical coordinates of a point are 5, π, 1. Find the Cartesian coordinates. 1) A) (, -5, 1) B) (, 5, 1) C), - 5, 1 D) (-5,, 1) ) The Cartesian coordinates of a point are (-1, -1, ). Find the spherical coordinates. ) A), 5π, π B), 5π 8, π C), 5π, π D), 5π, ) The spherical coordinates of a point are 1, π, π. Find the cylindrical coordinates. ) A) (, 1, ) B) (,, -1) C) (1, π, ) D), π, ) The cylindrical coordinates of a point are (,, 9). Find the spherical coordinates. ) A) 9, π, B) (9,, ) C) 9,, π D) 9,, π Make the required change in the given equation. 5) z = x + y to cylindrical coordinates A) r tan θ = 1 B) z = r C) z = r(cos θ + sin θ) D) r = cot φ csc φ 5) 6) z = cot θ to Cartesian coordinates 6) A) x + y = z B) z = x y C) z r = x y D) z = x r ) r = to spherical coordinates ) A) ρ = csc φ B) ρ(1 - cos φ) = 9 C) cos φ = ρ D) ρ = sin φ 8) ρ = cos θ sin φ to Cartesian coordinates A) x + y + z = y B) x + y + z - z = C) x + y + z = x D) x + y + z - x = 8)
Write the equation for the plane. 9) The plane through the point P(,, ) and normal to n = i + j + 6k. A) -x - y - 6z = 6 B) x + y + z = 6 C) x + y + 6z = 6 D) -x - y - z = 6 9) ) The plane through the points P(5, -, -18), Q(-, 8, 1) and R(-1, -5, ). A) 8x + y + z = -1 B) x + y + 8z = 1 C) 8x + y + z = 1 D) x + y + 8z = -1 ) 1) Find an equation for the level curve of the function f(x, y) = 6 - x - y that passes through the point, 5. A) x - y = 8 B) x + y = - 8 C) x + y = 8 D) x + y = 1) ) Find an equation for the level surface of the function f(x, y, z) = x + y + z that passes through the point, 1,. A) x + y + z = 1 B) x + y + z = ± 1 C) x + y + z = 169 D) x + y + z = 1 ) ) Find an equation for the level surface of the function f(x, y, z) = ln xy z that passes through the ) point e1, e, e. A) ln xy = 1 z 9 B) xy z = e11 C) xy z = e9 D) ln xy z = 9 Find all the first order partial derivatives for the following function. ) f(x, y) = ln y 8 x A) f x = - x ; f y = 8 y C) f x = 8 y ; f y = x B) f x = -ln y 8 x ; f y = ln 8y x D) f x = -ln x ; f y = ln 8 y ) e-x 5) f(x, y) = x + y A) f x = - xe-x (x + y) ; f y = - B) f x = e -x(x + y + x) (x + y) ; f y = ye-x (x + y) C) f x = - e -x(x + y + x) (x + y) ; f y = - D) f x = - e -x(x + y + x) (x + y) ; f y = - ye-x (x + y) ye-x (x + y) ye-x (x + y) 5) 5
Find all the second order partial derivatives of the given function. 6) f(x, y) = x + y - ex+y A) f x = 1 - e x+y; f y = - e x+y; B) f x = - e x+y; f y = - e x+y; C) f x = - y ex+y; f y = -x ex+y; D) f x = + e x+y; f y = e x+y; f y x = f x y = -e x+y f y x = f x y = -e x+y f y x = f x y = - y ex+y f y x = f x y = e x+y 6) ) f(x, y) = ln (xy - x) A) f x = xy - x y - 1 (xy - x) B) f x = xy - x y - 1 (xy - x) C) f x = xy - x y - 1 (xy - x) D) f x = xy - x y - 1 (xy - x) ; f y = ; f y = - ; f y = - ; f y = - x (xy - x) ; f y x = f x y = x (xy - x) ; f y x = f x y = - x (xy - x) ; f y x = f x y = - x (xy - x) ; f y x = f x y = - x (xy - x) x (xy - x) x (xy - x) x (xy - x) ) 8) Evaluate dw dt A) - 1 π at t = xy π for the function w(x, y, z) = z ; x = sin t, y = cost, z = t. B) - 1 π C) - 9 1 π D) 1 π 8) Use implicit differentiation to find the specified derivative at the given point. 9) Find dy dx at the point (1, 1) for x + y + xy =. A) - 1 B) 1 1 C) - 1 D) - 1 1 9) 5) Find z yz at the point (8, 1, -1) for ln y x - e xy+z =. A) e 9-1 1-8e9 B) 1-8e 9 1 - e9 C) 8e 9-1 1 - e9 D) 1 - e 9 1-8e9 5) Find the derivative of the function at the given point in the direction of A. 51) f(x, y, z) = 9x - y - z, (, -9, -6), A = i - 6j - k A) 1 B) 5 C) D) 9 51) 6
Compute the gradient of the function at the given point. 5) f(x, y) = ln(-6x - 8y), (-9, -) A) - 9 86 i - j B) - i - j C) - i - 9 86 j D) 1 86 i + 1 86 j 5) Provide an appropriate response. 5) Find the direction in which the function is increasing most rapidly at the point P. f(x, y, z) = xy - ln(z), P(1,, ) A) 1 1 (i + j - k) B) 1 1 (i + j - k) C) 1 1 (i - j + k) D) 1 (i + j - k) 1 5) 5) Write an equation for the tangent line to the curve x - xy + y = 6 at the point (-1, 1). A) y = x - B) y = x + 1 C) x + y = 1 D) x - y + = 5) 55) Find parametric equations for the normal line to the surface -x - 6y + 5z = 1 at the point (1, -1, ). A) x = -t + 1, y = -6t - 1, z = 5t + B) x = -t - 1, y = -6t + 1, z = 5t - C) x = -t -, y = t - 6, z = -t + 5 D) x = t -, y = -t - 6, z = t + 5 55) 56) Write parametric equations for the tangent line to the curve of intersection of the surfaces z = 5x + y and z = x + y + at the point (1, 1, 9). A) x = -t + 1, y = 11t + 1, z = t + 9 B) x = -9t + 1, y = 11t + 1, z = t + 9 C) x = -t + 1, y = 9t + 1, z = t + 9 D) x = -9t + 1, y = 9t + 1, z = t + 9 56) 5) If the length, width, and height of a rectangular solid are measured to be 6,, and 5 inches respectively and each measurement is accurate to within.1 inch, estimate the maximum percentage error in computing the volume of the solid. A) 5.6% B).% C) 8.% D) 6.% 5) Find all local extreme values of the given function and identify each as a local maximum, local minimum, or saddle point. 58) f(x, y) = x - 18x + y + 1y - 6 58) A) f(9, -6) = -1, local minimum B) f(-9, 6) = 5, local maximum C) f(-9, -6) = 1, saddle point D) f(9, 6) = 1, saddle point Find the extreme values of the function subject to the given constraint. 59) f(x, y) = x + y, x + y = A) Maximum: at (, 1); minimum: - at (, -1) B) Maximum: 8 at (, 1); minimum: - at (, -1) C) Maximum: 8 at (, 1); minimum: -1 at (1, -) D) Maximum: at (, 1); minimum: -1 at (1, -) 59) 6) A rectangular box with square base and no top is to have a volume of ft. What is the least amount of material required? A) 8 ft B) ft C) 6 ft D) ft 6)
61) What is the largest possible volume of an open metal box made from 5 ft of tin? Ignore the thickness of the tin. A) 1. ft B) 6.5 ft C) 1.5 ft D) 15. ft 61) Find the indicated limit or state that it does not exist. x 6) lim cos (x, y) (, ) x + y 6) A) π B) C) 1 D) No limit 6) Find the mass of a thin triangular plate bounded by the coordinate axes and the line x + y = 1 if δ(x, y) = x + y. A) 5 B) C) D) 1 6) Find the area of the region specified by the integral(s). - x 6) dy dx A) 8 B) 8 C) 16 D) 6) 65) Integrate f(x, y) = ln(x + y) over the region 1 x + y 16. x + y A) π(ln ) B) π ln C) π(ln ) D) π ln 66) Find the mass of the rectangular solid of density δ(x, y, z) = xyz defined by x, y 5, z. A) 15 B) 15 C) 5 D) 98 65) 66) Find the volume of the indicated region. 6) the region bounded by the paraboloid z = 81 - x - y and the xy-plane A) π B) 9 6561 π C) 18π D) π 6) Change the Cartesian integral to an equivalent polar integral, and then evaluate. 68) 5-5 A) 5π 5 - x dy dx B) 15π C) π D) 5π 68) 8
Evaluate the cylindrical coordinate integral. 6 9 r 69) z dz r dr dθ A),5 B) 61,1 C),5 D) 1,6,55 69) ) Find the center of mass of the thin semicircular region of constant density δ = bounded by the x-axis and the curve y = 5 - x. A) x =, y = π B) x =, y = π C) x =, y = 1 π D) x =, y = 5 π ) 1) Let D be the region bounded below by the cone z = x + y and above by the sphere z = 5 - x - y. Set up the triple integral in cylindrical coordinates that gives the volume of using the order of integration dz dr dθ. A) C) π/ 5 5 - r r dz dr dθ B) π/ 5/ 5 - r r dz dr dθ D) π π 5 5/ 5 - r r dz dr dθ 5 - r r dz dr dθ 1) Find the volume of the indicated region. ) the region bounded below by the xy-plane, laterally by the cylinder r = 8 cos θ, and above by the plane z = A) 11π B) 1π C) 98π D) 8π ) (x, y) (x, y, z) Find the Jacobian or (as appropriate) using the given equations. (u, v) (u, v, w) ) x = u cos v, y = u sin v A) 1v B) 6v C) 1u D) 6u ) Use the given transformation to evaluate the integral. ) u = x + y, v = -x + y; x dx dy, R where R is the parallelogram bounded by the lines y = -x + 1, y = -x +, y = x +, y = x + 5 A) - B) C) 8 D) -8 ) Evaluate the line integral of f(x,y) along the curve C. 5) f(x, y) = x + y, C: y = x +, x A) 9 5 B) 9 C) 1 5 D) 18 5 5) 9
Find the work done by F over the curve in the direction of increasing t. 6) F = -5yi + 5xj + 8zk; C: r(t) = cos ti + sin tj, t 9 A) W = 9 B) W = 5 C) W = 5 D) W = 6) Test the vector field F to determine if it is conservative. ) F = xyzi + xyzj + xyzk A) Conservative B) Not conservative ) Find the potential function f for the field F. 8) F = 1 z i - 5j - x z k 8) A) f(x, y, z) = x z - 5 + C B) f(x, y, z) = x z + C C) f(x, y, z) = x z - 5y + C D) f(x, y, z) = x z - 5y + C Find the divergence of the field F. 9) F = -6x6i + 5xyj + 5xzk A) -6x5 + 1x - 6 B) -6x5 + 1x C) -6x5 + 5y + 5z D) -6 9) Evaluate the work done between point 1 and point for the conservative field F. 8) F = (y + z)i + xj + xk; P1(,, ), P(9, 1, 8) A) W = B) W = 9 C) W = 16 D) W = 18 8) Find the mass of the wire that lies along the curve r and has density δ. 81) r(t) = 6i + (5 - t)j + tk, t π ; δ = 5(1 + sin 9t) A) 5 9 + 5π units B) 1 9 + 5π units C) 5π units D) 1π units 81) Evaluate. The differential is exact. (,, ) 8) (xy - xz) dx + xy dy - xz dz (,, ) A) B) C) 19 D) 8 8) Apply Greenʹs Theorem to evaluate the integral. 8) (y + 6) dx + (x + 1) dy C C: The triangle bounded by x =, x + y = 1, y = A) B) - C) D) 8) 1
8) (6y dx + 5y dy) C C: The boundary of x π, y sin x A) -1 B) C) D) - 8) Find the potential function f for the field F. 85) F = (y - z)i + (x + y - z)j - (x + y)k A) f(x, y, z) = x + y - xz - yz + C B) f(x, y, z) = xy + y - x - y + C C) f(x, y, z) = xy + y - xz - yz + C D) f(x, y, z) = x(y + y) - xz - yz + C 85) 11
Answer Key Testname: CAL FINAL REVIEW 1) D ) A ) A ) A 5) A 6) B ) C 8) C 9) C 1) C 11) C 1) B 1) C 1) D 15) C 16) D 1) A 18) C 19) D ) B 1) A ) D ) D ) B 5) A 6) D ) A 8) C 9) D ) D 1) A ) C ) C ) B 5) B 6) B ) A 8) D 9) C ) C 1) C ) C ) B ) A 5) D 6) B ) D 8) C 9) D 5) B 1
Answer Key Testname: CAL FINAL REVIEW 51) C 5) B 5) B 5) D 55) A 56) C 5) B 58) A 59) A 6) A 61) B 6) D 6) D 6) B 65) C 66) B 6) D 68) A 69) A ) A 1) D ) A ) C ) A 5) A 6) C ) A 8) D 9) B 8) C 81) C 8) C 8) C 8) D 85) C 1