The position vector of a particle is r(t). Find the requested vector. 12) The velocity at t = 0 for r(t) = cos(2t)i + 9ln(t -
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1 Find an equation for the sphere with the given center and radius. ) Center (,, 5), radius = Find the angle between u and v in radians. ) u = j - k, v = i - 9j - k Find an equation for the line that passes through the given point and satisfies the given conditions. ) P = (-, -5); perpendicular to v = -i + 9j Find the vector projv u. ) v = 7i - j + k, u = -j + k Find the angle between the planes. 5) x + y + z = - and x + 9y + z = - Calculate the requested distance. ) The distance from the point S(7,, -5) to the line x = - + t, y = -9 + t, z = + t 7) The distance from the point S(5,, ) to the plane x + y + z = 7 Write the equation for the plane. ) The plane through the point P(-, -7, 9) and perpendicular to the line x = 7 + 7t, y = t, z = 7 - t. Find parametric equations for the line described below. 9) The line through the point P(-,, ) and perpendicular to the plane x + y + z = 5 The vector r(t) is the position vector of a particle at time t. Find the angle between the velocity and the acceleration vectors at time t =. )r(t) = e7ti + (9 + e-7t)j + (5 cos t)k If r(t) is the position vector of a particle in the plane at time t, find the indicated vector. )r(t) = (-5t - 9)i + 5 t j. Find the velocity vector of the particle. The position vector of a particle is r(t). Find the requested vector. ) The velocity at t = for r(t) = cos(t)i + 9ln(t - )j - t k Calculate the arc length of the indicated portion of the curve r(t). )r(t) = (7 - t)i + (5 + t)j + (9t - 7)k, - t - Find the length of the indicated portion of the curve. )r(t) = (etcos t)i + (etsin t)j + 5etk, -ln t Find the unit tangent vector of the given curve. 5)r(t) = ( sin 5t)i + ( cos 5t)j Find the curvature of the curve r(t). )r(t) = ( + cos t - sin t)i + (7 + sin t + cos t)j + k Find the curvature of the space curve. 7)r(t) = -7i + (t + )j +(ln(cos t) + )k Find the principal unit normal vector N for the curve r(t). )r(t) = (9 + t)i + ( + ln(cos t))k, -π/ < t < π/ For the curve r(t), write the acceleration in the form att + ann. 9)r(t) = ( + t)/i + ( - t)/j + tk Find two paths of approach from which one can conclude that the function has no limit as (x, y) approaches (, ). ) f(x, y) = Find the limit. xy x + y ) lim (x, y) (, ) e -x - y Determine whether the given function satisfies Laplace's equation. ) f(x, y, z) = cos (x) sin (y) e( z) Determine whether the given function satisfies the wave equation. ) w(x, t) = sin (5x + 5ct)
2 Find all the first order partial derivatives for the following function. ) f(x, y, z) = cos y xz Find all the second order partial derivatives of the given function. 5) f(x, y) = cos xy Find the extreme values of the function subject to the given constraint. ) f(x, y) = x + y, xy = Determine the order of integration and then evaluate the integral. sin y 5) dy dx y x Solve the problem. ) Evaluate w at (u,v) = (, ) for the function u w(x, y) = xy - y; x = u - v, y = uv. ) ey dy dx x/ Use implicit differentiation to find the specified derivative at the given point. 7) Find x y at the point, π, 7 for e x cos yz =. Compute the gradient of the function at the given point. ) f(x, y, z) = ln(x + y + z), (,, ) Answer the question. 9) Find the direction in which the function is increasing or decreasing most rapidly at the point Po. f(x, y, z) = x y + z, Po(,, ) Find the derivative of the function at the given point in the direction of A. ) f(x, y, z) = -x + y - 9z, (-9,, -), A = i - j - k Solve the problem. ) Write an equation for the tangent line to the curve x - xy + y = at the point (-, ). ) Find parametric equations for the normal line to the surface z = ln(x + y + ) at the point (,, ). Find the volume of the indicated region. 7) The region bounded by the paraboloid z = - x - y and the xy-plane ) The solid cut from the first octant by the surface z = 5 - x - y Integrate the function f over the given region. 9) f(x, y) = over the square x, y xy ) f(x, y) = x + y over the trapezoidal region bounded by the x-axis, y-axis, line x =, and line y = - x + Write an equivalent double integral with the order of integration reversed. π/ sin x ) (x + y) dy dx Express the area of the region bounded by the given line(s) and/or curve(s) as an iterated double integral. ) The curves y = x(x - ) and y = x(x - )(x - ) ) The lines x =, y = x, and y = 5 ) Find the equation for the tangent plane to the surface z = e9x + y at the point (,, ).
3 Change the Cartesian integral to an equivalent polar integral, and then evaluate. ) - y (x + y) dx dy Find the area of the region specified in polar coordinates. 5) One petal of the rose curve r = 7 sin θ Evaluate the integral by changing the order of integration in an appropriate way. 97 ) dx dy dz x(y + ) z 7) y π sin z sin x x Evaluate the integral. 9 r ) r dθ dr dz 9) π/ dz dx dy π/ ρ sin φ dφ dρ dθ Evaluate the cylindrical coordinate integral. π/ /r 5) z cos θ dz r dr dθ /r Evaluate the spherical coordinate integral. π/ π/ 5) ρ sin φ dρ dφ dθ Set up the iterated integral for evaluating f(r, θ, z) dz r dr dθ D over the given region D. 5) D is the right circular cylinder whose base is the circle r = cos θ in the xy-plane and whose top lies in the plane z = - x - y. 55) D is the solid right cylinder whose base is the region in the xy-plane that lies inside the cardioid r = + cos θ and outside the circle r = 7, and whose top lies in the plane z = 9. Use the given transformation to evaluate the integral. 5) u = -x + y, v = x + y; (x + y) dx dy, R where R is the parallelogram bounded by the lines y = x +, y = x + 7, y = -x +, y = -x + Find the Jacobian using the given equations. 57) x = u, y = uv (x, y) (x, y, z) or (as appropriate) (u, v) (u, v, w) Evaluate the line integral of f(x,y) along the curve C. 5) f(x, y) = cos x + sin y, C: y = x, x π Evaluate the line integral along the curve C. 59) x + y + z ds, C is the curve r(t) = ti + ( 5 C cos t)j + ( sin t)k, t π Find the volume of the indicated region. 5) The region enclosed by the sphere x + y + z = and the cylinder (x- ) + y = Find the mass of the wire that lies along the curve r and has density δ. )r(t) = (7 cos t)i + (7 sin t)j + 7tk, t π; δ = 5) The region bounded above by the sphere x + y + z = and below by the cone z = x + y
4 Calculate the flux of the field F across the closed plane curve C. )F = (x+y)i + xyj; the curve C is the closed counterclockwise path around the rectangle with vertices at (, ), (7, ), (7, ), and (, ) Evaluate the surface integral of the function g over the surface S. 7) g(x,y,z) = x + y + z; S is the surface of the cube formed from the coordinate planes and the planes x =, y =, and z = Calculate the work done by the force F along the path C. )F = -5zi + xj + 7yk; C: r(t) = ti + tj + tk, t Find the gradient field F of the function f. ) f(x, y, z) = x + y + z x7 Find the potential function f for the field F. )F = xyzi + xy9zj + xyz7k Using Green's Theorem, calculate the area of the indicated region. 5) The circle r(t) = (7 cos t)i + (7 sin t)j, t π ) The area bounded above by y = 7 and below by y = 7 x Find the flux of the vector field F across the surface S in the indicated direction. 7)F = i + yz7j - y7zk, S is the rectangular surface x =, - y, and - z, direction i Find the surface area of the surface S. 7) S is the portion of the surface x + z = that lies above the rectangle 7 x and y in the x-y plane 7) S is the portion of the paraboloid z = - x - y that lies above the ring x + y 9 in the x-y plane Evaluate the surface integral of g over the surface S. 7) S is the plane x + y + z = above the rectangle x 5 and y ; g(x,y,z) = z Apply Green's Theorem to evaluate the integral. 75) S is the parabolic cylinder y = 5x, x and z ; g(x, y, z) = x 7) ) (y + x) dx + (y + x) dy C C: The circle (x - ) + (y - ) = C (y + 5) dx + (x + ) dy C: The triangle bounded by x =, x + y =, y = Using Green's Theorem, find the outward flux of F across the closed curve C. 9)F = (-y - ey cos x)i + (y - ey sin x)j ; C is the right lobe of the lemniscate r = cos θ that lies in the first quadrant. Find the flux of the vector field F across the surface S in the indicated direction. 7)F = xi + yj + zk; S is portion of the plane x + y + z = 5 for which x and y ; direction is outward (away from origin) 77)F = x5yi + yj - zk; S is the portion of the parabolic cylinder z = - y for which z and x ; direction is outward (away from the x-y plane) Calculate the area of the surface S. 7) S is the portion of the sphere x + y + z = between z = - 9 and z = 9 79) S is the cap cut from the paraboloid z = 9-5 x - 5y by the cone z = x + y
5 Use Stokes' Theorem to calculate the circulation of the field F around the curve C in the indicated direction. )F = 5yi + xj - zk ; C: the portion of the plane x + y + 9z = in the first quadrant )F = xi + xj + 7zk; C: the cap cut from the upper hemisphere x + y + z = (z ) by the cylinder x + y = 5
6 Answer Key Testname: CAL-FINAL-PT ) x + y + z - z = ). ) -x + 9y = ) 5 59 i j k 5). ) 7) 7,5 ) 7x + 9y - z = - 9) x = t -, y = t +, z = t ) π )v(t) = (-t)i + 5 t j )v() = -j ) 77 ) 5)T(t) = (sin 5t)i - (cos 5t)j ) κ = 7) κ = cos t )N(t) = (-sin t)i - (cos t)k 9)a = - t N ) Answers will vary. One possibility is Path : x = t, y = t ; Path : x = t, y = t ) e- ) Yes ) Yes ) f x = - cosy xz ; f y = - siny xz ; f z = - cos y xz 5) f x = -y cos xy; f y = - x[xy cos (xy) + sin(x y)]; ) - 7) 7 ) 9 i + j + 9 k f y x = f x y = - y[xy cos (xy) + sin(xy)]; 9) i + j 5 + k ) ) x - y + = ) x=, y =, z = t ) z = ) Maximum: none; minimum: at (, ± ) 5) - cos ) (e - ) 7) π ) 5 9) ln ) ) ) ) 5 π/ (x + y) dx dy sin- y x(x - )(x - ) dy dx ),π 5) 9 π x(x - ) y/ dx dy ) ln 97 7) ( - cos ) ) 9) π 5 ln - 5) 5) 7π 5) (π- ) 9 5) 7π( - ) π cos θ 5) 55) 5) 5 π 7 + cos θ - r(cos θ + sin θ) f(r, θ, z) dz r dr d 9 f(r, θ, z) dz r dr dθ
7 Answer Key Testname: CAL-FINAL-PT 57) u 5) 59) π + ) π units ) 9 ) W = )F = -5x - 7y - 7z i + y x x7 j + z x7 k ) f(x, y, z) = xyz + C 5) 9π ) 7) -π ) 9) 7) 7 7) 7 7) 5 7) π [ ] 7) ) - 7) 9 77) 7) π π 79) 5 - ) - ) π 7
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