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mathematics
calculus 6th edition

Questions and Answers of Calculus 6th edition

Find the gradient vector field ∇f of f and sketch it.f(x, y) = x2 – y
Use a calculator or CAS to evaluate the line integral correct to four decimal places.∫c x sin(y + z) ds, where C has parametric equations x = t2, y = t3, z = t4, 0 ≤ t ≤ 5
Find the gradient vector field ∇f of f and sketch it.f(x, y) = √x2 + y2
Use a calculator or CAS to evaluate the line integral correct to four decimal places.∫c ze–xy ds, where C has parametric equations x = t, y = t2, z = e–t, 0 ≤ t ≤ 1
Plot the gradient vector field of f together with a contour map of f. Explain how they are related to each other.f(x, y) = sin x + sin y
Use Stokes Theorem to evaluate ∫C F · dr. In each case C is oriented counterclockwise as viewed from above.F(x, y, z) = e–xi + exj + ezk, C is the boundary of the part of the plane 2x + y + 2z =
Use a computer to graph the parametric surface. Get a printout and indicate on it which grid curves have u constant and which have v constant.r(u, v) = 〈u + v, u2, v2〉, –1 ≤ u ≤ 1, –1 ≤
Use Green’s Theorem to evaluate the line integral along the given positively oriented curve.∫c xe–2x dx + (x4 + 2x2y2) dy, C is the boundary of the region between the circles x2 + y2 = 1 and x2
Sketch the vector field F by drawing a diagram like Figure 5 or Figure 9.F(x, y, z) = x kFigure 5Figure 9 y F (0, 3) 0 F (2,2) F (1,0) X
Use the Divergence Theorem to calculate the surface integral ∫∫S F · ds; that is, calculate the flux of F across S.F(x, y, z) = x3yi – x2y2 j – x2yz k, S is the surface of the solid bounded
Determine whether or not F is a conservative vector field. If it is, find a function f such that F = ∇f.F(x, y) = (In y + 2xy3)i + (3x2y2 + x/y) j
Use Stokes Theorem to evaluate ∫C F · dr. In each case C is oriented counterclockwise as viewed from above.F(x, y, z) = yz i + 2xzj + exyk, C is the circle x2 + y2 = 16, z = 5Data from Stokes
Evaluate the line integral, where C is the given curve.∫c xyz ds. C: x = 2 sin t, y = t, z = –2 cos t, 0 ≤ t ≤ π
Use the Divergence Theorem to calculate the surface integral ∫∫S F · ds; that is, calculate the flux of F across S.F(x, y, z) = xy sin zi + cos(xz)j + y cos z k, S is the ellipsoid x2/a2 + y2/b2
Sketch the vector field F by drawing a diagram like Figure 5 or Figure 9.F(x, y, z) = j – iFigure 5Figure 9 y F (0, 3) 0 F (2,2) F (1,0) X
Determine whether or not F is a conservative vector field. If it is, find a function f such that F = ∇f.F(x, y) = (xy cosh xy + sinh xy)i + (x2 cosh xy) j
If F = Pi + Qj, how do you test to determine whether F is conservative? What if F is a vector field on R3?
Use Stokes Theorem to evaluate ∫C F · dr. In each case C is oriented counterclockwise as viewed from above.F(x, y, z) = xyi + 2zj + 3yk, C is the curve of intersection of the plane x + z = 5 and
Evaluate the line integral, where C is the given curve.∫c xyz2 ds. C is the line segment from (–1, 5, 0) to (1, 6, 4)
Use a computer to graph the parametric surface. Get a printout and indicate on it which grid curves have u constant and which have v constant.r(u, v) = 〈cos u sin v, sin u sin v, cos v + In
Evaluate the surface integral.S is the part of the cone z2 = x2 + y2 that lies between the planes z = 1 and z = 3 JJS x²z² ds.
Use the Divergence Theorem to calculate the surface integral ∫∫S F · ds; that is, calculate the flux of F across S.F(x, y, z)=x4i – x3z2j + 4xy2zk, S is the surface of the solid bounded by the
Match the vector fields F with the plots labeled I–IV. Give reasons for your choices.F(x, y) = 〈1, sin y〉 I ترا III -3 1 1 " . 1 1 F 3 -3 3 NIZ. -3 3 برا II ch IV ņ 5 S 5 1/12 5 5
(a) Find a function f such that F = ∇f(b) Use part (a) to evaluate ∫C F • dr along the given curve C.F(x, y) = x2i + y2j. C is the are of the parabola y = 2x2 from (–1, 2) to (2, 8)
Evaluate the line integral, where C is the given curve.∫c (2x + 9z) ds, C: x = t y = t2, z = t3, 0 ≤ t ≤ 1
Evaluate the surface integral.∫∫s zds S is the surface x = y + 2z2, 0 ≤ y ≤ 1,0 ≤ z ≤ 1
Use a computer to graph the parametric surface. Get a printout and indicate on it which grid curves have u constant and which have v constant.x = u sin u cos v, y = u cos u cos v, z = u sin v
Use the Divergence Theorem to calculate the surface integral ∫∫S F · ds; that is, calculate the flux of F across S.F(x, y, z) = 4x3zi + 4y3zj + 3z4k, S is the sphere with radius R and center the
Use Green’s Theorem to evaluate ʃC F · dr. (Check the orientation of the curve before applying the theorem.)F(x, y) = (ex + x2y, ey – xy2), C is the circle x2 + y2 = 25 oriented clockwiseData
Match the vector fields F with the plots labeled I–IV. Give reasons for your choices.F(x, y) = 〈x – 2, x + 1〉 I ترا III -3 1 1 " . 1 1 F 3 -3 3 NIZ. -3 3 برا II ch IV ņ 5 S 5 1/12 5 5
(a) Find a function f such that F = ∇f(b) Use part (a) to evaluate ∫C F • dr along the given curve C.F(x, y) = xy2i + x2yj, C: r(t) = 〈t + sin1/2πt, t + cos1/2πt〉, 0 ≤ t ≤ 1
Verify that Stokes Theorem is true for the given vector field F and surface S.F(x, y, z) = y2i + xj + z2k, S is the part of the paraboloid z = x2 + y2 that lies below the plane z = 1, oriented
Evaluate the line integral, where C is the given curve.∫c x2y√z dz, C: x = t3, y = t, z = t2, 0 ≤ t ≤ 1
Use Green’s Theorem to evaluate ʃC F · dr. (Check the orientation of the curve before applying the theorem.)F(x, y) = 〈y – In(x2 + y2), 2 tan–1(y/x)〉, C is the circle (x – 2)2 + (y –
Match the vector fields F with the plots labeled I–IV. Give reasons for your choices.F(x, y) = 〈y, 1/x〉 I ترا III -3 1 1 " . 1 1 F 3 -3 3 NIZ. -3 3 برا II ch IV ņ 5 S 5 1/12 5 5
(a) Find a function f such that F = ∇f(b) Use part (a) to evaluate ∫C F • dr along the given curve C.F(x, y) = y2/1 + x2 i + 2y arctan x j, C: r(t) = t2i + 2t j, 0 ≤ t ≤ 1
Determine whether or not the vector field is conservative. If it is conservative, find a function f such that F = ∇f.F(x, y, z) = xyz2 i + x2yz2j + x2y2z k
Evaluate the line integral, where C is the given curve.∫c z dx + x dy + y dz, C: x = t2, y = t3, z = t2, 0 ≤ t ≤ 1
Verify Green’s Theorem by using a computer algebra system to evaluate both the line integral and the double integral.P(x, y) = y2ex, Q(x, y) = x2ey, C consists of the line segment from (–1, 1) to
Evaluate the surface integral.∫∫S y2dS is the part of the sphere x2 + y2 + z2 = 4 that lies inside the cylinder x2 + y2 = 1 and above the xy-plane
Match the vector fields F on R3 with the plots labeled I–IV. Give reasons for your choices.F(x, y, z)= i + 2j + 3 k I 1 z0 -1 8 -0 z -1- -1 01 y -1 0 II 0 Z 1 -1 IV z 0 -1 -1 0 1 10-1 x
Match the equations with the graphs labeled I–VI and give reasons for your answers. Determine which families of grid curves have u constant and which have v constantr(u, v) = sin vi + cos u sin 2v
Match the vector fields F on R3 with the plots labeled I–IV. Give reasons for your choices.F(x, y, z)= i + 2j + zk I 1 z0 -1 8 -0 z -1- -1 01 y -1 0 II 0 Z 1 -1 IV z 0 -1 -1 0 1 10-1 x
Determine whether or not the vector field is conservative. If it is conservative, find a function f such that F = ∇f.F(x, y, z) = 2xy i + (x2 + 2yz)j + y2k
Evaluate the line integral, where C is the given curve.∫c (x + yz) dx + 2x dy + xyz dz, C consists of line segments from (1, 0, 1) to (2, 3, 1) and from (2, 3, 1) to (2, 5, 2)
(a) Find a function f such that F = ∇f(b) Use part (a) to evaluate ∫C F • dr along the given curve C.F(x, y, z) = (2xz + y2)i + 2xyj + (x2 + 3z2) k, C: x = t2, y = t + 1, z = 2t – 1, 0 ≤ t
Determine whether or not the vector field is conservative. If it is conservative, find a function f such that F = ∇f.F(x, y, z)= ez i + j + xez k
Evaluate the line integral, where C is the given curve.∫c x2 dx + y2 dy + z2 dz, C consists of line segments from (0, 0, 0) to (1, 2, –1) and from (1, 2, –1) to (3, 2, 0)
(a) Find a function f such that F = ∇f(b) Use part (a) to evaluate ∫C F • dr along the given curve C.F(x, y, z) = y2 cos z i + 2xy cos z j – xy2 sin z k, C: r(t) = t2i + sin tj + tk, 0 ≤ t
Match the equations with the graphs labeled I–VI and give reasons for your answers. Determine which families of grid curves have u constant and which have v constantx = (1 –|u|) cos v, y = (1 –
Determine whether or not the vector field is conservative. If it is conservative, find a function f such that F = ∇f.F(x, y, z) = ye–xi + e–x j + 2zk
(a) Find a function f such that F = ∇f(b) Use part (a) to evaluate ∫C F • dr along the given curve C.F(x, y, z) = eyi + xeyj + (z + 1)ezk, C: r(t) = ti + t2j + t3k, 0 ≤ t ≤ 1
Find a parametric representation for the surface.The plane that passes through the point (1, 2, –3) and contains the vectors i + j – k and i – j + k
Show that the line integral is independent of path and evaluate the integral.∫c tan y dx + x sec2y dy, C is any path from (1, 0) to (2, π/4)
Find a parametric representation for the surface.The lower half of the ellipsoid 2x2 + 4y2 + z2 = 1
Is there a vector field G on R3 such that curl G = 〈xyz, – y2z, y2z〉? Explain.
Show that, under conditions to be stated on the vector fields F and G, curl(F x G) = F div G - G div F + (GV)F- (F.V)G
Show that the line integral is independent of path and evaluate the integral.∫c (1 – ye–x) dx + e–xdy, C is any path from (0, 1) to (1, 2)
Plot the vector field and guess where div F > 0 and where div F < 0. Then calculate div F to check your guess.F(x, y) = 〈xy, x + y2〉
Find the gradient vector field of f.f(x, y) = xexy
Find a parametric representation for the surface.The part of the hyperboloid x2 + y2 – z2 = 1 that lies to the right of the xz-plane
Find the work done by the force field F in moving an object from P to Q. lF(x, y) = 2y3/2 i + 3x√y j; P(1, 1), Q(2, 4)
Find the gradient vector field of f.f(x, y) = tan(3x – 4y)
Find a parametric representation for the surface.The part of the elliptic paraboloid x + y2 + 2z2= 4 that lies in front of the plane x = 0
Find the work done by the force field F in moving an object from P to Q. F(x, y) = e–yi – xe–yj; P(0, 1), Q(2, 0)
Use a calculator or CAS to evaluate the line integral correct to four decimal places.∫c F · dr, where F(x, y) = xy i + sin y j and r(t) = eti + e–t2j, 1 ≤ t ≤ 2
Find the gradient vector field of f.f(x, y, z) = x cos(y/z)
Find a parametric representation for the surface.The part of the sphere x2 + y2 + z2 = 16 that lies between the planes z = –2 and z = 2
Find a parametric representation for the surface.The part of the cylinder y2 + z2 = 16 that lies between the planes x = 0 and x = 5
Plot the gradient vector field of f together with a contour map of f. Explain how they are related to each other.f(x, y) = sin(x + y)
Determine whether or not the given set is (a) Open,(b) Connected, (c) Simply-connected.{(x, y) |x > 0, y > 0}
Find parametric equations for the surface obtained by rotating the curve y = e–x, 0 ≤ x ≤ 3, about the x-axis and use them to graph the surface.
Determine whether or not the given set is (a) Open,(b) Connected, (c) Simply-connected.{(x, y) | x ≠ 0}
Find parametric equations for the surface obtained by rotating the curve x = 4y2 – y4, –2 ≤ y ≤ 2, about the y-axis and use them to graph the surface.
Evaluate ∫∫∫S xyz ds correct to four decimal places, where S is the surface z = xy, ≤ 0 ≤ 1, 0 ≤ y ≤ 1.
Determine whether or not the given set is (a) Open,(b) Connected, (c) Simply-connected.{(x, y) |1 < x2 + y2 < 4}
Find the exact value of ∫∫∫S x2yz ds, where S is the surface in Exercise 31.Data from Exercises 31Evaluate ∫∫∫S xyz ds correct to four decimal places, where S is the surface z = xy, ≤ 0
Let(a) Show that ∂P/∂y = ∂Q/∂x.(b) Show that ∫c F · dr is not independent of path. [Compute ∫c1 F · dr and ∫c2 F · dr where C1 and C2 are the upper and lower halves of the circle
Determine whether or not the given set is (a) Open,(b) Connected, (c) Simply-connected.{(x, y) | x2 + y2 ≤ 1 or 4 x2 + y2 ≤ 9}
Find an equation of the tangent plane to the given parametric surface at the specified point. If you have software that graphs parametric surfaces, use a computer to graph the surface and the tangent
Find an equation of the tangent plane to the given parametric surface at the specified point. If you have software that graphs parametric surfaces, use a computer to graph the surface and the tangent
Find an equation of the tangent plane to the given parametric surface at the specified point. If you have software that graphs parametric surfaces, use a computer to graph the surface and the tangent
Find the area of the surface.The part of the plane 2x + 5y + z = 10 that lies inside the cylinder x2 + y2 = 9
Find the work done by the force field F(x, y) = x sin yi + yj on a particle that moves along the parabola y = x2 from (–1, 1) to (2, 4).
Find the work done by the force field F(x, y, z) = 〈y + z,x + z,x + y〉 on a particle that moves along the line segment from (1, 0, 0) to (3, 4, 2).
Find the area of the surface.The part of the surface z = 1 + 3x + 2y2 that lies above the triangle with vertices (0, 0), (0, 1), and (2, 1)
Find the area of the surface.The part of the paraboloid x = y2 + z2 that lies inside the cylinder y2 + z2 = 9
Find the area of the surface correct to four decimal places by expressing the area in terms of a single integral and using your calculator to estimate the integral.The part of the surface z =
Find the area of the surface.The part of the surface y = 4x + z2 that lies between the planes x = 0, x = 1, z = 0, and z = 1
Verify that the conclusion of Clairaut's Theorem holds, that is, uxy = uyx.u = x sin(x + 2y)
Verify that the conclusion of Clairaut's Theorem holds, that is, uxy = uyx.u = x4y2 – 2xy5
Sketch the graph of the function.f(x, y) = 1 – y2
Use the Chain Rule to find dz/dt or dw/dt.z = √1 + x2 + y2, x = ln t, y = cos t
Find an equation of the tangent plane to the given surface at the specified point.z = √xy, (1, 1, 1)
Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint(s).kf(x, y) = x2 + y2; xy = 1
Sketch the graph of the function.f(x, y) = x2 + (y – 2)2
Find the directional derivative of f at the given point in the direction indicated by the angle θ.f(x, y) = x2y3 – y4, (2, 1), θ = π/4
Use the Chain Rule to find dz/dt or dw/dt.z = tan–1(y/x), x = et, y = 1 – e–t
Find an equation of the tangent plane to the given surface at the specified point.z = y ln x, (1, 4, 0)
Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint(s).f(x, y) = 4x + 6y; x2 + y2 = 13

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