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mathematics
precalculus 1st

Questions and Answers of Precalculus 1st

Prove the following identities, where C is a simple closed smooth oriented curve. f f(x) dx + g(y) dy = 0, where ƒ and g have continuous C derivatives on the region enclosed by C
R3 Given the force field F, find the work required to move an object on the given oriented curve. F = (-y, x, z) on the helix r(t) = (2 cos t, 2 sin t, t/2π), for 0 ≤ t ≤ 2π
Prove the following identities, where C is a simple closed smooth oriented curve. фах & dy = 0 C
R3 Given the force field F, find the work required to move an object on the given oriented curve. F(x, y, z) on the tilted ellipse r(t): = for 0 ≤ t ≤ 2π (4 cos t, 4 sin t, 4 cos t),
Evaluate the following line integrals using a method of your choice. V(e cos y) dr, where C is the line segment from (0, 0) to (In 2, 2π)
Evaluate the following line integrals using a method of your choice. fr.. F. dr, where F = = (2xy + z², x², 2xz) and C is the circle č r(t) = (3 cos t, 4 cos t, 5 sin t), for 0 ≤ t ≤ 2π.
For the following vector fields, compute(a) The circulation on (b) The outward flux across the boundary of the given region. Assume boundary curves have counterclockwise orientation. F = V(√x² +
Specify the component functions of a vector field F in R2 with the following properties. Solutions are not unique.At all points except (0, 0), F has unit magnitude and points away from the origin
For the following vector fields, compute(a) The circulation on (b) The outward flux across the boundary of the given region. Assume boundary curves have counterclockwise orientation. F = R = (x +
Evaluate the following line integrals using a method of your choice. pe (cos y dx + sin y dy), where C is the square with vertices C (±1, ±1) oriented counterclockwise
Specify the component functions of a vector field F in R2 with the following properties. Solutions are not unique.The flow of F is counterclockwise around the origin, increasing in magnitude with
For the following vector fields, compute(a) The circulation on (b) The outward flux across the boundary of the given region. Assume boundary curves have counterclockwise orientation. F (y cos x,
Consider the following potential functions and graphs of their equipotential curves.a. Find the associated gradient field F = ∇φ.b. Show that the vector field is orthogonal to the equipotential
Specify the component functions of a vector field F in R2 with the following properties. Solutions are not unique.F is everywhere normal to the line x = y.
Consider the following potential functions and graphs of their equipotential curves.a. Find the associated gradient field F = ∇φ.b. Show that the vector field is orthogonal to the equipotential
For the following vector fields, compute(a) The circulation on (b) The outward flux across the boundary of the given region. Assume boundary curves have counterclockwise orientation. F ¹3). {(r,
Consider the following potential functions and graphs of their equipotential curves.a. Find the associated gradient field F = ∇φ.b. Show that the vector field is orthogonal to the equipotential
Specify the component functions of a vector field F in R2 with the following properties. Solutions are not unique.F is everywhere normal to the line x = 2.
Consider the following potential functions and graphs of their equipotential curves.a. Find the associated gradient field F = ∇φ.b. Show that the vector field is orthogonal to the equipotential
Given the following vector fields and oriented curves C, evaluate ∫c F • T ds. F = (x, y) x² + y² on the line r(t) = (t, 4t), for 1 ≤ t ≤ 10
For the following vector fields, compute (a) The circulation on(b) The outward flux across the boundary of the given region. Assume boundary curves are oriented counterclockwise. F = (xy, -x + 2y); R
Given the following vector fields and oriented curves C, evaluate ∫c F • T ds. F = (x, y) on the curve r(t) = (1², 312), for 1 ≤t≤2 (x² + y²)³/2
Given the following vector fields and oriented curves C, evaluate ∫c F • T ds. F = (x, y) (x² + y²)3/2 on the line segment from (2, 2) to (10, 10)
For the following vector fields, compute (a) The circulation on(b) The outward flux across the boundary of the given region. Assume boundary curves are oriented counterclockwise. F = (2x + y, x -
For the following vector fields, compute (a) The circulation on(b) The outward flux across the boundary of the given region. Assume boundary curves are oriented counterclockwise. F = (x, y); R is
Find the gradient field F = ∇φ for the following potential functions φ. p(x, y, z) = e sin (x + y)
Find the gradient field F = ∇φ for the following potential functions φ. p(x, y, z) = (x² + y² + z²)-¹/2
For the following vector fields, compute (a) The circulation on(b) The outward flux across the boundary of the given region. Assume boundary curves are oriented counterclockwise. F = (-y, x); R is
Determine whether the following statements are true and give an explanation or counterexample.a. The work required to move an object around a closed curve C in the presence of a vector force field is
Given the following vector fields and oriented curves C, evaluate ∫c F • T ds. F = (y,x) on the line segment from (1, 1) to (5, 10)
Given the following vector fields and oriented curves C, evaluate ∫c F • T ds. F = (-y, x) on the semicircle r(t) = (4 cos t, 4 sin t), for 0 ≤t≤ T
Find the gradient field F = ∇φ for the following potential functions φ. p(x, y, z) = (x² + y² + z²)/2
Find the gradient field F = ∇φ for the following potential functions φ. p(x, y, z)= In (1 + x² + y² + z²)
Use Green’s Theorem to evaluate the following line integrals. Unless stated otherwise, assume all curves are oriented counterclockwise. The flux line integral of F= (e, ex), where C is the
Given the following vector fields and oriented curves C, evaluate ∫c F • T ds. F = (x, y) on the parabola r(t) = (4t, t²), for 0 ≤ t ≤ 1
Find the gradient field F = ∇φ for the following potential functions φ. p(x, y) = tan¹ (y/x)
Use Green’s Theorem to evaluate the following line integrals. Unless stated otherwise, assume all curves are oriented counterclockwise. The circulation line integral of F = (x² + y², 4x + y³),
Evaluate the line integral ∫C ∇φ. dr for the following functions w and oriented curves C in two ways.a. Use a parametric description of C and evaluate the integral directly.b. Use the
Use Green’s Theorem to evaluate the following line integrals. Unless stated otherwise, assume all curves are oriented counterclockwise. pray- - g dx, where (f, g) = (x², 2y²) and C is the
Use a scalar line integral to find the length of the following curves. r(t) = (30 sin t, 40 sin t, 50 cos t), for 0 ≤ t ≤ 2π
Evaluate the line integral ∫C ∇φ. dr for the following functions w and oriented curves C in two ways.a. Use a parametric description of C and evaluate the integral directly.b. Use the
Use Green’s Theorem to evaluate the following line integrals. Unless stated otherwise, assume all curves are oriented counterclockwise. f dy - g dx, where (f, g) = (0, xy) and C is the
Use Green’s Theorem to evaluate the following line integrals. Unless stated otherwise, assume all curves are oriented counterclockwise. (2x-3y) dy (3x + 4y) dx, where C is the unit circle -
Find the gradient field F = ∇φ for the following potential functions φ. p(x, y) = x/y
Use a scalar line integral to find the length of the following curves. r(t) = (20 sin t/4, 20 cos t/4, 1/2), for 0 ≤ t ≤ 2
Use Green’s Theorem to evaluate the following line integrals. Unless stated otherwise, assume all curves are oriented counterclockwise. √ (2x + e³²) dy − (4y2 + ex²) dx, where C is the
R3 Convert the line integral to an ordinary integral with respect to the parameter and evaluate it. [xe xez ds; C is r(t) = (t, 2t, -4t), for 1 ≤t≤ 2.
Evaluate the line integral ∫C ∇φ. dr for the following functions w and oriented curves C in two ways.a. Use a parametric description of C and evaluate the integral directly.b. Use the
Find the gradient field F = ∇φ for the following potential functions φ. p(x, y) = x²y-y²x
Find the gradient field F = ∇φ for the following potential functions φ. p(x, y) = Vxy
Consider the following regions R and vector fields F.a. Compute the two-dimensional divergence of the vector field.b. Evaluate both integrals in Green’s Theorem and check for consistency.c. State
R3 Convert the line integral to an ordinary integral with respect to the parameter and evaluate it. C ху - ds; C is the line segment from (1, 4, 1) to (3, 6, 3). Z
R3 Convert the line integral to an ordinary integral with respect to the parameter and evaluate it. Jo 0 ≤ t ≤ 2T. (yz) ds; C is the helix r(t) = (3 cos t, 3 sin t, t), for
Evaluate the line integral ∫C ∇φ. dr for the following functions w and oriented curves C in two ways.a. Use a parametric description of C and evaluate the integral directly.b. Use the
Consider the following regions R and vector fields F.a. Compute the two-dimensional divergence of the vector field.b. Evaluate both integrals in Green’s Theorem and check for consistency.c. State
Evaluate the line integral ∫C ∇φ. dr for the following functions w and oriented curves C in two ways.a. Use a parametric description of C and evaluate the integral directly.b. Use the
Evaluate the line integral ∫C ∇φ. dr for the following functions w and oriented curves C in two ways.a. Use a parametric description of C and evaluate the integral directly.b. Use the
Find the gradient field F = ∇φ for the potential function φ. Sketch a few level curves of φ and a few vectors of F. p(x, y) = x + y, for x ≤ 2, y ≤ 2
Find the gradient field F = ∇φ for the potential function φ. Sketch a few level curves of φ and a few vectors of F. p(x, y) = 2xy, for x ≤ 2, y ≤2
R3 Convert the line integral to an ordinary integral with respect to the parameter and evaluate it. xyz, ds; C is the line segment from (0, 0, 0) to (1, 2, 3).
Consider the following regions R and vector fields F.a. Compute the two-dimensional divergence of the vector field.b. Evaluate both integrals in Green’s Theorem and check for consistency.c. State
R3 Convert the line integral to an ordinary integral with respect to the parameter and evaluate it. So = (x -y + 2z) ds; C is the circle r(t) = č for 0 ≤ t ≤ 2TT. (1, 3 cos t, 3 sin t),
Determine whether the following vector fields are conservative on the specified region. If so, determine a potential function. Let R* and D* be open regions of R2 and R3, respectively, that do not
Determine whether the following vector fields are conservative on the specified region. If so, determine a potential function. Let R* and D* be open regions of R2 and R3, respectively, that do not
Find the gradient field F = ∇φ for the potential function φ. Sketch a few level curves of φ and a few vectors of F. p(x, y) = √x² + y², for x² + y² ≤ 9, (x, y) = (0,0)
Consider the following regions R and vector fields F.a. Compute the two-dimensional divergence of the vector field.b. Evaluate both integrals in Green’s Theorem and check for consistency.c. State
R3 Convert the line integral to an ordinary integral with respect to the parameter and evaluate it. (x+y+z) ds; C is the circle r(t) = = for 0≤ t ≤ 2TT. (2 cos t, 0, 2 sin t),
Find the gradient field F = ∇φ for the potential function φ. Sketch a few level curves of φ and a few vectors of F. p(x, y) = x² + y², for x² + y² ≤ 16
Consider the following regions R and vector fields F.a. Compute the two-dimensional divergence of the vector field.b. Evaluate both integrals in Green’s Theorem and check for consistency.c. State
Evaluate the following integrals in cylindrical coordinates. Polo 9-12 [S. L. (² 0 0 (x² + y2)3/2 dz dy dx
Use a change of variables to evaluate the following integrals. fav dV; D is bounded by the planes y - 2x = 0, y 2x = 1, D z 3y = 0, z 3y = 1, z 4x = 0, and z 4x = 3.
Use a change of variables to evaluate the following integrals. Iz zdV; D is bounded by the paraboloid z = 16 − x² – 4y² D and the xy-plane. Use x = 4u cos v, y = 2u sin v, z = w.
Find the mass and center of mass of the thin constant-density plates shown in the figure. (-2,2) (-4,0) (2, 2) (4,0)
Evaluate the following integrals in spherical coordinates. IT 0 p² sin o dp do de 4 Elo 2
Evaluate the following integrals in cylindrical coordinates. dx dz dy z(z² + zx + 1) 0 DJTJ V1-2
Evaluate the following integrals in spherical coordinates. 2π pπ/4 p2 sec o "JIJ 0 (p³) p² sin o dp do dº 4 X y
Find the mass and center of mass of the thin constant-density plates shown in the figure. (-4, 2) (-2,0) (-2, -1). (-4,-4) (4,2) -(2,0) -(2, -1) (4,-4)
Use integration in cylindrical coordinates to find the volume of the following regions. The region bounded by the plane z = √29 and the hyperboloid z = √4 + x² + y² V29 z = √4 + x² + y² Xx
Use a change of variables to evaluate the following integrals. If a dV; D is bounded by the upper half of the ellipsoid D x²/9 + y²/4 + z² = 1 and the xy-plane. Use x = 3u, y = 2v, z = w.
Evaluate the following integrals in spherical coordinates. 2π π/3 π/6 Jo 2 csc 4 p² sin o dp do do X 2014 2 FIM y
Use spherical coordinates to find the volume of the following regions. The region bounded by the cylinders r = 1 and r = 2, and the cones = 7/6 and = π/3 4 X 2 1 16 Elm y
Use spherical coordinates to find the volume of the following regions. The region bounded by the sphere p sphere p = 1, z ≥ 0 X = 2 cos and the hemi-
Use spherical coordinates to find the volume of the following regions. The cardioid of revolution D = {(p, q,0): 0 ≤ p ≤ 1 + cos 0,0 ≤ ≤ π,0 ≤ 0 ≤ 2πT }
Use integration in spherical coordinates to find the volume of the following regions. The rose petal of revolution D = {(p, q,0): 0 ≤ p ≤ 4 sin 24, 0 ≤ ≤ π/2,0 ≤ 0 ≤ 2π} X
Evaluate the following integrals in spherical coordinates. π/44 sec p II. 0 2 sec 4 p² sin o dp do do
Use integration in spherical coordinates to find the volume of the following regions. The cardioid of revolution D = {(p, q,0): 0 ≤ p ≤ (1 cos p)/2,0 ≤ X ZA ≤ 7,0 ≤ 0 ≤ 2}
Use integration in spherical coordinates to find the volume of the following regions. The region above the cone = 7/4 and inside the sphere p = 4 cos p * 2
Use spherical coordinates to find the volume of the following regions. That part of the ball p ≤ 4 that lies between the planes z = 2 and z = 2√3 X 2 z = 2√3 z = 2
Use spherical coordinates to find the volume of the following regions. The region outside the cone p = 4 cos p = π N A π/4 and inside the sphere 2 E
Use spherical coordinates to find the volume of the following regions. The region inside the cone z = (x² + y²)¹/² that lies between the planes z = 1 and z = 2 x z = 2 z = 1
Consider the following two- and three-dimensional regions. Specify the surfaces and curves that bound the region, choose a convenient coordinate system, and compute the center of mass assuming
Evaluate the following iterated integrals. SS 0 J1 3y √x + y² dx dy
Explain the meaning of logb x.
Assume logb x = 0.36, logb y = 0.56, and logb z = 0.83. Evaluate the following expressions. logb √xy N
Assume logb x = 0.36, logb y = 0.56, and logb z = 0.83. Evaluate the following expressions. log.x²
Assume logb x = 0.36, logb y = 0.56, and logb z = 0.83. Evaluate the following expressions. logb b²x5/2 Vy
Assume that b > 0 and b ≠ 1. Show that log1/b x = - logb x.
Graph the following functions. f(x) 3x 1 - x + 1 if x < 1 if x ≥ 1
Graph the following functions. f(x) = 3x - 1 -2x + 1 if x ≤ 0 if x > 0
Graph the following functions. -2x 1 if x < -1 f(x) = 1 if -1 ≤ x ≤ 1 2x 1 ifx > 1

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