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mathematics
precalculus

Questions and Answers of Precalculus

Use Green’s Theorem to evaluate ʃC F • dr. (Check the orientation of the curve before applying the theorem.)F(x, y) = (y cos x - xy sin x, xy + x cos x), C is the triangle from (0, 0) to (0, 4)
Use Green’s Theorem to evaluate the line integral along the given positively oriented curve.ʃC (1 - y3) dx + (x3 + ey2) dy, C is the boundary of the region between the circles x2 + y2 = 4 and x2 +
Use Green’s Theorem to evaluate the line integral along the given positively oriented curve.ʃC y2 dx - x2 dy, C is the circle x2 + y2 = 4
Use Green’s Theorem to evaluate the line integral along the given positively oriented curve.ʃC y4 dx + 2xy3 dy, C is the ellipse x2 + 2y2 = 2
Use Green’s Theorem to evaluate the line integral along the given positively oriented curve.ʃC (y + e√x) dx + (2x + cos y2) dy, C is the boundary of the region enclosed by the parabolas y = x2
Use Green’s Theorem to evaluate the line integral along the given positively oriented curve.ʃC (x2 + y2) dx + (x2 - y2) dy, C is the triangle with vertices (0, 0), (2, 1), and (0, 1)
Use Green’s Theorem to evaluate the line integral along the given positively oriented curve.ʃC (x2 + y2) dx + (x2 - y2) dy, C is the triangle with vertices (0, 0), (2, 1), and (0, 1)
Evaluate the line integral by two methods: (a) directly and (b) using Green’s Theorem.ʃC yex dx + 2ex dy, C is the rectangle with vertices (0, 0), (3, 0), (3, 4), and (0, 4)
Evaluate the line integral by two methods: (a) directly and (b) using Green’s Theorem.∮c x2y2 dx + xy dy, C consists of the arc of the parabola y = x2 from (0, 0) to (1, 1) and the line segments
Evaluate the line integral by two methods: (a) directly and (b) using Green’s Theorem.∮c xy dx + x2y3 dy, C is the triangle with vertices (0, 0), (1, 0), and (1, 2)
Evaluate the line integral by two methods: (a) directly and (b) using Green’s Theorem.∮c y dx - x dy, C is the circle with center the origin and radius 4
Determine whether or not the given set is (a) open, (b) connected, and (c) simply-connected.{(x, y) | (x, y) ≠ (2, 3)}
Determine whether or not the given set is (a) open, (b) connected, and (c) simply-connected.{(x, y) | 1 < x2 9+ y2 < 4, y > 0
Determine whether or not the given set is (a) open, (b) connected, and (c) simply-connected.{(x, y) | 1 < |x| < 2}
Determine whether or not the given set is (a) open, (b) connected, and (c) simply-connected.{(x, y) | 0 < y < 3}
Use Exercise 29 to show that the line integral ∫C y dx + x dy + xyz dz is not independent of path.
Let F = ∇f , where f (x, y) = sin(x - 2y). Find curves C1 and C2 that are not closed and satisfy the equation. F· dr = 1 (a) (b) |F· dr = 0
If F(x, y) = sin y i + (1 + x cos y) j, use a plot to guess whether F is conservative. Then determine whether your guess is correct.
Is the vector field shown in the figure conservative?Explain. УА х
Is the vector field shown in the figure conservative?Explain. y. х
Find the work done by the force field F in moving an object from P to Q.F(x, y) = (2x + y) i + x j; P(1, 1), Q(4, 3)
Find the work done by the force field F in moving an object from P to Q.F(x, y) = x3 i + y3 j; P(1, 0), Q(2, 2)
Suppose an experiment determines that the amount of work required for a force field F to move a particle from the point (1, 2) to the point (5, -3) along a curve C1 is 1.2 J and the work done by F in
Suppose you’re asked to determine the curve that requires the least work for a force field F to move a particle from one point to another point. You decide to check first whether F is conservative,
Show that the line integral is independent of path and evaluate the integral. Se sin y dx + (x cos y – sin y) dy, C is any path from (2, 0) to (1, 77)
(a) Find a function f such that F = ∇f and (b) use part (a) to evaluate ∫C F • dr along the given curve C.F(x, y, z) = sin y i + (x cos y + cos z) j - y sin z k, C: r(t) = sin t i + t j + 2t k,
Show that the line integral is independent of path and evaluate the integral.
(a) Find a function f such that F = ∇f and (b) use part (a) to evaluate ∫C F • dr along the given curve C.F(x, y, z) = yzexz i + exz j + xyexz k, C: r(t) = (t2 + 1) i + (t2 - 1) j + (t2 - 2t)
(a) Find a function f such that F = ∇f and (b) use part (a) to evaluate ∫C F • dr along the given curve C.F(x, y, z) = (y2z + 2xz2) i + 2xyz j + (xy2 + 2x2z) k, C: x = √t , y = t + 1, z = t2,
(a) Find a function f such that F = ∇f and (b) use part (a) to evaluate ∫C F • dr along the given curve C.F(x, y, z) = yz i + xz j + (xy + 2z) k, C is the line segment from (1, 0, -2) to (4, 6,
(a) Find a function f such that F = ∇f and (b) use part (a) to evaluate ∫C F • dr along the given curve C.
(a) Find a function f such that F = ∇f and (b) use part (a) to evaluate ∫C F • dr along the given curve C. F(x, y) = x²y³ i + x³y² j, C: r(t) = (t – 2t, t³ + 2t), .3,,2 0
(a) Find a function f such that F = ∇f and (b) use part (a) to evaluate ∫C F • dr along the given curve C. F(x, y) = (3 + 2xy³) i + 2x²yj, Cis the arc of the hyperbola y = 1/x from (1, 1) to
The figure shows the vector field F(x, y) = (2xy, x2)and three curves that start at (1, 2) and end at (3, 2).(a) Explain why ∫C F • dr has the same value for all three curves.(b) What is this
Determine whether or not F is a conservative vector field. If it is, find a function f such that F = ∇f.F(x, y) = y2exy i + (1 + xy)exy j
Determine whether or not F is a conservative vector field. If it is, find a function f such that F = ∇f.F(x, y) = (ln y + y/x) i + (ln x + x/y) j
Determine whether or not F is a conservative vector field. If it is, find a function f such that F = ∇f.F(x, y) = (y2 cos x + cos y) i + (2y sin x - x sin y) j
Determine whether or not F is a conservative vector field. If it is, find a function f such that F = ∇f.F(x, y) = (2xy + y-2) i + (x2 - 2xy-3) j, y > 0
Determine whether or not F is a conservative vector field. If it is, find a function f such that F = ∇f.F(x, y) = (yex + sin y) i + (ex + x cos y) j
Determine whether or not F is a conservative vector field. If it is, find a function f such that F = ∇f.F(x, y) = yex i + (ex + ey) j
Determine whether or not F is a conservative vector field. If it is, find a function f such that F = ∇f.F(x, y) = (y2 - 2x) i + 2xy j
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 + y2) i + (x2 + 2xy) j
A table of values of a function f with continuous gradient is given. Find ʃC ∇f • dr, where C has parametric equationsx = t2 + 1 y = t3 + t 0 < t < 1 х 4 1 2. 2. 3. 2.
The figure shows a curve C and a contour map of a function f whose gradient is continuous.find SeVf• dr.
Experiments show that a steady current I in a long wire produces a magnetic field B that is tangent to any circle that lies in the plane perpendicular to the wire and whose center is the axis of the
An object moves along the curve C shown in the figure from (1, 2) to (9, 8). The lengths of the vectors in the force field F are measured in newtons by the scales on the axes. Estimate the work done
If C is a smooth curve given by a vector function r(t), a < t < b, show that r- dr = {[Ir(b)P° – |r(a)|]
If C is a smooth curve given by a vector function r(t), a < t < b, and v is a constant vector, show that v. dr - v: [r(ь) — r(а)] [r(b) – r(a)] Ус
The base of a circular fence with radius 10 m is given by x = 10 cos t, y = 10 sin t. The height of the fence at position (x, y) is given by the function h(x, y) = 4 + 0.01(x2 - y2), so the height
(a) Show that a constant force field does zero work on a particle that moves once uniformly around the circle x2 + y2 = 1.(b) Is this also true for a force field F(x) = kx, where k is a constant and
Suppose there is a hole in the can of paint in Exercise 45 and 9 lb of paint leaks steadily out of the can during the man’s ascent. How much work is done?
A 160-lb man carries a 25-lb can of paint up a helical staircase that encircles a silo with a radius of 20 ft. If the silo is 90 ft high and the man makes exactly three complete revolutions climbing
An object with mass m moves with position function r(t) = a sin t i + b cos t j + ct k, 0 < t < π/2. Find the work done on the object during this time period.
The position of an object with mass m at time t is r(t) = at2 i + bt3 j, 0 < t < 1.(a) What is the force acting on the object at time t?(b) What is the work done by the force during the time
The force exerted by an electric charge at the origin on a charged particle at a point (x, y, z) with position vector r = (x, y, z) is F(r) = Kr/|r|3 where K is a constant. (See Example 16.1.5.) Find
Find the work done by the force field F(x, y, z) = (x - y2 , y - z2, z - x2) on a particle that moves along the line segment from (0, 0, 1) to (2, 1, 0).
Find the work done by the force field F(x, y) = x2 i + yex j on a particle that moves along the parabola x = y2 + 1 from (1, 0) to (2, 1).
Find the work done by the force fieldF(x, y) = x i + ( y + 2) jin moving an object along an arch of the cycloidr(t) = (t - sin t) i + (1 - cos t) j 0 < t < 2π
If a wire with linear density p(x, y, z) lies along a space curve C, its moments of inertia about the x-, y-, and z-axes are defined asFind the moments of inertia for the wire in Exercise 35. I, = [
If a wire with linear density p(x, y) lies along a plane curve C, its moments of inertia about the x- and y-axes are defined asFind the moments of inertia for the wire in Example 3. 1. - [y°p(x, y)
Find the mass and center of mass of a wire in the shape of the helix x = t, y = cos t, z = sin t, 0 < t < 2π, if the density at any point is equal to the square of the distance from the origin.
(a) Write the formulas similar to Equations 4 for the center of mass (x, y, z) of a thin wire in the shape of a space curve C if the wire has density function p(x, y, z).(b) Find the center of mass
A thin wire has the shape of the first-quadrant part of the circle with center the origin and radius a. If the density function is p(x, y) = kxy, find the mass and center of mass of the wire.
A thin wire is bent into the shape of a semicircle x2 + y2 = 4, x > 0. If the linear density is a constant k, find the mass and center of mass of the wire.
(a) Find the work done by the force field F(x, y) = x2 i + xy j on a particle that moves once around the circle x2 + y2 = 4 oriented in the counterclockwisedirection.(b) Use a computer algebra system
Find the exact value of ʃC x3y2z ds, where C is the curve with parametric equations x = e2-t cos 4t, y = e2-t sin 4t, z = e2-t, 0 < t < 2π.
Use a graph of the vector field F and the curve C to guess whether the line integral of F over C is positive, negative, or zero. Then evaluate the line integral.F(x, y) = x/√x2 + y2 i + y/√x2 +
Use a graph of the vector field F and the curve C to guess whether the line integral of F over C is positive, negative, or zero. Then evaluate the line integral.F(x, y) = (x - y) i + xy j, C is the
Evaluate the line integral ʃC F • dr, where C is given by the vector function r(t).F(x, y, z) = x i + y j + xy k, r(t) = cos t i + sin t j + t k, 0 < t < π
Evaluate the line integral ʃC F • dr, where C is given by the vector function r(t).F(x, y, z) = sin x i + cos y j + xz k, r(t) = t3 i - t2 j + t k, 0 < t < 1
Evaluate the line integral ʃC F • dr, where C is given by the vector function r(t).F(x, y, z) = (x + y2) i + xzj + (y + z) k, r(t) = t2 i + t3 j - 2t k, 0 < t < 2
Evaluate the line integral ʃC F • dr, where C is given by the vector function r(t).F(x, y) = xy2 i - x2 j, r(t) = t3 i + t2 j, 0 < t < 1
The figure shows a vector field F and two curves C1 and C2. Are the line integrals of F over C1 and C2 positive, negative, or zero? Explain. ул ¡C2}
Evaluate the line integral, where C is the given curve.ʃC (y + z) dx + (x + z) dy + (x + y) dz, C consists of line segments from (0, 0, 0) to (1, 0, 1) and from (1, 0, 1) to (0, 1, 2)
Evaluate the line integral, where C is the given curve.ʃC z2 dx + x2 dy + y2 dz, C is the line segment from (1, 0, 0) to (4, 1, 2)
Evaluate the line integral, where C is the given curve.ʃC y dx + z dy + x dz, C: x = √t , y = t, z = t2, 1 < t < 4
Evaluate the line integral, where C is the given curve.ʃC xyeyz dy, C: x = t, y = t2, z = t3, 0 < t < 1
Evaluate the line integral, where C is the given curve.ʃC (x2 + y2 + z2)ds, C: x = t, y = cos 2t, z = sin 2t, 0 < t < 2π
Evaluate the line integral, where C is the given curve.ʃC xeyz ds, C is the line segment from (0, 0, 0) to (1, 2, 3)
Evaluate the line integral, where C is the given curve.ʃC y2z ds, C is the line segment from (3, 1, 2) to (1, 2, 5)
Evaluate the line integral, where C is the given curve.ʃC x2y ds, C: x = cos t, y = sin t, z = t, 0 < t < ∇/2
Evaluate the line integral, where C is the given curve.ʃC x2 dx + y2 dy, C consists of the arc of the circle x2 + y2 = 4 from (2, 0) to (0, 2) followed by the line segment from (0, 2) to (4, 3)
Evaluate the line integral, where C is the given curve.ʃC (x + 2y) dx + x2 dy, C consists of line segments from (0, 0) to (2, 1) and from (2, 1) to (3, 0)
Evaluate the line integral, where C is the given curve.ʃC ex dx, C is the arc of the curve x = y3 from (-1, -1) to (1, 1)
Evaluate the line integral, where C is the given curve.ʃC (x2y + sin x) dy, C is the arc of the parabola y = x2 from (0, 0) to (π, π2)
Evaluate the line integral, where C is the given curve.ʃC xey ds, C is the line segment from (2, 0) to (5, 4)
Evaluate the line integral, where C is the given curve.ʃC xy4 ds, C is the right half of the circle x2 + y2 = 16
Evaluate the line integral, where C is the given curve.ʃC (x/y) ds, C: x = t3, y = t4, 1 < t < 2
(a) Sketch the vector field F(x, y) = i + x j and then sketch some flow lines. What shape do these flow lines appear to have?(b) If parametric equations of the flow lines are x = x(t), y = y(t), what
The flow lines (or streamlines) of a vector field are the paths followed by a particle whose velocity field is the given vector field. Thus the vectors in a vector field are tangent to the flow
At time t = 1, a particle is located at position (1, 3). If it moves in a velocity fieldF(x, y) = (xy - 2, y2 - 10)find its approximate location at time t = 1.05.
A particle moves in a velocity field V(x, y) = (x2, x + y2). If it is at position (2, 1) at time t = 3, estimate its location at time t = 3.01.
Match the functions f with the plots of their gradient vector fields labeled I–IV. Give reasons for your choices.f (x, y) = sin√x2 + y2 II 4 4 4 4 -4 -4 -4 -4 Ш IV 4 4 -4 -4 4 -4 -4 4-
Match the functions f with the plots of their gradientvector fields labeled I–IV. Give reasons for your choices.f (x, y) = (x + y)2 II 4 4 4 4 -4 -4 -4 -4 Ш IV 4 4 -4 -4 4 -4 -4 4-
Match the functions f with the plots of their gradient vector fields labeled I–IV. Give reasons for your choices.f(x, y) = x(x + y) II 4 4 4 4 -4 -4 -4 -4 Ш IV 4 4 -4 -4 4 -4 -4 4-
Match the functions f with the plots of their gradient vector fields labeled I–IV. Give reasons for your choices.f (x, y) = x2 + y2 II 4 4 4 4 -4 -4 -4 -4 Ш IV 4 4 -4 -4 4 -4 -4 4-
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) = ln(1 + x2 + 2y2)
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) = cos x - 2 sin y
Find the gradient vector field ∇f of f and sketch it.f (x, y) = 1/2 (x2 - y2)
Find the gradient vector field ∇f of f and sketch it.f (x, y) = 1/2(x - y)2

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