State Green’s Theorem.
> The regression equation for the Old Faithful data is y ̂= 12.481x + 33.683. Use this to predict the time until the next eruption for each eruption duration. (Recall from Section 9.1, Example 6, that x and y have a significant linear correlation.) 1. 2 mi
> Use the data to a. find the coefficient of determination r2 and interpret the result, and b. find the standard error of estimate se and interpret the result The table shows the total assets (in billions of dollars) of individual retirement accounts (IR
> Use the data to a. find the coefficient of determination r2 and interpret the result, and b. find the standard error of estimate se and interpret the result The table shows the amounts of crude oil (in thousands of barrels per day) produced by the Unit
> Find the critical values x_L^2 and x_R^2 for a two-tailed test when n = 51 and α = 0.01.
> Find the critical value x_0^2 for a left-tailed test when n = 30 and α = 0.05
> Find the critical value x_0^2 for a right-tailed test when n = 18 and α = 0.01.
> Use the data to a. find the coefficient of determination r2 and interpret the result, and b. find the standard error of estimate se and interpret the result The U.S. voting age populations (in millions) and the number of ballots cast (in millions) for
> The table shows the metacarpal bone lengths (in centimeters) and the heights (in centimeters) of 12 adults. The equation of the regression line is y Ì‚= 1.707x + 94.380. a. Find the coefficient of determination r2 and interpret the results
> 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) = eyz i + xzeyz j + xyeyz k
> 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) = i + sin z j + y cos z k
> Verify Green’s Theorem by using a computer algebra system to evaluate both the line integral and the double integral. P (x, y) = x3y4, Q (x, y) = x5y4, C consists of the line segment from (- π/2, 0) to ( π/2, 0) followed by the arc of the curve y = cos
> (a). Define the gradient vector ∇f for a function f of two or three variables. (b). Express Du f in terms of ∇f. (c). Explain the geometric significance of the gradient.
> 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) = y2z3 i + 2xyz3 j + 3xy2z2 k
> Let f be a scalar field and F a vector field. State whether each expression is meaningful. If not, explain why. If so, state whether it is a scalar field or a vector field. (a). curl f (b). grad f (c). div F (d). curl (grad f) (e). grad F (f). grad (d
> The vector field F is shown in the xy-plane and looks the same in all other horizontal planes. (In other words, F is independent of z and its z-component is 0.) (a). Is div F positive, negative, or zero? Explain. (b). Determine whether curl F = 0. If n
> The vector field F is shown in the xy-plane and looks the same in all other horizontal planes. (In other words, F is independent of z and its z-component is 0.) (a). Is div F positive, negative, or zero? Explain. (b). Determine whether curl F = 0. If n
> Verify that Green’s Theorem is true for the line integral ∫C xy2 dx - x2 y dy, where C consists of the parabola y = x2 from (-1, 1) to (1, 1) and the line segment from (1, 1) to (-1, 1).
> Show that F is a conservative vector field. Then find a function f such that F = ∇f. F(x, y) = (1 + xy)e" i + (e' + x*e")j
> Suppose F is a vector field on R3. (a). Define curl F. (b). Define div F. (c). If F is a velocity field in fluid flow, what are the physical interpretations of curl F and div F?
> Find the area of the part of the surface z = x2 + 2y that lies above the triangle with vertices (0, 0), (1, 0), and (1, 2).
> Find the work done by the force field F (x, y, z) = z i + x j + y k in moving a particle from the point (3, 0, 0) to the point (0, π/2, 3) along (a). a straight line (b). the helix x = 3 cos t, y = t, z = 3 sin t
> If f is a harmonic function, that is, ∇2 f = 0, show that the line integral ∫ fy dx - fx dy is independent of path in any simple region D.
> If f and t are twice differentiable functions, show that ∇2(f g) = f ∇2g + g ∇2 f + 2 ∇f ∙ ∇g
> If C is any piecewise-smooth simple closed plane curve and f and t are differentiable functions, show that ∫C f (x) dx + g (y) dy = 0
> A spring with a mass of 2 kg has damping constant 16, and a force of 12.8 N keeps the spring stretched 0.2 m beyond its natural length. Find the position of the mass at time t if it starts at the equilibrium position with a velocity of 2.4 m/s.
> A series circuit contains a resistor with R = 40 V, an inductor with L = 2 H, a capacitor with C = 0.0025 F, and a 12-V battery. The initial charge is Q = 0.01 C and the initial current is 0. Find the charge at time t.
> Use power series to solve the differential equation y'' – xy' - 2y = 0
> Use power series to solve the initial-value problem y'' + xy' + y = 0 y (0) = 0 y'(0) = 1
> Solve the boundary-value problem, if possible. y'' + 4y' + 29y = 0, y (0) = 1, y (π) = -e-2 π
> (a). Write the definition of the line integral of a scalar function f along a smooth curve C with respect to arc length. (b). How do you evaluate such a line integral? (c). Write expressions for the mass and center of mass of a thin wire shaped like a cu
> Solve the boundary-value problem, if possible. y'' + 4y' + 29y = 0, y (0) = 1, y (π) = -1
> Solve the initial-value problem. 9y'' + y = 3x + e-x, y (0) = 1, y'(0) = 2
> Solve the initial-value problem. y'' - 5y' + 4y = 0, y (0) = 0, y'(0) = 1
> Solve the initial-value problem. y'' - 6y' + 25y = 0, y (0) = 2, y'(0) = 1
> Solve the initial-value problem. y'' + 6y' = 0, y (1) = 3, y'(1) = 12
> Solve the differential equation. d2y/dx2 + y = csc x, 0 < x < π/2
> Solve the differential equation. d2y/dx2 – dy/dx - 6y = 1 + e-2x
> Solve the differential equation. d2y/dx2 + 4y = sin 2x
> Solve the differential equation. d2y/dx2 - 2 dy/dx + y = x cos x
> Solve the differential equation. d2y/dx2 + dy/dx - 2y = x2
> What is a vector field? Give three examples that have physical meaning.
> Prove the following identity: ∇ (F ∙ G) = (F ∙ ∇) G + (G ∙ ∇) F + F × curl G + G × curl F
> Solve the differential equation. y'' + 8y' + 16y = 0
> Let C be a simple closed piecewise-smooth space curve that lies in a plane with unit normal vector n = 〈a, b, c〉 and has positive orientation with respect to n. Show that the plane area enclosed by C is } 6г — су)
> Evaluate the line integral. ∫C x ds, C is the arc of the parabola y = x2 from (0, 0) to (1, 1)
> Solve the differential equation. 4y'' - y = 0
> Solve the differential equation. d2y/dx2 - 4 dy/dx + 5y = e2x
> Solve the differential equation. y'' + 3y = 0
> Write expressions for the area enclosed by a curve C in terms of line integrals around C.
> Solve the differential equation. 3y'' + 4y' - 3y = 0
> State the Fundamental Theorem for Line Integrals.
> The figure depicts the sequence of events in each cylinder of a four-cylinder internal combustion engine. Each piston moves up and down and is connected by a pivoted arm to a rotating crankshaft. Let P (t) and V (t) be the pressure and volume within a cy
> Solve the differential equation. Y'' – y' - 6y = 0
> Use the Divergence Theorem to calculate the surface integral ∫∫S F dS; that is, calculate the flux of F across S. S is the ellipsoid x2 + 2y2 + 3z2 = 4 F(х, у, 2) — хе'і + (г — е')j — хуk,
> Use the Divergence Theorem to calculate the surface integral ∫∫S F dS; that is, calculate the flux of F across S. S is the sphere with center the origin and radius 2 F(x, y, z) = (x³ + y³) i+ (y' + z')j + (z²
> Use the Divergence Theorem to calculate the surface integral ∫∫S F dS; that is, calculate the flux of F across S. S is the surface of the solid bounded by the cylinder y2 + z2 = 1 and the planes x = -1 and x = 2
> Use the Divergence Theorem to calculate the surface integral ∫∫S F dS; that is, calculate the flux of F across S. S is the surface of the box enclosed by the planes x = 0, x = a, y = 0, y = b, z = 0, and z = c, wh
> Use the Divergence Theorem to calculate the surface integral ∫∫S F dS; that is, calculate the flux of F across S. S is the surface of the box bounded by the coordinate planes and the planes x = 3, y = 2, and z = 1
> Use Stokes’ Theorem to evaluate ∫∫S curl F dS. F (x, y, z) = tan-1(x2/z2) i + x2y j + x2z2 k, S is the cone x = √y2 + z2, 0 < x < 2, oriented in the direction of the positive x-axis
> Use Stokes’ Theorem to evaluate ∫∫S curl F dS. F (x, y, z) = zey i + x cos y j + xz sin y k, S is the hemisphere x2 + y2 + z2 = 16, y > 0, oriented in the direction of the positive y-axis
> Prove each identity, assuming that S and E satisfy the conditions of the Divergence Theorem and the scalar functions and components of the vector fields have continuous second-order partial derivatives. || (S Vg) • n dS || (SV°g+ Vf• Vg) đV E
> Prove each identity, assuming that S and E satisfy the conditions of the Divergence Theorem and the scalar functions and components of the vector fields have continuous second-order partial derivatives. V(E) = || F· dS, where F(x, y, z) = xi+ yj+ z k
> (a). Define the line integral of a vector field F along a smooth curve C given by a vector function r (t). (b). If F is a force field, what does this line integral represent? (c). If F =〈P, Q, R〉, what is the connection between the line integral of F and
> Use the Divergence Theorem to evaluate ∫∫S (2x + 2y + z2) dS where S is the sphere x2 + y2 + z2 = 1.
> Verify that div E = 0 for the electric field E (x) = ∈Q |x |3 x.
> Suppose S and C satisfy the hypotheses of Stokes’ Theorem and f, t have continuous second-order partial derivatives. Use Exercises 24 and 26 in Section 16.5 to show the following. Exercises 24: Prove the identity, assuming that the ap
> Find (a) the curl and (b) the divergence of the vector field. F (x, y, z) = x3yz2 j + y4z3 k
> A vector field F is shown. Use the interpretation of divergence derived in this section to determine whether div F is positive or negative at P1 and at P2. -2 2 -2 tttt / 2. ! 11 !
> Evaluate the surface integral. ∫∫S (x + y + z) dS, S is the part of the half-cylinder x2 + z2 = 1, z > 0, that lies between the planes y = 0 and y = 2
> Use the Divergence Theorem to evaluate ∫∫S F ∙ dS, where F (x, y, z) = z2x i + (1/3 y3 + tan z) j + (x2z + y2) k and S is the top half of the sphere x2 + y2 + z2 = 1. [Hint: Note that S is not a closed surface. First compute integrals over S1 and S2, wh
> Use the Divergence Theorem to calculate the surface integral ∫∫S F dS; that is, calculate the flux of F across S. F = |r | r, where r = x i + y j + z k, S consists of the hemisphere z = √1 - x2 - y2 and the disk x2 + y2 < 1 in the xy-plane
> Use the Divergence Theorem to calculate the surface integral ∫∫S F dS; that is, calculate the flux of F across S. S is the surface of the solid bounded by the cylinder x2 + y2 = 4 and the planes z = y - 2 and z =
> Use the Divergence Theorem to calculate the surface integral ∫∫S F dS; that is, calculate the flux of F across S. S is the surface of the solid bounded by the paraboloid z = 1 - x2 - y2 and the xy-plane F(x, y
> (a). What is a conservative vector field? (b). What is a potential function?
> Use the Divergence Theorem to calculate the surface integral ∫∫S F dS; that is, calculate the flux of F across S. S is the surface of the tetrahedron enclosed by the coordinate planes and the plane x/a + y/b + z/c
> Verify that the Divergence Theorem is true for the vector field F on the region E. F (x, y, z) = 3x i + xy j + 2xz k, E is the cube bounded by the planes x = 0, x = 1, y = 0, y = 1, z = 0, and z = 1
> Evaluate the surface integral. ∫∫S x2yz dS, S is the part of the plane z = 1 + 2x + 3y that lies above the rectangle [0, 3g] × [0, 2]
> Use Stokes’ Theorem to evaluate ∫C F ∙ dr. In each case C is oriented counterclockwise as viewed from above. F (x, y, z) = i + (x + yz) j + (xy - √z) k, C is the boundary of the part of the plane 3x + 2y + z = 1 in the first octant
> Use Stokes’ Theorem to evaluate ∫C F ∙ dr. In each case C is oriented counterclockwise as viewed from above. F (x, y, z) = (x + y2) i + (y + z2) j + (z + x2) k, C is the triangle with vertices (1, 0, 0), (0, 1, 0), and (0, 0, 1)
> Use Stokes’ Theorem to evaluate ∫∫S curl F dS. F (x, y, z) = exy i + exz j + x2z k, S is the half of the ellipsoid 4x2 + y2 + 4z2 = 4 that lies to the right of the xz-plane, oriented in the direction of the positive y-axis
> Evaluate the surface integral. ∫∫S (x + y + z) dS, S is the parallelogram with parametric equations x = u + v, y = u - v, z = 1 + 2u + v, 0 < u < 2, 0 < v < 1
> Let r = x i + y j + z k and r = |r |. Verify each identity. (a). ∇r = r/r (b). ∇ × r = 0 (c). ∇ (1/r) = -r/r3 (d). = ln r = r/r2
> Evaluate the surface integral ∫∫S F ∙ dS for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward) orientation. F (x, y, z) = x i + y j + 5 k, S is the boundary
> Evaluate the surface integral ∫∫S F ∙ dS for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward) orientation. F (x, y, z) = x i + 2y j + 3z k, S is the cube wi
> In what ways are the Fundamental Theorem for Line Integrals, Green’s Theorem, Stokes’ Theorem, and the Divergence Theorem similar?
> Evaluate the surface integral ∫∫S F ∙ dS for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward) orientation. F (x, y, z) = yz i + zx j + xy k, S is the surfac
> Evaluate the surface integral ∫∫S F ∙ dS for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward) orientation. F (x, y, z) = y j - z k, S consists of the parabo
> Evaluate the surface integral ∫∫S F ∙ dS for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward) orientation. F (x, y, z) = y i - x j + 2z k, S is the hemisphe
> Evaluate the surface integral ∫∫S F ∙ dS for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward) orientation. F (x, y, z) = x i + y j + z2 k, S is the sphere w
> Evaluate the surface integral ∫∫S F ∙ dS for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward) orientation. F (x, y, z) = -x i - y j + z3 k, S is the part of
> Evaluate the surface integral ∫∫S F ∙ dS for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward) orientation. F (x, y, z) = xy i + yz j + zx k, S is the part o
> Evaluate the surface integral ∫∫S F ∙ dS for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward) orientation. F (x, y, z) = z i + y j + x k, S is the helicoid
> Evaluate the surface integral. ∫∫S xz dS, S is the boundary of the region enclosed by the cylinder y2 + z2 = 9 and the planes x = 0 and x + y = 5
> Evaluate the surface integral. ∫∫S (x2z + y2z) dS, S is the hemisphere x2 + y2 + z2 = 4, z > 0
> Evaluate the surface integral. ∫∫S y2 dS, S is the part of the sphere x2 + y2 + z2 = 1 that lies above the cone z = √x2 + y2
> State the Divergence Theorem.
> Verify that Stokes’ Theorem is true for the given vector field F and surface S. F (x, y, z) = y i + z j + x k, S is the hemisphere x2 + y2 + z2 = 1, y > 0, oriented in the direction of the positive y-axis
> Evaluate the surface integral. ∫∫S y2z2 dS, S is the part of the cone y = √x2 + z2 given by 0 < y < 5
> Evaluate the surface integral. ∫∫S z2 dS, S is the part of the paraboloid x = y2 + z2 given by 0 < x < 1
> Evaluate the surface integral. ∫∫S y dS, S is the surface z = 2/3 (x3/2 + y3/2), 0 < x < 1, 0 < y < 1
