Find the positively oriented simple closed curve C for which the value of the line integral ∫C (y3 – y) dx - 2x3 dy is a maximum.
> Solve the differential equation. 9y'' + 4y = 0
> Solve the differential equation. 4y'' + 4y' + y = 0
> Solve the differential equation. y'' + y' - 12y = 0
> Solve the differential equation. y'' + 2y = 0
> 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. Sſ Da f ds = {[[ v?fav v²fdV E
> Evaluate the surface integral. ∫∫S F ∙ dS, where F (x, y, z) = xz i - 2y j + 3x k and S is the sphere x2 + y2 + z2 = 4 with outward orientation
> 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. curl F· dS = 0
> Solve the boundary-value problem, if possible. y'' + 6y' = 0, y (0) = 1, y (1) = 0
> 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. || a·n ds = 0, where a is a constant vector %3
> Solve the initial-value problem. 4y'' + 4y' + 3y = 0, y (0) = 0, y' (0) = 1
> Solve the initial-value problem. y'' – y' - 12y = 0, y (1) = 0, y' (1) = 1
> Solve the initial-value problem. 3y'' - 2y' - y = 0, y (0) = 0, y' (0) = -4
> Solve the initial-value problem. 9y'' + 12y' + 4y = 0, y (0) = 1, y' (0) = 0
> Solve the initial-value problem. y'' - 2y' - 3y = 0, y (0) = 2, y'(0) = 2
> Solve the initial-value problem. y'' + 3 = 0, y (0) = 1, y' (0) = 3
> Use a computer algebra system to plot the vector field F (x, y, z) = sin x cos2y i + sin3y cos4z j + sin5z cos6x k in the cube cut from the first octant by the planes x = π/2, y = π/2, and z = π/2. Then compute the flux across the surface of the cube.
> Evaluate the surface integral. ∫∫S (x2z + y2z) dS, where S is the part of the plane z = 4 + x + y that lies inside the cylinder x2 + y2 = 4
> 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) = ey tan z i + y√3 - x2 j + x sin y k, S is the surface of the solid that lies above the xy-plane and below the surface z = 2
> Use the Divergence Theorem to calculate the surface integral ∫∫S F dS; that is, calculate the flux of F across S. F = |r |2 r, where r = x i + y j + z k, S is the sphere with radius R and center the origin
> Solve the differential equation. 3 d2V/dt2 + 4 dV/dt + 3V = 0
> 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
> Define the linearization of f at (a, b). What is the corresponding linear approximation? What is the geometric interpretation of the linear approximation?
> (a). Write the definition of the triple integral of f over a rectangular box B. (b). How do you evaluate ∫∫∫B f (x, y, z) dV? (c). How do you define ∫∫∫B f (x, y, z) dV if E is a bounded solid region that is not a box? (d). What is a type 1 solid region?
> What is a function of three variables? How can you visualize such a function?
> 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) = (y2 - 2x) i + 2xy j
> How do you change from rectangular coordinates to polar coordinates in a double integral? Why would you want to make the change?
> Evaluate the surface integral. ∫∫S z dS, where S is the part of the paraboloid z = x2 + y2 that lies under the plane z = 4
> (a). How do you define ∫∫D f (x, y) dA if D is a bounded region that is not a rectangle? (b). What is a type I region? How do you evaluate ∫∫D f (x, y) dA if D is a type I region? (c). What is a type II region? How do you evaluate ∫∫D f (x, y) dA if D is
> The figure shows a curve C and a contour map of a function f whose gradient is continuous. Find ∫C ∇f dr. yA 60 50 40 C 30 20 10
> What does lim (x, y) → (a, b) f (x, y) = L mean? How can you show that such a limit does not exist?
> If F is the vector field of Example 5, show that ∫C F ∙ dr = 0 for every simple closed path that does not pass through or enclose the origin.
> (a). What is a closed set in R2? What is a bounded set? (b). State the Extreme Value Theorem for functions of two variables. (c). How do you find the values that the Extreme Value Theorem guarantees?
> State the Second Derivatives Test.
> (a). If f has a local maximum at (a, b), what can you say about its partial derivatives at (a, b)? (b). What is a critical point of f?
> (a). Write an expression as a limit for the directional derivative of f at (x0, y0) in the direction of a unit vector u =〈a, b〉. How do you interpret it as a rate? How do you interpret it geometrically? (b). If f is differentiable, write an expression fo
> Match the vector fields F with the plots labeled I–IV. Give reasons for your choices. F (x, y) = 〈y, x - y〉 3 -3 -3 en
> 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.
> (a). Find an equation of the tangent plane at the point (4, -2, 1) to the parametric surface S given by (b). Use a computer to graph the surface S and the tangent plane found in part (a). (c). Set up, but do not evaluate, an integral for the surface a
> (a). If a transformation T is given by x = g (u, v), y = h (u, v), what is the Jacobian of T? (b). How do you change variables in a double integral? (c). How do you change variables in a triple integral?
> (a). How do you change from rectangular coordinates to cylindrical coordinates in a triple integral? (b). How do you change from rectangular coordinates to spherical coordinates in a triple integral? (c). In what situations would you change to cylindrica
> Determine whether or not the given set is (a) open, (b) connected, and (c) simply-connected. {(x, y) | 0 < y < 3}
> 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 4 -4 -4 4.
> 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.
> (a). Write formulas for the unit normal and binormal vectors of a smooth space curve r (t). (b). What is the normal plane of a curve at a point? What is the osculating plane? What is the osculating circle?
> Explain how the method of Lagrange multipliers works in finding the extreme values of f (x, y, z) subject to the constraint g (x, y, z) = k. What if there is a second constraint h (x, y, z) = c?
> What do the following statements mean? (a). f has a local maximum at (a, b). (b). f has an absolute maximum at (a, b). (c). f has a local minimum at (a, b). (d). f has an absolute minimum at (a, b). (e). f has a saddle point at (a, b).
> (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 differential equations do these functions satisf
> 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 lines. (a). Use a sketch of the vector field F (x, y) = x i
> At time t = 1, a particle is located at position (1, 3). If it moves in a velocity field F (x, y) = 〈xy - 2, y^2 - 10〉 find its approximate location at time t = 1.05.
> Find the area of the part of the sphere x2 + y2 + z2 = a2 that lies inside the cylinder x2 + y2 = ax.
> The figure shows the surface created when the cylinder y2 + z2 = 1 intersects the cylinder x2 + z2 = 1. Find the area of this surface. ZA x- y
> Find the area of the part of the sphere x2 + y2 + z2 = 4z that lies inside the paraboloid z = x2 + y2.
> 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
> (a). Show that the parametric equations x = a sin u cos v, y = b sin u sin v, z = c cos u, 0 < u
> (a). Set up, but do not evaluate, a double integral for the area of the surface with parametric equations (b). Eliminate the parameters to show that the surface is an elliptic paraboloid and set up another double integral for the surface area. (c). Use
> Match the vector fields F on R3 with the plots labeled I–IV. Give reasons for your choices. F (x, y, z) = x i + y j + 3 k III 1 -1 -101 y
> Match the vector fields F with the plots labeled I–IV. Give reasons for your choices. F (x, y) = 〈y, y + 2〉 III 3 -3 3 -3 هر ار ا دهه ا
> Let F (x) = (r2 - 2r) x, where x = 〈x, y〉 and r − |x |. Use a CAS to plot this vector field in various domains until you can see what is happening. Describe the appearance of the plot and explain it by finding the points where F (x) = 0.
> (a). Sketch the curve C with parametric equations x = cos t y = sin t z = sin t 0 (b) Find ſc 2.xe²" dx + (2.x²e + 2y cot z) dy – y°csc?z dz.
> State the Chain Rule for the case where z = f (x, y) and x and y are functions of one variable. What if x and y are functions of two variables?
> (a). Write expressions for the partial derivatives fx (a, b) and fy (a, b) as limits. (b). How do you interpret fx (a, b) and fy (a, b) geometrically? How do you interpret them as rates of change? (c). If f (x, y) is given by a formula, how do you calcul
> 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 wire (as in the figure). Ampèreâ€
> 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 by F on the object. (meters) C 1 (meters)
> Find the area of the surface. The part of the sphere x2 + y2 + z2 = b2 that lies inside the cylinder x2 + y2 = a2, where 0 < a < b
> Suppose f is a continuous function defined on a rectangle R = [a, b] × [c, d]. (a). Write an expression for a double Riemann sum of f. If f (x, y) > 0, what does the sum represent? (b). Write the definition of ∫∫R f (x, y) dA as a limit. (c). What is the
> 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 varies from 3 m to 5 m. Suppose that 1 L of paint cover
> Find the area of the surface. The part of the paraboloid y = x2 + z2 that lies within the cylinder x2 + z2 = 16
> Find the area of the surface. The part of the surface x = z2 + y that lies between the planes y = 0, y = 2, z = 0, and z = 2
> 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. C is the parabola y = 1 + x2 from (-1, 2) to (1, 2) y F(x, y) = -i+ /x² + y? =j. Vx? +
> Let S be a smooth parametric surface and let P be a point such that each line that starts at P intersects S at most once. The solid angle Ω (S) subtended by S at P is the set of lines starting at P and passing through S. Let S (a) be
> 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 arc of the circle x2 + y2 = 4 traversed counterclockwi
> 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.
> Let S be the part of the sphere x2 + y2 + z2 = 25 that lies above the plane z = 4. If S has constant density k, find (a) the center of mass and (b) the moment of inertia about the z-axis.
> Sketch the vector field F by drawing a diagram like Figure 5 or Figure 9. Figure 9: F (x, y, z) = z i ZA
> (a). What does it mean to say that f is differentiable at (a, b)? (b). How do you usually verify that f is differentiable?
> How do you find a tangent plane to each of the following types of surfaces? (a). A graph of a function of two variables, z = f (x, y) (b). A level surface of a function of three variables, F (x, y, z) = k
> Find a formula for ∫∫S F dS similar to Formula 10 for the case where S is given by x = k (y, z) and n is the unit normal that points forward (that is, toward the viewer when the axes are drawn in the usual way).
> Find a formula for ∫∫S F dS similar to Formula 10 for the case where S is given by y = h (x, z) and n is the unit normal that points toward the left.
> (a). Suppose that F is an inverse square force field, that is, F (r) = cr/ |r |3 for some constant c, where r = x i + y j + z k. Find the work done by F in moving an object from a point P1 along a path to a point P2 in terms of the distances d1 and d2 fr
> Consider the boundary-value problem y'' - 2y' + 2y = 0, y (a) = c, y (b) = d. (a). If this problem has a unique solution, how are a and b related? (b). If this problem has no solution, how are a, b, c, and d related? (c). If this problem has infinitely m
> (a). What does it mean to say that ∫C F ∙ dr is independent of path? (b). If you know that ∫C F ∙ dr is independent of path, what can you say about F?
> If a, b, and c are all positive constants and y (x) is a solution of the differential equation ay'' + by' + cy = 0, show that lim x→∞ y (x) = 0.
> Let L be a nonzero real number. (a). Show that the boundary-value problem y'' + λy = 0, y (0) = 0, y (L) = 0 has only the trivial solution y – 0 for the cases λ = 0 and λ < 0. (b). For the case λ > 0, find the values of λ for which this Problem has a non
> Solve the boundary-value problem, if possible. y'' + 4y' + 20y = 0, y (0) = 1, y (π) = e-2π
> Solve the boundary-value problem, if possible. y'' + 4y' + 20y = 0, y (0) = 1, y (π) = 2
> Solve the boundary-value problem, if possible. 4y'' - 4y' + y = 0, y (0) = 4, y (2) = 0
> Prove the identity, assuming that the appropriate partial derivatives exist and are continuous. If f is a scalar field and F, G are vector fields, then f F, F ∙ G, and F × G are defined by curl (curl F) = grad (div F) - &
> Solve the boundary-value problem, if possible. y'' - 8y' + 17y = 0, y (0) = 3, y (π) = 2
> Solve the boundary-value problem, if possible. y'' + 4y' + 4y = 0, y (0) = 2, y (1) = 0
> Find a parametric representation for the surface. The part of the plane z = x + 3 that lies inside the cylinder x2 + y2 = 1
> Solve the boundary-value problem, if possible. y'' + 16y = 0, y (0) = -3, y (π/8) = 2
> Assume that the earth is a solid sphere of uniform density with mass M and radius R = 3960 mi. For a particle of mass m within the earth at a distance r from the earth’s center, the gravitational force attracting the particle to the center is Fr = -GMr m
> Find a parametric representation for the surface. The part of the cylinder x2 + z2 = 9 that lies above the xy-plane and between the planes y = -4 and y = 4
> State Kepler’s Laws.
> Solve the differential equation using (a) undetermined coefficients and (b) variation of parameters. y" – y' = e*
> Solve the differential equation using (a) undetermined coefficients and (b) variation of parameters. y" – 2y' + y = e2x
> (a). Are the points P1 and P2 sources or sinks for the vector field F shown in the figure? Give an explanation based solely on the picture. (b). Given that F (x, y) = 〈x, y2〉, use the definition of divergence to verify
> If S is a sphere and F satisfies the hypotheses of Stokes’ Theorem, show that ∫∫S curl F ∙ dS = 0.
> The figure shows a pendulum with length L and the angle θ from the vertical to the pendulum. It can be shown that θ, as a function of time, satisfies the nonlinear differential equation where g is the acceleration due to gravi
