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.
> 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
> 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.
> 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.
> 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
> A particle moves along line segments from the origin to the points s1, 0, 0d, s1, 2, 1d, s0, 2, 1d, and back to the origin under the influence of the force field F (x, y, z) = z2 i + 2xy j + 4y2 k Find the work done.
> Graph the two basic solutions along with several other solutions of the differential equation. What features do the solutions have in common? d'y 2 dx? dy y= 0 %3D dx
> Graph the two basic solutions along with several other solutions of the differential equation. What features do the solutions have in common? d²y +2- dx? + 2y = 0 dx
> Discuss two applications of second-order linear differential equations.
> Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. 1. If F is a vector field, then div F is a vector field. 2. If F is a vector field, then curl F is
> Graph the two basic solutions along with several other solutions of the differential equation. What features do the solutions have in common? d²y dy 4 4 + y = 0 dx? dx
> Verify that Stokes’ Theorem is true for the given vector field F and surface S. F (x, y, z) = 2y i + x j - 2 k, S is the cone z2 = x2 + y2, 0 < z < 4, oriented downward
> Graph the particular solution and several other solutions. What characteristics do these solutions have in common? y'' + 4y = e-x
> Show that if ω0 ≠ ω, but ω /ω 0 is a rational number, then the motion described by Equation 6 is periodic.
> As in Exercise 9, consider a spring with mass m, spring constant k, and damping constant c = 0, and let ω = √k/m. If an external force F (t) = F0 cos t is applied (the applied frequency equals the natural frequency), use the
> Suppose a spring has mass m and spring constant k and let ω = √k/m. Suppose that the damping constant is so small that the damping force is negligible. If an external force F (t) = F0 cos 0t is applied, where ω0 ≠ ω , use the method of undetermined coe
> A spring has a mass of 1 kg and its damping constant is c = 10. The spring starts from its equilibrium position with a velocity of 1 m/s. Graph the position function for the following values of the spring constant k: 10, 20, 25, 30, 40. What type of damp
> A spring has a mass of 1 kg and its spring constant is k = 100. The spring is released at a point 0.1 m above its equilibrium position. Graph the position function for the following values of the damping constant c: 10, 15, 20, 25, 30. What type of dampi
> Identify the surface with the given vector equation. r (s, t) =〈3 cos t, s, sin t〉, -1 < s < 1
> Use Stokes’ Theorem to evaluate ∫∫S curl F dS. F (x, y, z) = xyz i + xy j + x2yz k, S consists of the top and the four sides (but not the bottom) of the cube with vertices (±1, ±1, ±1), oriented outward
> Solve the differential equation. y'' - 2y' + 10y = 0
> Verify that the Divergence Theorem is true for the vector field F on the region E. F (x, y, z) = 〈x2, 2y, z〉, E is the solid cylinder y2 + z2 < 9, 0 < x < 2
> A spring with a mass of 2 kg has damping constant 14, and a force of 6 N is required to keep the spring stretched 0.5 m beyond its natural length. The spring is stretched 1 m beyond its natural length and then released with zero velocity. Find the positi
> Use power series to solve the differential equation. y' = xy
> A spring has natural length 0.75 m and a 5-kg mass. A force of 25 N is needed to keep the spring stretched to a length of 1 m. If the spring is stretched to a length of 1.1 m and then released with velocity 0, find the position of the mass after t second
> What are the traces of a surface? How do you find them?
> (a). How do you find the distance from a point to a line? (b). How do you find the distance from a point to a plane? (c). How do you find the distance between two lines?
> (a). Describe a method for determining whether three points P, Q, and R lie on the same line. (b). Describe a method for determining whether four points P, Q, R, and S lie in the same plane.
> (a). How do you tell if two vectors are parallel? (b). How do you tell if two vectors are perpendicular? (c). How do you tell if two planes are parallel?
> Write a vector equation and a scalar equation for a plane.
