Calculate the value of the multiple integral. ∬D xy dA, where D = {(x, y) | 0 < y < 1, y2 < x < y + 2}
> Find the local maximum and minimum values and saddle points of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function. f (x, y) = x2 - xy + y2 +
> A pentagon is formed by placing an isosceles triangle on a rectangle, as shown in the figure. If the pentagon has fixed perimeter P, find the lengths of the sides of the pentagon that maximize the area of the pentagon.
> Shoe that ∫_0^∞(arctan πx-arctanx)/x dx=π/2 lnπ by first expressing the integral as an iterated integral.
> A package in the shape of a rectangular box can be mailed by the US Postal Service if the sum of its length and girth (the perimeter of a cross-section perpendicular to the length) is at most 108 in. Find the dimensions of the package with largest volume
> Find the points on the surface xy2z3 = 2 that are closest to the origin.
> Use Lagrange multipliers to find the maximum and minimum values of f subject to the given constraint(s). f (x, y, z) = x2 + 2y2 + 3z2; x + y + z = 1, x - y + 2z = 2
> Use Lagrange multipliers to find the maximum and minimum values of f subject to the given constraint(s). f (x, y, z) = xyz; x2 + y2 + z2 = 3
> (a). Evaluate ∬D 1/((x^2+y^2 )^(n/2) ) dA, where n is an integer and D is the region bounded by the circles with center the origin and radii r and R, 0 < r < R. (b). For what values of n does the integral in part (a) have a limit as r → +? (c). Find ∭E 1
> Use Lagrange multipliers to find the maximum and minimum values of f subject to the given constraint(s). f (x, y) = x2y; x2 + y2 = 1
> The Mean Value Theorem for double integrals says that if f is a continuous function on a plane region D that is of type I or II, then there exists a point sx0, y0 d in D such that Use the Extreme Value Theorem (14.7.8) and Property 15.2.11 of integrals
> Use the change of variables formula and an appropriate transformation to evaluate ∬R xy dA, where R is the square with vertices (0, 0), (1, 1), (2, 0), and (1, -1).
> Find the absolute maximum and minimum values of f on the set D. f (x, y) = e-x2-y2 (x2 + 2y2); D is the disk x2 + y2 < 4
> Find the absolute maximum and minimum values of f on the set D. f (x, y) = 4xy2 - x2y2 - xy3; D is the closed triangular region in the xy-plane with vertices (0, 0), (0, 6), and (6, 0)
> Suppose f is a differentiable function of one variable. Show that all tangent planes to the surface z = x f (y/x) intersect in a common point.
> Find the local maximum and minimum values and saddle points of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function. f (x, y) = (x2 + y) ey/2
> Find the local maximum and minimum values and saddle points of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function. f (x, y) = 3xy - x2y - xy2
> A lamp has three bulbs, each of a type with average lifetime 800 hours. If we model the probability of failure of a bulb by an exponential density function with mean 800, find the probability that all three bulbs fail within a total of 1000 hours.
> The joint density function for random variables X and Y is (a). Find the value of the constant C. (b). Find P (X 1). (c). Find P (X + Y S(x + y) if 0 <I< 3, 0< y< 2 f(x, y) = otherwise
> Find parametric equations of the tangent line at the point (-2, 2, 4) to the curve of intersection of the surface z = 2x2 - y2 and the plane z = 4.
> The contour map shows wind speed in knots during Hurricane Andrew on August 24, 1992. Use it to estimate the value of the directional derivative of the wind speed at Homestead, Florida, in the direction of the eye of the hurricane. 70 60 70 5565 U 8
> Use spherical coordinates to evaluate ∫_(-2)^2 ∫_0^(√(4-y^2 ) ∫_(-√(4-x^2-y^2 ) ^(√(4-x^2-y^2 ) y^2 √(x^2+y^2+z^2 ) dz dx dy
> Use polar coordinates to evaluate ∫_0^3 ∫ (-√(9-x^2 ) ^(√(9-x^2 ) (x^2+xy^2) dy dx
> Graph the surface z = x sin y, -3 < x < 3, -π < y < π, and find its surface area correct to four decimal places.
> Find the area of the part of the surface z = x2 + y that lies above the triangle with vertices (0, 0), (1, 0), and (0, 2).
> Find the extreme values of f subject to both constraints. f(x, y, z) = x + y + z; x² + z? = 2, x + y= 1
> (a). When is the directional derivative of f a maximum? (b). When is it a minimum? (c). When is it 0? (d). When is it half of its maximum value?
> (a). Find the centroid of a solid right circular cone with height h and base radius a. (Place the cone so that its base is in the xy-plane with center the origin and its axis along the positive z-axis.) (b). If the cone has density function ρ (x, y, z) =
> A lamina occupies the part of the disk x2 + y2 < a2 that lies in the first quadrant. (a). Find the centroid of the lamina. (b). Find the center of mass of the lamina if the density function is ρ (x, y) = xy2.
> Consider a lamina that occupies the region D bounded by the parabola x = 1 - y2 and the coordinate axes in the first quadrant with density function ρ (x, y) = y. (a). Find the mass of the lamina. (b). Find the center of mass. (c). Find the moments of ine
> Find the volume of the given solid. Above the paraboloid z = x2 + y2 and below the half-cone z = √(x^2 + y^2 )
> Find the volume of the given solid. One of the wedges cut from the cylinder x2 + 9y2 = a2 by the planes z = 0 and z = mx
> Find the volume of the given solid. Find the volume of the given solid. Bounded by the cylinder x2 + y2 = 4 and the planes z = 0 and y + z = 3
> Find the volume of the given solid. The solid tetrahedron with vertices (0, 0, 0), (0, 0, 1), (0, 2, 0), and (2, 2, 0)
> Find the volume of the given solid. Under the surface z = x2y and above the triangle in the xy-plane with vertices (1, 0), (2, 1), and (4, 0)
> Find the volume of the given solid. Under the paraboloid z = x2 + 4y2 and above the rectangle R = [0, 2] × [1, 4]
> Find the minimum value of f (x, y, z) = x2 + 2y2 + 3z2 subject to the constraint x + 2y + 3z = 10. Show that f has no maximum value with this constraint.
> The two legs of a right triangle are measured as 5 m and 12 m with a possible error in measurement of at most 0.2 cm in each. Use differentials to estimate the maximum error in the calculated value of (a) the area of the triangle and (b) the length of th
> Calculate the value of the multiple integral. ∭E yz dV, where E lies above the plane z = 0, below the plane z = y, and inside the cylinder x2 + y2 = 4
> Calculate the value of the multiple integral. ∭E z dV, where E is bounded by the planes y = 0, z = 0, x + y = 2 and the cylinder y2 + z2 = 1 in the first octant.
> Find the points on the hyperboloid x2 + 4y2 - z2 = 4 where the tangent plane is parallel to the plane 2x + 2y + z = 5.
> Use a computer to graph the surface z = x2 + y4 and its tangent plane and normal line at (1, 1, 2) on the same screen. Choose the domain and viewpoint so that you get a good view of all three objects.
> Find equations of (a) the tangent plane and (b) the normal line to the given surface at the specified point. sin (xyz) = x + 2y + 3z, (2, 21, 0)
> Find equations of (a) the tangent plane and (b) the normal line to the given surface at the specified point. xy + yz + zx = 3, (1, 1, 1)
> Find equations of (a) the tangent plane and (b) the normal line to the given surface at the specified point. x2 + 2y2 - 3z2 = 3, (2, -1, 1)
> Find equations of (a) the tangent plane and (b) the normal line to the given surface at the specified point. z = ex cos y, (0, 0, 1)
> Find equations of (a) the tangent plane and (b) the normal line to the given surface at the specified point. z = 3x2 - y2 + 2x, (1, -2, 1)
> The method of Lagrange multipliers assumes that the extreme values exist, but that is not always the case. Show that the problem of finding the minimum value of f (x, y) = x2 + y2 subject to the constraint xy = 1 can be solved using Lagrange multipliers,
> Calculate the value of the multiple integral. ∬D 1/(1+x^2 ) dA, where D is the triangular region with vertices (0, 0), (1, 1), and (0, 1).
> Calculate the value of the multiple integral. ∭E y/(1+x^2 dA, where D is bounded by y = √x, y = 0, x = 1
> Calculate the value of the multiple integral. ∬R yexy dA, where R = {(x, y) | 0 < x < 2, 0 < y < 3)
> Calculate the iterated integral by first reversing the order of integration. Vet 3 dx dy
> Calculate the iterated integral by first reversing the order of integration. CC cos(y?) dy dx Jo
> The speed of sound traveling through ocean water is a function of temperature, salinity, and pressure. It has been modeled by the function where C is the speed of sound (in meters per second), T is the temperature (in degrees Celsius), S is the salinit
> Sketch the solid consisting of all points with spherical coordinates (ρ, θ, σ) such that 0 < θ < π/2, 0 < σ < π/6, and 0 < ρ < 2 cos σ.
> Write the equation in cylindrical coordinates and in spherical coordinates. (a). x2 + y2 + z2 = 4 (b). x2 + y2 = 4
> If [x] denotes the greatest integer in x, evaluate the integral where R = {(x, y) | 1 || [x + y] dA R
> Identify the surfaces whose equations are given. (a). θ = π/4 (b). φ = π/4
> The spherical coordinates of a point are (8, π/4, π/6). Find the rectangular and cylindrical coordinates of the point.
> The rectangular coordinates of a point are (2, 2, -1). Find the cylindrical and spherical coordinates of the point.
> The cylindrical coordinates of a point are (2√(3 ), π/3, 2). Find the rectangular and spherical coordinates of the point.
> Write ∬R f (x, y) dA as an iterated integral, where R is the region shown and f is an arbitrary continuous function on R. y4 4, R -4 4 x
> Write ∬R f (x, y) dA as an iterated integral, where R is the region shown and f is an arbitrary continuous function on R. y. 4 R -4 -2 0 4 x 2.
> Calculate the iterated integral. ∫_0^1 ∫_0^y ∫_x^16xyz dz dx dy
> Calculate the iterated integral. ∫_0^π ∫_0^1 √(1-y^2 ) y sin x dz dy dx
> Calculate the iterated integral. ∫_0^1 ∫_x^(e^x) (3xy^2) dy dx
> Calculate the iterated integral. ∫_0^1 ∫_0^x (x^2) dy dx
> The plane x/a+y/b+z/c = 1 a > 0, b > 0, c > 0 cuts the solid ellipsoid into two pieces. Find the volume of the smaller piece. a? < 1 +
> Calculate the iterated integral. ∫_0^1 ∫_0^1 (ye^xy dx dy
> The Taylor polynomial approximation to functions of one variable that we discussed in Chapter 11 can be extended to functions of two or more variables. Here we investigate quadratic approximations to functions of two variables and use them to give insigh
> Find and sketch the domain of the function. f (x, y) = √(4 - x^2 - y^2 ) + √(1 - x^2 )
> A contour map is shown for a function f on the square R = [0, 3] × [0, 3]. Use a Riemann sum with nine terms to estimate the value of ∬R f (x, y) dA. Take the sample points to be the upper right corners of the squares.
> If f (x, y) has two local maxima, then f must have a local minimum. Find a linear approximation to the temperature function T (x, y) in Exercise 11 near the point (6, 4). Then use it to estimate the temperature at the point (5, 3.8). Exercise 11: If f
> If (2, 1) is a critical point of f and fxx (2, 1) fyy (2, 1) < [ fxy (2, 1)]2 Evaluate the limit or show that it does not exist. lim┬((x,y)→(1,1))2xy/(x^2+2y^2 )
> Make a rough sketch of a contour map for the function whose graph is shown. If f has a local minimum at (a, b) and f is differentiable at (a, b), then =f (a, b) = 0.
> If f (x, y) = ln y, then ∆f (x, y) = 1/y. Evaluate the limit or show that it does not exist. lim┬((x,y)→(1,1))2xy/(x^2+2y^2 )
> If f (x, y) = sin x + sin y, then -√2 A metal plate is situated in the xy-plane and occupies the rectangle 0 (a). Estimate the values of the partial derivatives Tx (6, 4) and Ty (6, 4). What are the units? (b). Estimate the value of
> If z = y + f (x2 - y2), where f is differentiable, show that Y ∂z/∂x + ∂z/∂y + = x
> Each of these extreme value problems has a solution with both a maximum value and a minimum value. Use Lagrange multipliers to find the extreme values of the function subject to the given constraint. f(x, y, z) = x* + y* + 24; x' + y? + z? = 1
> Use a tree diagram to write out the Chain Rule for the case where w = f (t, u, v), t = t (p, q, r, s), u = u (p, q, r, s), and v = v (p, q, r, s) are all differentiable functions.
> Suppose z = f (x, y), where x = g (s, t), y = h (s, t), t (1, 2) = 3, gs (1, 2) = -1, gt (1, 2) = 4, h (1, 2) = 6, hs (1, 2) = -5, ht (1, 2) = 10, fx (3, 6) = 7, and fy (3, 6) = 8. Find ∂z/∂s and ∂z/∂t when s = 1 and t = 2.
> If v = x2 sin y + yexy, where x = s + 2t and y = st, use the Chain Rule to find ∂v/∂s and ∂v/∂t when s = 0 and t = 1.
> If u = x2y3 + z4, where x = p + 3p2, y = pep, and z = p sin p, use the Chain Rule to find du/dp.
> Find the linear approximation of the function f (x, y, z) = x3 √(y^2+ z^2 ) at the point (2, 3, 4) and use it to estimate the number (1.98)3 √((3.01)^2 + (3.97)^2 ).
> Find du if u = ln (1 + se2t).
> Sketch the graph of the function. f (x, y) = 1 - y2
> If z = sin (x + sin t), show that az az ax ax at az a'z at ax?
> If z = xy + xey/x, show that x∂z/∂x + y ∂z/∂y = xy + z.
> Find all second partial derivatives of f. v = r cos (s + 2t)
> Each of these extreme value problems has a solution with both a maximum value and a minimum value. Use Lagrange multipliers to find the extreme values of the function subject to the given constraint. f(x, y, z) = x² + y² + z²; x* + y* + z* = 1
> Suppose that f is continuous on a disk that contains the point (a, b). Let Dr be the closed disk with center (a, b) and radius r. Use the Mean Value Theorem for double integrals (see Exercise 58) to show that Exercise 58: The Mean Value Theorem for do
> Find all second partial derivatives of f. f (x, y, z) = xkylzm
> Find all second partial derivatives of f. z = xe-2y
> Use the transformation x = u2, y = v2, z = w2 to find the volume of the region bounded by the surface √x + √y + √z = 1 and the coordinate planes.
> Use the transformation u = x - y, v = x + y to evaluate where R is the square with vertices (0, 2), (1, 1), (2, 2), and (1, 3). х — у -dA х+у
> Give five other iterated integrals that are equal to ∫_0^2 ∫_0^(y^3) ∫_0^(y^2) f (x,y,z) dz dx dy
> Rewrite the integral as an iterated integral in the order dx dy dz. *1-y °f(x, y, z) dz dy dx Jo
> Find all second partial derivatives of f. f (x, y) = 4x3 - xy2
> Find the first partial derivatives. S (u, v, w) = u arctan (v √w)
> Find the center of mass of the solid tetrahedron with vertices (0, 0, 0), (1, 0, 0), (0, 2, 0), (0, 0, 3) and density function ρ (x, y, z) = x2 + y2 + z2.
> Each of these extreme value problems has a solution with both a maximum value and a minimum value. Use Lagrange multipliers to find the extreme values of the function subject to the given constraint. f(x, y, z) = In(x² + 1) + In(y² + 1) + In(z² + 1)
