Calculate the iterated integral. ∫_0^1 ∫_0^1 (ye^xy dx dy
> 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. ∬D xy dA, where D = {(x, y) | 0 < y < 1, y2 < x < y + 2}
> 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 +
> 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)
> Pictured are a contour map of f and a curve with equation t (x, y) = 8. Estimate the maximum and minimum values of f subject to the constraint that t (x, y) = 8. Explain your reasoning. yA glx, y) = 8 40 50 60 70 30 20 10
> An alternative form of estimation is accomplished through the method of moments. This method involves equating the population mean and variance to the corresponding sample mean ¯x and sample variance s2 and solving for the parameters, the results being t
> It is argued that the resistance of wire A is greater than the resistance of wire B. An experiment on the wires shows the following results (in ohms): Assuming equal variances, what conclusions do you draw? Justify your answer. Wire B 0.135 Wire A
> A manufacturer of electric irons produces these items in two plants. Both plants have the same suppliers of small parts. A saving can be made by purchasing thermostats for plant B from a local supplier. A single lot was purchased from the local supplier,
> An anthropologist is interested in the proportion of individuals in two Indian tribes with double occipital hair whorls. Suppose that independent samples are taken from each of the two tribes, and it is found that 24 of 100 Indians from tribe A and 36 of
> An experiment was conducted to determine whether surface finish has an effect on the endurance limit of steel. There is a theory that polishing increases the average endurance limit (for reverse bending). From a practical point of view, polishing should
> The Department of Civil Engineering at Virginia Tech compared a modified (M-5 hr) assay technique for recovering fecal coliforms in storm water runoff from an urban area to a most probable number (MPN) technique. A total of 12 runoff samples were collect
> A health spa claims that a new exercise program will reduce a person’s waist size by 2 centimeters on average over a 5-day period. The waist sizes, in centimeters, of 6 men who participated in this exercise program are recorded before a
> A study was undertaken at Virginia Tech to determine if fire can be used as a viable management tool to increase the amount of forage available to deer during the critical months in late winter and early spring. Calcium is a required element for plants a
> It is claimed that a new diet will reduce a person’s weight by 4.5 kilograms on average in a period of 2 weeks. The weights of 7 women who followed this diet were recorded before and after the 2-week period. Test the claim about the d
> According to the Roanoke Times, McDonald’s sold 42.1% of the market share of hamburgers. A random sample of 75 burgers sold resulted in 28 of them being from McDonald’s. Use material in Section 9.10 to determine if this information supports the claim in
> Regular consumption of presweetened cereals contributes to tooth decay, heart disease, and other degenerative diseases, according to studies conducted by Dr. W. H. Bowen of the National Institute of Health and Dr. J. Yudben, Professor of Nutrition and Di
> Consider two estimators of σ2 for a sample x1, x2, . . . , xn, which is drawn from a normal distribution with mean μ and variance σ2. The estimators are the unbiased estimator s2 = 1 the maximum likelihood estimator
> Consider the observation X from the negative binomial distribution given in Section 5.4. Find the maximum likelihood estimator for p, assuming k is known.
> Consider a hypothetical experiment where a man with a fungus uses an antifungal drug and is cured. Consider this, then, a sample of one from a Bernoulli distribution with probability function where p is the probability of a success (cure) and q = 1 &ac
> Consider the independent observations x1, x2, . . . , xn from the gamma distribution discussed in Section 6.6. (a) Write out the likelihood function. (b) Write out a set of equations that, when solved, give the maximum likelihood estimators of α and β.
> Consider a random sample of x1, . . . , xn from a uniform distribution U(0, θ) with unknown parameter θ, where θ > 0. Determine the maximum likelihood estimator of θ.
> Consider a random sample of x1, x2, . . . , xn observations from a Weibull distribution with parameters α and β and density function / (a) Write out the likelihood function. (b) Write out the equations that, when solved, give the maximum likelihood
> Consider a random sample of x1, . . . , xn coming from the gamma distribution discussed in Section 6.6. Suppose the parameter α is known, say 5, and determine the maximum likelihood estimation for parameter β.
