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mathematics
precalculus
Calculus Of A Single Variable 11th Edition Ron Larson, Bruce H. Edwards - Solutions
In Exercises 3–8, evaluate the triple iterated integral. n/4 c6 6-r 10 Jo rz dz dr de
In Exercises 3–10, evaluate the triple iterated integral. Ap xp zp z Vy-912 гу/3 JJJ
In Exercises 3–8, evaluate the triple iterated integral. */2 *cos 0 *3+x3 10 10 2r sin 0 dz dr de
In Exercises 3–10, evaluate the triple iterated integral. SCC ² 10 0 2ze-x² dy dx dz
In Exercises 3–8, evaluate the triple iterated integral. π/2 e-p² p² dp de do
In Exercises 3–8, evaluate the triple iterated integral. 2 m/2 sin 10 Jo 0 p cos o dp do de
In Exercises 3–10, evaluate the triple iterated integral. r4 re² 1/xz In z dy dz dx
In Exercises 3–8, evaluate the triple iterated integral. *1/4 1/4 *cos H p² sin cos o dp de dø
In Exercises 9 and 10, use a computer algebra system to evaluate the triple iterated integral. */2 re" de dr dz
In Exercises 3–10, find the Jacobian ∂(x,y)/∂(u,v) for the indicated change of variables.x = u + 1, y = 9v
In Exercises 3–10, evaluate the triple iterated integral. ^{2_1 + 3x —3.0 10 x cos y dz dy dx
In Exercises 9 and 10, use a computer algebra system to evaluate the triple iterated integral. n/2 /2 (sin 8 0 0 2p² cos o dp de do
In Exercises 11 and 12, use a computer algebra system to evaluate the triple iterated integral. 9-y INI 9-y²Jo y dz dx dy
In Exercises 11 and 12, use a computer algebra system to evaluate the triple iterated integral. 3 2-(2y/3) 6-2y-3z -6573) "* CC 10 0 ze-x²y² dx dz dy
In Exercises 13–18, set up a triple integral for the volume of the solid. Do not evaluate the integral.The solid in the first octant bounded by the coordinate planes and the plane z = 7 - x - 2y
In Exercises 25–30, sketch the solid whose volume is given by the iterated integral. Then rewrite the integral using the indicated order of integration. Co ITS -1J0 Rewrite using dy dz dx. dz dy dx
In Exercises 19–24, use a triple integral to find the volume of the solid bounded by the graphs of the equations. X 4 x=4-y² Z = X 0=2 4 N 3.
In Exercises 19–24, use a triple integral to find the volume of the solid bounded by the graphs of the equations. 8 6 4 2 z = 2xy 2 X ترا 0 x 2 0 y 2
In Exercises 35–40, find the Jacobian д(x,y,z) д(u, y, w) for the indicated change of variables. If x = f(u, v, w), y = g(u, v, w), and z = h(u, v, w) then the Jacobian of x, y, and z with respect to u, v, and w is Әх əx ах ди ду ди ду ду ду du ду дw 3(x,y,z) д(u, y,
In Exercises 25–30, sketch the solid whose volume is given by the iterated integral. Then rewrite the integral using the indicated order of integration. 1-y² LIT Rewrite using dz dy dx. dz dx dy
In Exercises 25–30, sketch the solid whose volume is given by the iterated integral. Then rewrite the integral using the indicated order of integration. Clcl-x dz dx dy 1Jy² Rewrite using dx dz dy.
In Exercises 15–20, use cylindrical coordinates to find the volume of the solid.Solid inside both x2 + y2 + z2 = 36 and (x - 3)2 + y2 = 9
In Exercises 41–44, convert the integral from rectangular coordinates to both cylindrical and spherical coordinates, and evaluate the simplest iterated integral. -I/^ +1J JENT /1-A IN x dz dy dx
In Exercises 13–18, set up a triple integral for the volume of the solid. Do not evaluate the integral.The solid bounded by z = √1 - x2 - y2 and z = 0
In Exercises 19–24, use a triple integral to find the volume of the solid bounded by the graphs of the equations.z = 6x2, y = 3 - 3x, first octant
In Exercises 23–30, use a change of variables to find the volume of the solid region lying below the surface z = f (x, y) and above the plane region R.f(x, y) = 9xyR: region bounded by the square with vertices (1, 0), (0, 1), (1, 2), (2, 1)
In Exercises 45–50, sketch the region of integration. Then evaluate the iterated integral, changing the order of integration if necessary. (1/2 Jo Jy/2 e-x² dx dy
In Exercises 27–30, set up a double integral that represents the area of the surface given by z = f(x, y) that lies above the region R. ƒ(x, y) = cos(x² + y²), R = {(x, y): x² + y² ≤ EIN
Consider the surface f(x, y) = x + y. What is the relationship between the area of the surface that lies above the region R1 = {(x, y): x2 + y2 ≤ 1} and the area of the surface that lies above the region R2 = {(x, y): x2 + y2 ≤ 4}?
In Exercises 27–30, set up a double integral that represents the area of the surface given by z = f(x, y) that lies above the region R.f(x, y) = exy, R = {(x, y): 0 ≤ x ≤ 4, 0 ≤ y ≤ 10}
In Exercises 3–16, find the area of the surface given by z = f (x, y) that lies above the region R.f(x, y) = √x2 + y2, R = {(x, y): 0 ≤ f(x, y) ≤ 1}
In Exercises 3–16, find the area of the surface given by z = f (x, y) that lies above the region R.f(x, y) = 3 + 2x3/2R: rectangle with vertices (0, 0), (0, 4), (1, 4), (1, 0)
Write a double integral that represents the surface area of the portion of the plane z = 3 that lies above the rectangular region with vertices (0, 0), (4, 0), (0, 5), and (4, 5). Then find the surface area without integrating.
In Exercises 3 and 4, find the domain and range of the function. f(x, y) X y
In Exercises 3 and 4, find the domain and range of the function. f(x, y) = √√√36x² - y² - 2
In Exercises 13–18, find the limit (if it exists) and discuss the continuity of the function. In z lim (x, y, z) (-3, 1, 2) xyz
In Exercises 13–18, find the limit (if it exists) and discuss the continuity of the function. lim (x, y, z) (1,3,7) sin XZ 2y
In Exercises 5 and 6, describe and sketch the surface given by the function.f(x, y) = -2
In Exercises 1 and 2, evaluate the function at the given values of the independent variables. Simplify the results.f(x, y) = x2y - 3(a) f(0, 4)(b) f(2, -1)(c) f(-3, 2)(d) f(x, 7)
In Exercises 1 and 2, evaluate the function at the given values of the independent variables. Simplify the results.f(x, y) = 6 - 4x - 2y2(a) f(0, 2)(b) f(5, 0)(c) f(-1, -2)(d) f(-3, y)
In Exercises 31–34, find the four second partial derivatives. Observe that the second mixed partials are equal. h(x, y): X x + y
In Exercises 11 and 12, describe and sketch the graph of the level surface f(x, y, z) = c at the given value of c.f(x, y, z) = x2 - y + z2, c = 2
In Exercises 37–40, find the total differential. W || 3x + 4y y + 3z
In Exercises 5 and 6, describe and sketch the surface given by the function.g(x, y) = x
A manufacturer estimates that its production can be modeled by f(x, y) = 100x0.8y0.2 where x is the number of units of labor and y is the number of units of capital.(a) Find the production level when x = 100 and y = 200.(b) Find the production level when x = 500 and y = 1500.
In Exercises 19–26, find all first partial derivatives.f(x, y) = y3ey/x
In Exercises 19–26, find all first partial derivatives.z = ln(x2 + y2 + 1)
In Exercises 27–30, find all first partial derivatives, and evaluate each at the given point.f(x, y) = x2 - y, (0, 2)
In Exercises 27–30, find all first partial derivatives, and evaluate each at the given point.f(x, y) = xe2y, (-1, 1)
In Exercises 27–30, find all first partial derivatives, and evaluate each at the given point.f(x, y, z) = xy cos xz, (2, 3, -π/3)
In Exercises 27–30, find all first partial derivatives, and evaluate each at the given point.f(x, y, z) = √x2 + y - z2, (-3, -3, 1)
In Exercises 37–40, find the total differential.z = 5x4y3
In Exercises 37–40, find the total differential.z = x sin xy
In Exercises 37–40, find the total differential.w = 3xy2 - 2x3yz2
In Exercises 61–66, find the gradient of the function and the maximum value of the directional derivative at the given point. Z = y x² + y²² (1, 1)
In Exercises 53 and 54, differentiate implicitly to find dy/dx. xy² x + y = 3
In Exercises 61–66, find the gradient of the function and the maximum value of the directional derivative at the given point. z = ex cos y, 0, π 4
In Exercises 61–66, find the gradient of the function and the maximum value of the directional derivative at the given point. Z= X² x - y (2, 1)
In Exercises 45 and 46, show that the function is differentiable by finding values of ε1 and ε2 as designated in the definition of differentiability, and verify that both ε1 and ε2 approach 0 as (Δx, Δy) = (0, 0).f(x, y) = 6x - y2
In Exercises 61–66, find the gradient of the function and the maximum value of the directional derivative at the given point. w=xy-yz, (-1.1/22)
In Exercises 45 and 46, show that the function is differentiable by finding values of ε1 and ε2 as designated in the definition of differentiability, and verify that both ε1 and ε2 approach 0 as (Δx, Δy) = (0, 0).f(x, y) = xy2
In Exercises 61–66, find the gradient of the function and the maximum value of the directional derivative at the given point. w = e√x+y+z, (5, 0, 2)
In Exercises 47–50, find dw/dt(a) By using the appropriate Chain Rule (b) By converting w to a function of t before differentiating.w = x2z + y + z, x = et, y = t, z = t2
In Exercises 69–72, find an equation of the tangent plane to the surface at the given point. f(x, y) = /25 -², (2, 3, 4)
In Exercises 47–50, find dw/dt(a) By using the appropriate Chain Rule (b) By converting w to a function of t before differentiating.w = sin x + y2z + 2z, x = arcsin(t - 1), y = t3, z = 3
In Exercises 53 and 54, differentiate implicitly to find dy/dx.x3 - xy + 5y = 0
The table shows the yield y (in milligrams) of a chemical reaction after t minutes.(a) Use the regression capabilities of a graphing utility to find the least squares regression line for the data. Then use the graphing utility to plot the data and graph the model.(b) Use a graphing utility to plot
In Exercises 55 and 56, differentiate implicitly to find the first partial derivatives of z.x2 + xy + y2 + yz + z2 = 0
In Exercises 79–84, find all relative extrema and saddle points of the function. Use the Second Partials Test where applicable. 1 f(x, y) = xy + + X y
In Exercises 75 and 76, find the angle of inclination of the tangent plane to the surface at the given point.xy + yz2 = 32, (-4, 1, 6)
In Exercises 77 and 78, find the point(s) on the surface at which the tangent plane is horizontal.z = 9 - 2x2 + y3
In Exercises 77 and 78, find the point(s) on the surface at which the tangent plane is horizontal.z = 2xy + 3x + 5y
In Exercises 79–84, find all relative extrema and saddle points of the function. Use the Second Partials Test where applicable.f(x, y) = x6y6
In Exercises 7–12, sketch the region R and evaluate the iterated integral ∫R∫ f(x, y) dA. 2 (1 1000 (1 - 4.x + 8y) dydx
In Exercises 3–10, evaluate the integral. x fo (2x - y) dy
In Exercises 3–10, evaluate the integral. √4-² x²y dy
Sketch the region of integration represented by the double integral 27 (6 BD √3 f(r, 0)r dr de.
In Exercises 93–98, use Lagrange multipliers to find the indicated extrema, assuming that x and y are positive.Minimize f(x, y) = x2 + y2Constraint: x + y - 8 = 0
In Exercises 3–10, evaluate the integral. [² X y ln x - dx, y> 0
In Exercises 3–10, evaluate the integral. x Y dy X
In Exercises 93–98, use Lagrange multipliers to find the indicated extrema, assuming that x and y are positive.Maximize f(x, y) = 2x + 3xy + yConstraint: x + 2y = 29
In Exercises 93–98, use Lagrange multipliers to find the indicated extrema, assuming that x and y are positive.Minimize f(x, y) = x2 - y2Constraint: x - 2y + 6 = 0
In Exercises 3–10, evaluate the integral. [ (x2 + 3y2) dy
In Exercises 9–16, evaluate the double integral ∫R∫ f(r,θ) dA and sketch the region R. π (2 cos 0 If Jo Jo r dr de
In Exercises 9–16, evaluate the double integral ∫R∫ f(r,θ) dA and sketch the region R. 2π pl 10 Jo 6r² sin 0 dr de
In Exercises 93–98, use Lagrange multipliers to find the indicated extrema, assuming that x and y are positive.Maximize f(x, y) = 2xyConstraint: 2x + y = 12
In Exercises 3–10, evaluate the integral. 10 ye-y/x dy
In your own words, describe the process of using an inner partition to approximate the volume of a solid region lying above the xy-plane. How can the approximation be improved?
In Exercises 9–16, evaluate the double integral ∫R∫ f(r,θ) dA and sketch the region R. n/2 (sin 8 Jo r² dr de
In Exercises 7–12, sketch the region R and evaluate the iterated integral ∫R∫ f(x, y) dA. -3√√√9-x² (x + y) dy dx
In your own words, describe r-simple regions and θ-simple regions.
In Exercises 3–10, evaluate the integral. √1-y² |-√√1-3² (x² + y²) dx
In Exercises 3–6, find the mass of the lamina described by the inequalities, given that its density is ρ(x, y) = xy.0 ≤ x ≤ 2, 0 ≤ y ≤ 4 - x2
In Exercises 3–10, evaluate the integral. *n/2 sin³ x cos y dx
In Exercises 11–28, evaluate the iterated integral. n/4 cl Jo y cos x dy dx
In Exercises 11–28, evaluate the iterated integral. 2 010 (x + y) dy dx
In Exercises 11–28, evaluate the iterated integral. -1J-2 (x² - y²) dy dx
In Exercises 13–20, set up integrals for both orders of integration. Use the more convenient order to evaluate the integral over the plane region R. ff.xy da R: rectangle with vertices (0, 0), (0, 5), (3, 5), (3, 0)
In Exercises 11–28, evaluate the iterated integral. In 4 In 3 ex+y dy dx
In Exercises 13–20, set up integrals for both orders of integration. Use the more convenient order to evaluate the integral over the plane region R. Safsin R: rectangle with vertices (-7, 0), (7,0), (л, л/2), (-n, 1/2) sin x sin y dA
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