2.99 See Answer

Question: Find the first partial derivatives. F (α, β


Find the first partial derivatives.
F (α, β) = α2 ln (α2 + β2)


> Consider a process that attempts to prepare tyrosine using a Hell–Volhard–Zelinsky reaction: a. Identify the necessary starting carboxylic acid. b. When treated with Br2, the starting carboxylic acid can react with two equivalents to produce a compound

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> (a). A lamina has constant density ρ and takes the shape of a disk with center the origin and radius R. Use Newton’s Law of Gravitation (see Section 13.4) to show that the magnitude of the force of attraction that the lamina

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> Use spherical coordinates. (a). Find the volume of the solid that lies above the cone φ = π/3 and below the sphere ρ = 4 cos φ. (b). Find the centroid of the solid in part (a).

> Use a computer algebra system to find the mass, center of mass, and moments of inertia of the lamina that occupies the region D and has the given density function. D = {(x, y) | 0 < y < xe-x, 0 < x < 2j; θ (x, y) = x2y2

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> A lamina with constant density ρ (x, y) = ρ occupies the given region. Find the moments of inertia Ix and Iy and the radii of gyration x and y. The part of the disk x2 + y2 < a2 in the first quadrant

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> A lamina with constant density ρ (x, y) = ρ occupies the given region. Find the moments of inertia Ix and Iy and the radii of gyration x and y. The rectangle 0 < x < b, 0 < y < h

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> Sketch the region whose area is given by the integral and evaluate the integral. ∫_(π/4)^(3π/4) ∫_1^2r dr dθ

> Evaluate the double integral. ∬D y/(x^2+1) dA, D = {(x, y) | 0 < x < 4, 0 < y < √x}

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> Leonhard Euler was able to find the exact sum of the series in Problem 5. In 1736 he proved that In this problem we ask you to prove this fact by evaluating the double integral in Problem 5. Start by making the change of variables This gives a rotati

> Use spherical coordinates. Evaluate ∭E y2z2 dV, where E lies above the cone φ = π/3 and below the sphere ρ = 1.

> Sketch the solid whose volume is given by the integral and evaluate the integral. ∫_0^2 ∫_0^2π ∫_0^r r dz dθ dr

> Sketch the solid described by the given inequalities. 0 < θ < π/2, r < z < 2

> Identify the surface whose equation is given. r = 2 sin θ

> Sketch the solid whose volume is given by the integral and evaluate the integral. ∫_(-π/2)^(-π/2) ∫_0^2 ∫_0^(r^2)r dz dr dθ

> Describe in words the surface whose equation is given. θ= π/6

> (a). A lamp has two bulbs, each of a type with average lifetime 1000 hours. Assuming that we can model the probability of failure of a bulb by an exponential density function with mean μ = 1000, find the probability that both of the lamp’s bulbs fail wit

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> If D is the region bounded by the curves y = 1/2 x2 and y = ex, find the approximate value of the integral ∬D y2 dA. (Use a graphing device to estimate the points of intersection of the curves.)

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> Find the area of the part of the cone z2 = a2(x2 + y2) between the planes z = 1 and z = 2.

> Find the gradient of the function f (x, y, z) = x2eyz2.

> Calculate the value of the multiple integral. ∭H z3√(x^2+y^2+ z^2 ) dV, where H is the solid hemisphere that lies above the xy-plane and has center the origin and radius 1

> Calculate the value of the multiple integral. ∭E y2z2 dV, where E is bounded by the paraboloid x = 1 - y2 - z2 and the plane x = 0

> Calculate the value of the multiple integral. ∭T xy dV, where T is the solid tetrahedron with vertices (0, 0, 0), (1/3 , 0, 0), (0, 1, 0), and (0, 0, 1)

> Calculate the value of the multiple integral. ∬E xy dV, where E = {(x, y, z) | 0 < x < 3, 0 < y < x, 0 < z < x + y}

> Calculate the value of the multiple integral. ∭T x dA, where D is the region in the first quadrant that lies between the circles x2 1 y2 − 1 and x2 + y2 = 2

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> Calculate the value of the multiple integral. ∬D y dA, where D is the region in the first quadrant that lies above the hyperbola xy = 1 and the line y = x and below the line y = 2

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> 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) = e""; 2x² + y² + z? = 24

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> 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(x1, x2, ..., xn) = x1 + x2 + .. xỉ + x + ... +

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> In this project we find formulas for the volume enclosed by a hypersphere in n-dimensional space. 1. Use a double integral and trigonometric substitution, together with Formula 64 in the Table of Integrals, to find the area of a circle with radius r. 2.

> The figure shows the solid enclosed by three circular cylinders with the same diameter that intersect at right angles. In this project we compute its volume and determine how its shape changes if the cylinders have different diameters. 1. Sketch carefu

> Evaluate the iterated integral. ∫_0^1 ∫_0^1 ∫_0^(√(1-z^2 ))z/(y+1) dx dz dy

> Evaluate the iterated integral. ∫_1^2 ∫_0^2x ∫_0^lnxxe^(-y) dy dx dz

> If f is continuous, show that ∫_0^x ∫_0^y ∫_0^z f(t) dt dz dy =1/2 ∫_0^x (x-t)^2 f(t)dt

> Change from rectangular to spherical coordinates. (a). (1, 0, 3 ) (b). ( 3 , -1, - 3 )

> Change from rectangular to spherical coordinates. (a). (0, -2, 0) (b). (-1, 1, - 2 )

> Use spherical coordinates. Find the average distance from a point in a ball of radius a to its center.

> Use spherical coordinates. Evaluate ∭E √(x^2+y^2+z^2 ) dV, where E lies above the cone z = √(x^2+y^2 ) and between the spheres x2 + y2 + z2 − 1 and x2 + y2 + z2 = 4.

> Use spherical coordinates. Evaluate ∭E xe^(x^2+y^2+z^2 ) dV, where E is the portion of the unit ball x2 + y2 + z2 < 1 that lies in the first octant

> Set up the triple integral of an arbitrary continuous function f (x, y, z) in cylindrical or spherical coordinates over the solid shown. ZA y 2.

> Plot the point whose spherical coordinates are given. Then find the rectangular coordinates of the point. (a). (2, π/2, π/2) (b). (4, -π/4, π/3)

> Set up the triple integral of an arbitrary continuous function f (x, y, z) in cylindrical or spherical coordinates over the solid shown. ZA 3- 2 X. y

> Sketch the solid whose volume is given by the integral and evaluate the integral. ∫_0^(π/4) ∫_0^2π ∫_0^secφρ^2 sin φ dρ dθ dφ

> Sketch the solid whose volume is given by the integral and evaluate the integral. ∫_0^(π/6) ∫_0^(π/2) ∫_0^3 ρ^2 sin φ dρ dθ dφ

> (a). Show that when Laplace&acirc;&#128;&#153;s equation (&acirc;&#136;&#130;^2 u)/(&acirc;&#136;&#130;x^2 ) + (&acirc;&#136;&#130;^2 u)/(&acirc;&#136;&#130;y^2 ) + (&acirc;&#136;&#130;^2 u)/(&acirc;&#136;&#130;z^2 ) = 0 is written in cylindrical coordin

> Evaluate the triple integral. ∭E ez/y dV, where E = {(x, y, z) | 0 < y < 1, y < x < 1, 0 < z < xy}

> Plot the point whose spherical coordinates are given. Then find the rectangular coordinates of the point. (a). (6, π/3, π /6) (b). (3, π /2, 3 π/4)

> Evaluate the triple integral. ∭E y dV, where E = {(x, y, z) | 0 < x < 3, 0 < y < x, x - y < z < x + y}

2.99

See Answer