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mathematics
calculus 10th edition

Questions and Answers of Calculus 10th Edition

Meteorologists measure the atmospheric pressure in millibars. From these observations, they create weather maps on which the curves of equal atmospheric pressure (isobars), are drawn (see figure). On
Use Lagrange multipliers to find the indicated extrema, assuming that x and y are positive.Minimize: ƒ(x, y) = 3x² - y²Constraint: 2x - 2y + 5 = 0
Find the four second partial derivatives. Observe that the second mixed partials are equal.z = 2xey - 3ye-x
Use a computer algebra system to find the first and second partial derivatives of the function. Determine whether there exist values of x and y such that ƒx(x, y) = 0 and ƒy(x, y) = 0
(a) If ƒ(2, 3) = 4, can you conclude anything about Give reasons for your answer.(b) Ifcan you conclude anything about ƒ (2, 3)? Give reasons for your answer. lim (x, y)→(2, 3) f(x, y)?
Find the four second partial derivatives. Observe that the second mixed partials are equal.z =cos xy
The acidity of rainwater is measured in units called pH. A pH of 7 is neutral, smaller values are increasingly acidic, and larger values are increasingly alkaline. The map shows curves of equal pH
State the definition of continuity of a function of two variables.
Use a computer algebra system to find the first and second partial derivatives of the function. Determine whether there exist values of x and y such that ƒx(x, y) = 0 and ƒy(x, y) = 0
The figure shows the graph of ƒ(x, y) = ln x² + y². From the graph, does it appear that the limit at each point exists?(a) (-1, -1) (b) (0, 3)(c) (0, 0) (d) (2, 0)
Use a computer algebra system to find the first and second partial derivatives of the function. Determine whether there exist values of x and y such that ƒx(x, y) = 0 and ƒy(x, y) = 0
Use a computer algebra system to find the first and second partial derivatives of the function. Determine whether there exist values of x and y such that ƒx(x, y) = 0 and ƒy(x, y) = 0
The contour map shown in the figure was computer generated using data collected by satellite instrumentation. Color is used to show the “ozone hole” in Earth’s atmosphere. The purple and blue
Find the volume of the solid of intersection of the three cylinders x² + z² = 1, y² + z² = 1, and x² + y² = 1 (see figure). 3 3 2 -3+ 3 -3 3 Z -3↑ 3 - у
Evaluate the integral. 2x xy³ dy
A rectangular box with an open top has a length of x feet, a width of y feet, and a height of z feet. It costs $1.20 per square foot to build the base and $0.75 per square foot to build the sides.
Approximate the integral∫R∫ ƒ(x,y) dA by dividing the rectangle R with vertices (0, 0), (4, 0), (4, 2), and (0, 2) into eight equal squares and finding the sum where (xi, yi) is the center of
Find the Jacobian∂(x, y)/∂(u, v) for the indicated change of variables. x = − ² (u − v), y = ¹/(u + v)
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.If ƒ(x0, y0) = ƒ(x₁, y₁), then x0 = X₁ and y0 = y₁.
Evaluate the iterated integral. *5 π/2 (3 LIT -1J0 0 r cos 0 dr de dz
Evaluate the integral. S (x + 2y) dy
Show that the mixed partial derivatives ƒxyy, ƒyxy, and ƒyyx are equal.ƒ(x, y, z) = xyz
The region R for the integral ∫R ∫ ƒ(x, y) dA is shown. State whether you would use rectangular or polar coordinates to evaluate the integral. 4 3 2 1 y 1 R 2 نیا + 4 X
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.If ƒ is a function, then ƒ(ax, ay) = a²ƒ(x, y).
Show that the mixed partial derivatives ƒxyy, ƒyxy, and ƒyyx are equal.ƒ(x, y, z) = x² -3xy + 4yz + z³
Evaluate the triple iterated integral. ССС 0 (x + y + z) dx dz dy
Find the mass of the lamina described by the inequalities, given that its density is ρ(x, y) = xy.0 ≤ x ≤ 2, 0 ≤ y ≤ 2
Evaluate the triple iterated integral. ССС 0 (x + y + z) dx dz dy
Evaluate the integral. 2y (x² + y²) dx
Let a, b, c, and d be positive real numbers. The first octant of the plane ax + by + cz = d is shown in the figure. Show that the surface area of this portion of the plane is equal towhere A(R) is
Approximate the integral ∫R∫ ƒ(x,y) dA by dividing the rectangle R with vertices (0, 0), (4, 0), (4, 2), and (0, 2) into eight equal squares and finding the sum where (xi, yi) is the center of
Find the area of the surface given by z = ƒ(x, y) over the region R.ƒ(x, y) = 2x + 2yR: triangle with vertices (0, 0), (4, 0), (0, 4)
Evaluate the iterated integral. Jo TT/4 1-9J 9J +/ 0/ *6-r Jo rz dz dr de
Find the Jacobian ∂(x, y)/∂(u, v) for the indicated change of variables.x = au + bv, y = cu + dv
Evaluate the integral. V X dy
Find the mass of the lamina described by the inequalities, given that its density is ρ(x, y) = xy.0 ≤ x ≤ 3, 0 ≤ y ≤ 9 - x²
Find the area of the surface given by z = ƒ(x, y) over the region R.ƒ(x, y) = 15 + 2x - 3yR: square with vertices (0, 0), (3, 0), (0, 3), (3, 3)
Find the area of the surface given by z = ƒ(x, y) over the region R.ƒ(x, y) = 12 + 2x - 3yR = {(x, y): x² + y² ≤ 9}
Find the area of the surface given by z = ƒ(x, y) over the region R.ƒ(x, y) = y²R: square with vertices (0, 0), (3, 0), (0, 3), (3, 3)
Use polar coordinates to describe the region shown. -4 -2 2 -2 y 2 4 X
Find the area of the surface given by z = ƒ(x, y) over the region R.ƒ(x, y) = 3 + x3/2R: rectangle with vertices (0, 0), (0, 4), (3, 4), (3, 0)
Evaluate the triple iterated integral. J LSS x²y²z² dx dy dz J-1J-1
The region R for the integral ∫R ∫ ƒ(x, y) dA is shown. State whether you would use rectangular or polar coordinates to evaluate the integral. + -6 4 -2 2 R ++ -2 +F -4 2 X
The figure shows the region R bounded by the curvesUse the change of variables x = u¹/3v2/3 and y = u²/3v1/3 to find the area of the region R. x² x² y = √√x, y = √2x, y =, and y = = 3' 4
Evaluate the integral. X dx, y> 0
Evaluate the iterated integral. /2 (2 cos²0 (4-r² 0 r sin 0 dz dr de
Evaluate the iterated integral. 0 1+x 0 (3x + 2y) dy dx
Approximate the integral ∫R∫ ƒ(x,y) dA by dividing the rectangle R with vertices (0, 0), (4, 0), (4, 2), and (0, 2) into eight equal squares and finding the sum where (xi, yi) is the center of
Evaluate the triple iterated integral. *.xy Jo Jo Jo x dz dy dx
Prove that The figure shows the region R bound lim 11-0 SC Jo xnyn dx dy = 0.
Find the Jacobian ∂(x, y)/∂(u, v) for the indicated change of variables.x = u - v2, y = u + v
The region R for the integral ∫R ∫ ƒ(x, y) dA is shown. State whether you would use rectangular or polar coordinates to evaluate the integral. -4 4 2 -4 y R 2 4 X
Find the mass of the lamina described by the inequalities, given that its density is ρ(x, y) = xy.0 ≤ x ≤ 1,0 ≤ y ≤ √1 - x²
Evaluate the triple iterated integral. *.xy Jo Jo Jo x dz dy dx
Approximate the integral ∫R∫ ƒ(x,y) dA by dividing the rectangle R with vertices (0, 0), (4, 0), (4, 2), and (0, 2) into eight equal squares and finding the sum where (xi, yi) is the center of
Find the area of the surface given by z = ƒ(x, y) over the region R.ƒ(x, y) = 7 + 2x + 2yR = {(x, y): x² + y² ≤ 4}
Evaluate the integral. cos y y dx
Evaluate the iterated integral. TT/2 TT 0 0 10 e-P³ p² dp de do
Evaluate the iterated integral. 2x ff 10 x² (x² + 2y) dy dx
Sketch the region R and evaluate the iterated integral ∫R∫ ƒ(x,y) dA. ST 10 0 (1 + 2x + 2y) dy dx
Find the Jacobian ∂(x, y)/∂(u, v) for the indicated change of variables.x = uv - 2u, y = uv
Evaluate the iterated integral. 2T (T/4 (cos 0 JO 0 0 p² sin o dp do do
Evaluate the iterated integral. √9-x² [Sº 10 4x dy dx
Evaluate the triple iterated integral. *y/3 ᏝᏝ . 0 0 0 y²-9x² z dz. dx dy
The region R for the integral ∫R ∫ ƒ(x, y) dA is shown. State whether you would use rectangular or polar coordinates to evaluate the integral. 3 2 1 y -¹+ R 2 3 4
Find the mass of the lamina described by the inequalities, given that its density is ρ(x, y) = xy.x ≥ 0, 3 ≤ y ≤ 3 + √9 - x²
Evaluate the integral. √√√4-x2² x²y dy
Find the mass and center of mass of the lamina for each density.R: square with vertices (0, 0), (a, 0), (0, a),(a, a)(a) ρ = k (b) ρ = ky (c) ρ = kx
Find the area of the surface given by z = ƒ(x, y) over the region R.ƒ(x, y) = 9 - x²R: square with vertices (0, 0), (2, 0), (0, 2), (2, 2)
Find the Jacobian ∂(x, y)/∂(u, v) for the indicated change of variables.x = u cos θ - v sin θ, y = u sin θ + v cos θ
Evaluate the triple iterated integral. 0 " 2ze-x² dy dx dz
Evaluate the iterated integral. m/4 */4 *cos # 10 p² sin cos o dp do do
Use polar coordinates to describe the region shown. -8 -4 12 4 -4 y 4 00 X
Evaluate the iterated integral. 2y [²³ 10 (9+ 3x² + 3y²) dx dy
Sketch the region R and evaluate the iterated integral ∫R∫ ƒ(x,y) dA. *π/2 SS Jo Jo sin² x cos² y dy dx
Evaluate the integral. (x² + 3y²) dy
Find the Jacobian ∂(x, y)/∂(u, v) for the indicated change of variables.x = u + a, y = v + a
Find the mass and center of mass of the lamina for each density.R: rectangle with vertices (0, 0), (a, 0), (0, b), (a, b)(a) ρ = kxy (b) ρ = k(x² + y²)
Use a computer algebra system to approximate the iterated integral. 0 Jo 0 JO π/2 re" do dr dz
Evaluate the integralsandAre the results the same? Why or why not? *1 S.S." Jo x - y (x + y)² dx dy
Sketch the region R and evaluate the iterated integral ∫R∫ ƒ(x,y) dA. 6 (3 Jy/2 (x + y) dx dy
Evaluate the integral. ex y ln x -dx, y > 0
Use polar coordinates to describe the region shown. y 10 8 6 4 2. + -22- + 2 4 6 00 8 + 10
Evaluate the triple iterated integral. 0 "/2 *1-x 0 0 x cos y dz dy dx
Find the Jacobian ∂(x, y)/∂(u, v) for the indicated change of variables.x = eu sin v, y = eu cos v
Find the mass and center of mass of the lamina for each density.R: triangle with vertices (0, 0), (0, a),(a, a)(a) ρ = k (b) ρ = ky (c) ρ = kx
Use an iterated integral to find the area of the region bounded by the graphs of the equations.x + 3y = 3, x = 0, y = 0
Sketch the region R and evaluate the iterated integral ∫R∫ ƒ(x,y) dA. 1 Jay x²y² dx dy
Use a computer algebra system to approximate the iterated integral. */2 TT (sin 8 ( (2 cos d)p² dp de do 0
Evaluate the integral. √1-y² J-√1-y² (x² + y²) dx
Use polar coordinates to describe the region shown. -4 -2 4 -4 y - X
Evaluate the triple iterated integral. π/2 (y/2 1/y 0 sin y dz dx dy
Show that the volume of a spherical block can be approximated by ΔV ≈ ρ2 sin Ø Δρ ΔØ Δθ.
Find the area of the surface given by z = ƒ(x, y) over the region R. f(x, y) = 2 + 3y³/2 R = {(x, y): 0 ≤ x ≤ 2,0 ≤ y ≤2-x}
Find the Jacobian ∂(x, y)/∂(u, v) for the indicated change of variables.x = u/v, y = u + v
Find the mass and center of mass of the lamina for each density.R: triangle with vertices (0, 0), (a/2, a),(a, 0)(a) ρ = k(b) ρ = kxy
Use an iterated integral to find the area of the region bounded by the graphs of the equations.y = 6x - x², y = x² - 2x
Evaluate the integral. So x²e-x² dx
Sketch theimage S in the uv-plane of the region R in the xy-plane using thegiven transformations. x = 3u + 2v y = 3v y 3 نرا 2 (0, 0) (2, 3) R 2 (3, 0) 3 X
Sketch the solid region whose volume is given by the iterated integral, and evaluate the iterated integral. π/2 (3 p-p² [²² Jo Jo Jo r dz dr de

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