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mathematics
calculus 10th edition

Questions and Answers of Calculus 10th Edition

Find the four second partial derivatives. Observe that the second mixed partials are equal.ƒ(x, y) = 3x² - xy + 2y³
Use Lagrange multipliers to find the minimum distance from the curve or surface to the indicated point. Surface Plane: x + y + z = 1 Point (2, 1, 1)
Find the domain and range of the function. f(x, y) = √36x² - y²
Find the domain and range of the function. f(x,y) = X y
UseLagrange multipliers to find the indicated extrema, assumingthat x and y are positive.Maximize: ƒ(x, y) = xyConstraint: x + y = 10
Find and simplify the function values.ƒ(x, y) = 3x²y(a) (1, 3) (b) (-1, 1) (c) (-4, 0)(d) (x, 2)
Use Lagrange multipliers to find the indicated extrema, assuming that x and y are positive.Minimize: ƒ(x, y) = 2x + yConstraint:xy = 32
Find and simplify the function values.ƒ(x, y) = 6 - 4x - 2y²(a) (0, 2)(b) (5, 0)(c) (-1, -2)(d) (-3, y)
Use Lagrange multipliers to find the indicated extrema, assuming that x and y are positive.Minimize ƒ(x, y) = x² + y²Constraint: x + 2y - 5 = 0
A principal of $2000 is deposited in a savings account that earns interest at a rate of r (written as a decimal) compounded continuously. The amount A(r, t) after t years isUse this function of two
Use Lagrange multipliers to find the indicated extrema, assuming that x and y are positive.Maximize ƒ(x, y) = x² - y²Constraint:2y - x² = 0
Use Lagrange multipliers to find the indicated extrema, assuming that x and y are positive. Maximize f(x, y) = √√√6x² - y² Constraint: x + y2 = 0
Use Lagrange multipliers to find the indicated extrema, assuming that x and y are positive.Maximize ƒ(x, y) = 2x + 2xy + yConstraint:2x + y = 100
Use Lagrange multipliers to find the indicated extrema, assuming that x and y are positive. Minimize f(x, y) = √√√x² + y² 2 Constraint: 2x + 4y - 15 = 0
Describethe level curves of the function. Sketch a contour map of thesurface using level curves for the given c-values.z = 3 - 2x + y, c = 0, 2, 4, 6, 8
Use Lagrange multipliers to find the indicated extrema, assuming that x and y are positive.Minimize ƒ(x, y) = 3x + y + 10Constraint:x²y = 6
Describe the level curves of the function. Sketch a contour map of the surface using level curves for the given c-values.z = 2x² + y², c = 1, 2, 3, 4, 5
Consider the function ƒ(x, y) = x² + y².(a) Sketch the graph of the surface given by ƒ.(b) Make a conjecture about the relationship between the graphs of ƒ and g(x, y) = ƒ(x, y) + 2. Explain
Find the limit (if it exists) and discuss the continuity of the function. xy lim (x, y) (1.1) x² + y²
UseLagrange multipliers to find the indicated extrema, assumingthat x, y, and z are positive.Minimize ƒ(x, y, z) = x² + y² + z²Constraint: x + y + z - 9= 0
Sketch the graph of the level surface ƒ(x, y, z) = c at the given value of c.ƒ(x, y, z) = x² - y + z², c = 2
Find the limit (if it exists) and discuss the continuity of the function. xy lim (x, y) (1, 1) x² - y²
Use Lagrange multipliers to find the indicated extrema, assuming that x, y, and z are positive.Maximize ƒ(x, y, z) = xyzConstraint: x + y +z - 3 = 0
Find the limit (if it exists) and discuss the continuity of the function. x²y lim (x, y) (0,0) x4 + y²
Sketch the graph of the level surface ƒ(x, y, z) = c at the given value of c.ƒ(x, y, z) = 4x² - y² + 4z², c = 0
Find the limit (if it exists) and discuss the continuity of the function. lim (x, y) (0,0) 1 + x² V+ xe giữ
Use Lagrange multipliers to find the indicated extrema, assuming that x, y, and z are positive.Minimize ƒ(x, y, z) = x² + y² + z²Constraint: x + y + z = 1
Use Lagrange multipliers to find the indicated extrema, assuming that x, y, and z are positive.Maximize ƒ(x, y, z) = x + y + zConstraint:x² + y² + z² = 1
UseLagrange multipliers to find any extrema of the functionsubject to the constraint x² + y² ≤ 1.ƒ(x, y) = x² + 3xy + y²
Use Lagrange multipliers to find any extrema of the function subject to the constraint x² + y² ≤ 1.ƒ(x, y) = e-xy/4
Use Lagrange multipliers to find the minimum distance from the curve or surface to the indicated point. Curve Line: x + y = 1 Point (0, 0)
UseLagrange multipliers to find the indicated extrema of ƒ subjectto two constraints. In each case, assume that x, y, and z arenonnegative.Maximize ƒ(x, y, z) = xyzConstraints: x + y + z = 32, x -
Find all first partial derivatives.ƒ(x, y) = 5x³ + 7y - 3
Use Lagrange multipliers to find the indicated extrema of ƒ subject to two constraints. In each case, assume that x, y, and z are nonnegative.Minimize ƒ(x, y, z) = x² + y² + z²Constraints: x +
Find all first partial derivatives.ƒ(x, y) = 4x² - 2xy + y²
Use Lagrange multipliers to find the minimum distance from the curve or surface to the indicated point. Curve Line: 2x + 3y = -1 Point (0, 0)
Find all first partial derivatives.ƒ(x, y) = ex cos y
Find the gradient of the function and the maximum value of the directional derivative at the given point. x² + y² (1, 1)
Find the gradient of the function and the maximum value of the directional derivative at the given point. z = ex cos y, (0.7)
(a) Find the gradient of the function at P (b) Find a unit normal vector to the level curve ƒ(x, y) = c at P(c) Find the tangent line to the level curve ƒ(x, y) = c at P(d) Sketch the level
Find the slopes of the surface in the x- and y-directions at the given point. g(x, y) = 4 — x² - y² (1, 1, 2) X 2
Find the gradient of the function and the maximum value of the directional derivative at the given point. Z x² x - y (2, 1)
Find the maximumproduction level P when the total cost of labor (at $72 per unit)and capital (at $60 per unit) is limited to $250,000, where x isthe number of units of labor and y is the number of
(a) Find the gradient of the function at P (b) Find a unit normal vector to the level curve ƒ(x, y) = c at P(c) Find the tangent line to the level curve ƒ(x, y) = c at P(d) Sketch the level
Find the maximum production level P when the total cost of labor (at $72 per unit) and capital (at $60 per unit) is limited to $250,000, where x is the number of units of labor and y is the number of
Find the minimum cost ofproducing 50,000 units of a product, where x is the numberof units of labor (at $72 per unit) and y is the number of unitsof capital (at $60 per unit).P(x, y) =
Demonstrate the result of Exercise 51 forw = (x - y) sin(y - x).Data from in Exercise 51Show thatfor w = ƒ(x, y), x = u - v, and y = v - u. aw ди дw + = 0 Əv
Find the minimum cost of producing 50,000 units of a product, where x is the number of units of labor (at $72 per unit) and y is the number of units of capital (at $60 per unit).P(x, y) =
A can buoy is to be made of three pieces, namely, a cylinder and two equal cones, the altitude of each cone being equal to the altitude of the cylinder. For a given area of surface, what shape will
Describe the level curves of the function. Sketch a contour map of the surface using level curves for the given -values.z = x² + 4y², c = 0, 1, 2, 3, 4
Find an equation of the tangent plane to the surface at the given point.z = x² + y² + 2, (1, 3, 12)
Find the first partial derivatives with respect to x, y, and z. W = 7xz x + y
Describe the level curves of the function. Sketch a contour map of the surface using level curves for the given -values.ƒ(x, y) = xy, c = ±1, ±2,..., ±6
Find an equation of the tangent plane to the surface at the given point.9x² + y² + 4z² = 25, (0, -3, 2)
Find an equation of the tangent plane to the surface at the given point.z = -9 + 4x - 6y - x² - y², (2, -3, 4)
Show that any tangent plane to the conepasses through the origin. z² = a²x² + b²y²
Find an equation of the tangent plane to the surface at the given point.ƒ(x, y) = √25 - y², (2, 3, 4)
Examine the function for relative extrema and saddle points. 1 f(x,y) = xy + + + + = x y
Find an equation of the tangent plane and find a set of symmetric equations for the normal line to the surface at the given point.ƒ(x, y) = x²y, (2, 1, 4)
A company manufactures two types of bicycles, a racing bicycle and a mountain bicycle. The total revenue from x₁ units of racing bicycles and x₂ units of mountain bicycles iswhere x1 and x₂ are
Find an equation of the tangent plane and find a set of symmetric equations for the normal line to the surface at the given point.z = √9 - x² - y², (1, 2, 2)
Find the angle of inclination θ of the tangent plane to the surface x² + y² + z² = 14 at the point (2, 1, 3).
Examine the function for relative extrema and saddle points.ƒ(x, y) = - x² - 4y² + 8x - 8y - 11
Examine the function for relative extrema and saddle points.ƒ(x, y) = x² - y² - 16x - 16y
Examine the function for relative extrema and saddle points.ƒ(x, y) = 2x² + 6xy +9y² + 8x + 14
Examine the function for relative extrema and saddle points.ƒ(x, y) = -8x² + 4xy - y² + 12x + 7
An agronomist used four test plots to determine the relationship between the wheat yield y (in bushels per acre) and the amount of fertilizer x (in pounds per acre). The results are shown in the
Find the minimum distance from the point (2, 1, 4) to the surface x + y + z = 4.
Find three positive integers, x, y, and z, such that the product is 64 and the sum is a minimum.
The Doyle Log Rule is one of several methods used to determine the lumber yield of a log (in board-feet) in terms of its diameter d (in inches) and its length L (in feet). The number of board-feet
Use spherical coordinates to find the limit. Let x = p sin o cos 0, y = ρ sin Ø sin θ, y = ρ sin Ø sin θ, and z = ρ cos Ø and note that (x, y, z) → (0, 0, 0) implies ρ → 0+. XYZ lim (x,
Use spherical coordinates to find the limit. Let x = p sin o cos 0, y = ρ sin Ø sin θ, y = ρ sin Ø sin θ, and z = ρ cos Ø and note that (x, y, z) → (0, 0, 0) implies ρ → 0+. lim (x, y,
A corporation manufactures digital cameras at two locations. The cost of producing x, units at location 1 is C₁ = 0.05x21 + 15x₁ + 5400 and the cost of pro- ducing x₂ units at location 2 is
Consider the function(a) Show that ƒ is continuous at the origin.(b) Show that ƒx and ƒy exist at the origin, but that the directional derivatives at the origin in all other directions do not
Find the least squares regression line for the points. Use the regression capabilities of a graphing utility to verify your results. Use the graphing utility to plot the points and graph the
Find the least squares regression line for the points. Use the regression capabilities of a graphing utility to verify your results. Use the graphing utility to plot the points and graph the
Consider the functionand the unit vectorDoes the directional derivative of ƒ at P(0, 0) in the direction ofu exist? If ƒ(0, 0) were defined as 2 instead of 0, would thedirectional derivative exist?
UseLagrange multipliers to find the indicated extrema, assumingthat x and y are positive.Minimize: ƒ(x, y) = x² + y²Constraint: x + y - 8 = 0
The temperature T (in degrees Celsius) at any point (x, y) in a circular steel plate of radius 10 meters iswhere x and y are measured in meters. Sketch some of the isothermal curves. T = 600 0.75x²
Find the four second partial derivatives. Observe that the second mixed partials are equal.z = x² - 2xy + 3y²
Find the following limit. lim (x, y)→(0, 1) tan 1 x² + 1 x² + (y 1)²_ -
Use Lagrange multipliers to find the indicated extrema, assuming that x and y are positive.Maximize: ƒ(x, y) = xyConstraint:x + 3y - 6 = 0
The electric potential V at any point (x, y) is V(x, y) 5 25 + x² + y²
Find the four second partial derivatives. Observe that the second mixed partials are equal.z = x4 - 3x²y² + y4
For the functiondefine ƒ(0, 0) such that ƒ is continuous at the origin. f(x, y) = xy (x² - y²) \x² + y²
Use Lagrange multipliers to find the indicated extrema, assuming that x and y are positive.Maximize: ƒ(x, y) = 2x + 3xy + yConstraint: x + 2y = 29
Find the four second partial derivatives. Observe that the second mixed partials are equal.z = √x² + y²
Use Lagrange multipliers to find the indicated extrema, assuming that x and y are positive.Minimize: ƒ(x, y) = x² - y²Constraint: x - 2y + 6 = 0
Prove that if ƒ is continuous and ƒ(a, b) < 0, then there exists a δ-neighborhood about (a, b) such that ƒ(x, y) < 0for every point (x, y) in the neighborhood.
Prove thatas where ƒ(x, y) approaches L1 and g(x, y) approaches L2 as(x, y) → (a, b). lim [f(x, y) + g(x, y)] = L₁ + L₂ (x, y)→(a, b)
Show that the Cobb-Douglas production functioncan be rewritten as z = Z Cxayl-a
According to the Ideal Gas Law,where P is pressure, Vis volume, Tis temperature (in kelvins), and k is a constant of proportionality. A tank contains 2000 cubic inches of nitrogen at a pressure of 26
Find the four second partial derivatives. Observe that the second mixed partials are equal.z = ln(x - y)
Use the Cobb- Douglas production function to show that when the number of units of labor and the number of units of capital are doubled, the production level is also doubled.
Use Lagrange multipliers to find the indicated extrema, assuming that x and y are positive.Maximize: ƒ(x, y) = 2xyConstraint:2x + y = 12
Define the limit of a function of two variables. Describe a method for showing thatdoes not exist. lim (x, y)—(x, y)√(x, y)
Find the four second partial derivatives. Observe that the second mixed partials are equal.z = ex tan y
Find the four second partial derivatives. Observe that the second mixed partials are equal. z = arctan y X
A water line is to be built from point P to point S and must pass through regions where construction costs differ (see figure). The cost per kilometer in dollars is 3k from P to Q, 2k from Q to R,
The table shows the net sales x (in billions of dollars), the total assets y (in billions of dollars), and the shareholder's equity z (in billions of dollars) for Apple for the years 2006 through

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