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mathematics
applied calculus

Questions and Answers of Applied Calculus

In Exercises, both first partial derivatives of the function f (x, y) are zero at the given points. Use the second-derivative test to determine the nature of f (x, y) at each of these points. If the
The amount of space required by a particular firm is f (x, y) = 1000√6x2 + y2, where x and y are, respectively, the number of units of labor and capital utilized. Suppose that labor costs $480 per
Find a function f (x, y) that has the curve y = 2/x2 as a level curve.
In Exercises, both first partial derivatives of the function f (x, y) are zero at the given points. Use the second-derivative test to determine the nature of f (x, y) at each of these points. If the
Three hundred square inches of material are available to construct an open rectangular box with a square base. Find the dimensions of the box that maximize the volume.
Find a function f (x, y) that has the line y = 3x - 4 as a level curve.
In Exercises, find all points (x, y) where f (x, y) has a possible relative maximum or minimum.f (x, y) = x3 + 3x2 + 3y2 - 6y + 7
In Exercises, both first partial derivatives of the function f (x, y) are zero at the given points. Use the second-derivative test to determine the nature of f (x, y) at each of these points. If the
Let f (x, y, z) = xzeyz. Find af af ах ду and af az
Four hundred eighty dollars are available to fence in a rectangular garden. The fencing for the north and south sides of the garden costs $10 per foot, and the fencing for the east and west sides
Let f (x, y, z) = zex/y. Find af af ах ду af and - дz
Draw the level curve of the function f (x, y) = xy containing the point (1/2,4).
In Exercises, find all points (x, y) where f (x, y) has a possible relative maximum or minimum.f (x, y) = x2 + 3xy - y2 - x - 8y + 4
The crime rate in a certain city can be approximated by a function f (x, y, z), where x is the unemployment rate, y is the number of social services available, and z is the size of the police force.
The function f (x, y) = 1/2x2 + 2xy + 3y2 - x + 2y has a minimum at some point (x, y). Find the values of x and y where this minimum occurs.
Find the two positive numbers whose product is 25 and whose sum is as small as possible.
Draw the level curve of the function f (x, y) = x - y containing the point (0, 0).
In Exercises, find all points (x, y) where f (x, y) has a possible relative maximum or minimum.f (x, y) = -x2 + 2y2 + 6x - 8y + 5
The function f (x, y) = 2x + 3y + 9 - x2 - xy - y2 has a maximum at some point (x, y). Find the values of x and y where this maximum occurs.
Find the values of x, y, and z that minimize xy + xz - 2yz subject to the constraint x + y + z = 2.
Draw the level curves of heights 0, 1, and 2 for the functions in Exercises.f (x, y) = -x2 + 2y
Find all points (x, y) where f (x, y) has a possible relative maximum or minimum.f (x, y) = x4 - 2xy - 7x2 + y2 + 3
Find the values of x, y and z that minimize xy + xz - yz subject to the constraint x + y + z = 1.
Draw the level curves of heights 0, 1, and 2 for the functions in Exercises.f (x, y) = 2x - y
An ecologist wished to know whether certain species of aquatic insects have their ecological range limited by temperature. He collected the data in Table 8, relating the average daily temperature at
A dealer in a certain brand of electronic calculator finds that (within certain limits) the number of calculators she can sell per week is given by f(p, t) = - p + 6t - .02 pt, where p is the price
Let f (p, q) = 1 - p(1 + q). Find af af and. dq др
Find all points (x, y) where f (x, y) has a possible relative maximum or minimum.f (x, y) = x3 + x2y - y
Find the values of x and y that maximize xy subject to the constraint x2 - y = 3.
The value of residential property for tax purposes is usually much lower than its actual market value. If y is the market value, the assessed value for real estate taxes might be only 40% of y.
Find all points (x, y) where f (x, y) has a possible relative maximum or minimum. f(x, y) = x³ – 2y³ − 5x + 6y − 5
Calculate the volumes over the following regions R bounded above by the graph of f (x, y) = x2 + y2.R is the region bounded by the lines x = 0, x = 1 and the curves y = 0 and y = 3√x.
Let f (r, y, x) be the real estate tax function of Exercise 13.(a) Determine the real estate tax on a property valued at $100,000 with a homeowner’s exemption of $5000, assuming a tax rate of $2.20
Let f (x, y) = x5 - 2 x3y + 1/2y4. Findand 225 225 22f дх2 дуг” ахду 2
Find δf/δx and δf/δy for each of the following functions. ` f(x, y) = √x² + y²
Let f (L, K ) = 3√LK. Find δf/δL.
Calculate the volumes over the following regions R bounded above by the graph of f (x, y) = x2 + y2.R is the rectangle bounded by the lines x = 1, x = 3, y = 0, and y = 1.
Table 5 gives the number of students enrolled at the University of Illinois, at Urbana-Champaign (UIUC), for the fall semesters 2012–2015.(a) Find the least-squares line for these data.(b) The
Give a formula for evaluating a double integral in terms of an iterated integral.
Find the values of x and y that minimize 3x2 - 2xy + x - 3y + 1 subject to the constraint x - 3y = 1.
Let R be the rectangle consisting of all points (x, y) such that 0 ≤ x ≤ 2, 2 ≤ y ≤ 3. Calculate the following double integrals. Interpret each as a volume. JS R ex dx dy
Find all points (x, y) where f (x, y) has a possible relative maximum or minimum. 15 f(x, y) = ¹/³x² + 6xy − 3y² + 3x + 6y 4
Let f (x, y, z) = (x + y)z. Evaluate af ду -at (x, y, z) = (2, 3, 4). =
Find δf/δx and δf/δy for each of the following functions. f(x, y) = x - y x + y
Give a geometric interpretation forwhere f (x, y) ≥ 0. ff f(x, y) dx dy, R
Find the values of x and y that minimize 18x2 + 12xy + 4y2 + 6x - 4y + 5 subject to the constraint 3x + 2y - 1 = 0.
The present value of A dollars to be paid t years in the future (assuming a 5% continuous interest rate) is P(A, t) = Ae-0.05t. Find and interpret P(100, 13.8).
In the remaining exercises, use one or more of the three methods discussed in this section (partial derivatives, formulas, or graphing utilities) to obtain the formula for the least-squares line.
Let R be the rectangle consisting of all points (x, y) such that 0 ≤ x ≤ 2, 2 ≤ y ≤ 3. Calculate the following double integrals. Interpret each as a volume. Herr R -x-y dx dy
Let f (x, y) = x3y + 8. Compute af af (1, 2) and (1, 2). ду ax
Complete Table 3 and find the values of A and B for the straight line that provides the best least-squares fit to the data. Table 3 X 1 2 3 4 5 Σ. = 2 Ν 3 7 9 12 Σ= X} Σ.xy = r2 2x2 =
Let f (x, y, λ) = xy + λ(5 - x - y). Find af af ах ду and af - а)
Let R be the rectangle consisting of all points (x, y) such that 0 ≤ x ≤ 2, 2 ≤ y ≤ 3. Calculate the following double integrals. Interpret each as a volume. ff (xy + y²³) dx dy R
Find all points (x, y) where f (x, y) has a possible relative maximum or minimum.f (x, y) = 2x3 + 2x2y - y2 + y
What is the least-squares line approximation to a set of data points? How is the line determined?
Find the values of x and y that minimize 2x2 - 2xy + y2 - 2x + 1 subject to the constraint x - y = 3.
Let R be the rectangle consisting of all points (x, y) such that 0 ≤ x ≤ 2, 2 ≤ y ≤ 3. Calculate the following double integrals. Interpret each as a volume. JJxy R xy² dx dy
Let f (x, y) = 10x2/5y3/5. Show that f (3a, 3b) = 3f (a, b).
Find δf/δx and δf/δy for each of the following functions.f (x, y) = ln(xy)
Complete Table 2 and find the values of A and B for the straight line that provides the best least-squares fit to the data. Table 2 X 1 2 3 4 Σr = 7 6 4 3 Σy = xy Σχν = 2 2.2 –
Find all points (x, y) where f (x, y) has a possible relative maximum or minimum.f (x, y) = -8y3 + 4xy + 4x2 + 9y2
Outline how the method of Lagrange multipliers is used to solve an optimization problem.
Let f (x, y, z) = x3 - yz2. Find af af ах ду and af əz
Find δf/δx and δf/δy for each of the following functions. f(x, y) = ex 1 + e"
Calculate the following iterated integrals. L'exdy) de exty dy dx 0
Find the values of x and y that minimize 2x2 + xy + y2 - y subject to the constraint x + y = 0.
Find δf/δx and δf/δy for each of the following functions.f (x, y) = xex2y2
Consider the Cobb–Douglas production function f (x, y) = 20x1/3y2/3. Compute f (8, 1), f (1, 27), and f (8, 27). Show that, for any positive constant k, f (8k, 27k) = k f (8, 27).
Let f (x, y) = x/(x - 2y). Find af af and əx dy
Find all points (x, y) where f (x, y) has a possible relative maximum or minimum.f (x, y) = -8y3 + 4xy + 9y2 - 2y
Find a formula C (x, y, z) that gives the cost of materials for the closed rectangular box in Fig. 7(a), with dimensions in feet. Assume that the material for the top and bottom costs $3 per square
State the second-derivative test for functions of two variables.
Find the values of x and y that minimize x2 - 2xy + 2y2 subject to the constraint 2x - y + 5 = 0.
In Exercises, use partial derivatives to obtain the formula for the best least-squares fit to the data points.(1, 5), (2, 7), (3, 6), (4, 10)
Let f (x, y) = ex/y. Find af Jx af and . dy
Calculate the following iterated integrals. •2x L^ ( L ²³ ( x + 1) dy ) dx y) X
Explain how to find possible relative extreme points for a function of several variables.
Let f (x, y) = 3x - 1/2y4 + 1. Find af əx af and ду
Find the values of x and y that minimize xy + y2 - x - 1 subject to the constraint x - 2y = 0.
Find δf/δx and δf/δy for each of the following functions.f (x, y) = (2x - y + 5)2
Find δf/δx and δf/δy for each of the following functions. f(x, y) = 1 x + y
In Exercises, use partial derivatives to obtain the formula for the best least-squares fit to the data points.(1, 9), (2, 8), (3, 6), (4, 3)
Find all points (x, y) where f (x, y) has a possible relative maximum or minimum.f (x, y) = x3 + y2 - 3x + 6y
Find the values of x, y that maximizesubject to the constraint 3 -2x² - 2xy-y² + x + 2y,
Give an example of a Cobb–Douglas production function. What is the marginal productivity of labor? Of capital?
Find the values of x, y that minimize x2 + xy + y2 - 2x - 5y, subject to the constraint 1 - x + y = 0.
Calculate the following iterated integrals. Ꭰ X xy dy ) . p Ax dx
Find δf/δx and δf/δy for each of the following functions. f(x, y) = X y + y x
Let f (x, y) = 3x2 + xy + 5y2. Find af Jx and af dy
Let f (x, y) = xy. Show that f (2, 3 + k) - f (2, 3) = 2k.
In Exercises, use partial derivatives to obtain the formula for the best least-squares fit to the data points.(1, 8), (2, 4), (4, 3)
Find all points (x, y) where f (x, y) has a possible relative maximum or minimum.f (x, y) = 4x2 + 4xy - 3y2 + 4y - 1
Solve the following exercises by the method of Lagrange multipliers.Minimizesubject to the constraint 3x - y - 1 = 0. 1 4x2 – 3xy + y2 + , -
Interpret δf/δy(2, 3) as a rate of change.
Calculate the following iterated integrals. L² ² ½ v ³ x dy ) dx
Let f (x, y) = xy. Show that f (2 + h, 3) - f (2, 3) = 3h.
In Exercises, use partial derivatives to obtain the formula for the best least-squares fit to the data points.(1, 2), (2, 5), (3, 11)
Find all points (x, y) where f (x, y) has a possible relative maximum or minimum.f (x, y) = 3x2 + 8xy - 3y2 - 2x + 4y - 1
What expression involving a partial derivative gives an approximation to f (a + h, b) - f (a, b)?
Find δf/δx and δf/δy for each of the following functions.f (x, y) = xexy

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