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mathematics
precalculus

Questions and Answers of Precalculus

Use a computer algebra system to find the exact value of the integral where R = [0, 1] x [0, 1]. Then use the CAS to draw the solid whose volume is given by the integral. SRx*y*e* dA, JJR
Graph the solid that lies between the surface z = 2xy/)x2 + 1) and the plane z = x + 2y and is bounded by the planes x = 0, x = 2, y = 0, and y = 4. Then find its volume.
Find the volume of the solid enclosed by the paraboloid z = 2 + x2 + (y - 2)2 and the planes z = 1, x = 1, x = -1, y = 0, and y = 4.
Find the volume of the solid in the first octant bounded by the cylinder z = 16 - x2 and the plane y = 5.
Find the volume of the solid enclosed by the surface z = 1 + x2yey and the planes z = 0, x = ±1, y = 0, and y = 1.
Find the volume of the solid enclosed by the surface z = x2 + xy2 and the planes z = 0, x = 0, x = 5, and y = ±2.
Find the volume of the solid lying under the elliptic paraboloid x2y4 + y2/9 + z = 1 and above the rectangle R = [-1, 1] x [-2, 2].
Find the volume of the solid that lies under the hyperbolic paraboloid z = 3y2 - x2 + 2 and above the rectangle R = [-1, 1] x [1, 2].
Find the volume of the solid that lies under the plane 4x+ 6y - 2z + 15 = 0 and above the rectangle R= {(x, y) | –1
Sketch the solid whose volume is given by the iterated integral. C(2 - x? – y²) dy dx Jo Jo
Sketch the solid whose volume is given by the iterated integral. (4 - x - 2y) dx dy
Calculate the double integral. 1 R=[1,3]× [1, 2] -dA, 1 + x + y
Calculate the double integral. yey dA, R= [0, 2] × [0, 3] —ху
Calculate the double integral. R=[0, 1] × [0, 1] -dA, 1+ ху
Calculate the double integral. x sin(x + y) dA, R= [0, T/6] × [0, 1/3]
Calculate the double integral. tan 0 = dA, R={(0, t) | 0 < 0 < T/3, 0 < t
Calculate the double integral. xy2 - dA, R= {(x, y) | 0< x< 1, –3 < y< 3} x? + 1
Calculate the double integral. (y + xy?) dA, R = {(x, y) | 0 < x < 2, 1 < y< 2}
Calculate the double integral. x sec'y dA, R = {(x, y) | 0 < x < 2,0 < y< #/4}
Calculate the iterated integral. II Vs +t ds dt Jo
Calculate the iterated integral. '1 C v(u + v°)* du dv
Calculate the iterated integral. C xy/r? + y² dy dx
Calculate the iterated integral. *7/2 1? sin' ф dф dt *3
Calculate the iterated integral. ye*- dx dy C2
Calculate the iterated integral. *2 х dy dx х
Calculate the iterated integral. In y dy dx лл ху (5
Calculate the iterated integral.
Calculate the iterated integral.
Calculate the iterated integral. (x + e¯») dx dy JO
Calculate the iterated integral. dx dy (х + у)? o Jo
Calculate the iterated integral. (6x²y – 2x) dy dx C 1 Jo Jo
Calculate the iterated integral.f (x, y) = y√x + 2
Find and f (x, y) = x + 3x2y2 So f(x, y) dx 22 Si f(x, y) dy •3
Evaluate the double integral by first identifying it as the volume of a solid. Sle (4 – 2y) dA, R= [0, 1] × [0, 1]
Evaluate the double integral by first identifying it as the volume of a solid. Sle (2x + 1) dA, R= {(x, y) | 0 < x < 2,0 < y < 4}
Evaluate the double integral by first identifying it as the volume of a solid. S Sle V2 dA, R= {(x, y) | 2 < x < 6, – 1 < y < 5} %3D
The contour map shows the temperature, in degrees Fahrenheit, at 4:00 pm on February 26, 2007, in Colorado. (The state measures 388 mi west to east and 276 mi south to north.) Use the Midpoint Rule
A 20-ft-by-30-ft swimming pool is filled with water. The depth is measured at 5-ft intervals, starting at one corner of the pool, and the values are recorded in the table. Estimate the volume of
Let V be the volume of the solid that lies under the graph of f(x, y) = √52 - x2 - y2 and above the rectangle given by 2 < x < 4, 2 < y < 6. Use the lines x = 3 and y = 4 to divide R
(a) Estimate the volume of the solid that lies below the surface z = 1 + x2 + 3y and above the rectangle R = [1, 2] x [0, 3]. Use a Riemann sum with m = n = 2 and choose the sample points to be lower
(a) Use a Riemann sum with m = n = 2 to estimate the value of where R = [0, 2] x [0, 1]. Take the samplepoints to be upper right corners.(b) Use the Midpoint Rule to estimate the integral
If R = [0, 4] 3 [-1, 2], use a Riemann sum with m = 2, n = 3 to estimate the value of Take the sample points to be (a) the lower right corners and (b) the upper left corners of the
a) Estimate the volume of the solid that lies below the surface z = xy and above the rectangleR = {(x, y) | 0 < x < 6, 0 < y < 4}Use a Riemann sum with m = 3, n = 2, and take the sample
Find the maximum and minimum values of f subject to the given constraints. Use a computer algebra system to solve the system of equations that arises in using Lagrange multipliers.(If your CAS finds
Find the maximum and minimum values of f subject to the given constraints. Use a computer algebra system to solve the system of equations that arises in using Lagrange multipliers.(If your CAS finds
The plane 4x - 3y + 8z = 5 intersects the cone z2 = x2 + y2 in an ellipse. (a) Graph the cone and the plane, and observe the elliptical intersection.(b) Use Lagrange multipliers to find the
The plane x + y + 2z = 2 intersects the paraboloid z = x2 + y2 in an ellipse. Find the points on this ellipse that are nearest to and farthest from the origin.
Find the maximum and minimum volumes of a rectangular box whose surface area is 1500 cm2 and whose total edge length is 200 cm.
Use Lagrange multipliers to give an alternate solutionIf the length of the diagonal of a rectangular box must be L, what is the largest possible volume?
Use Lagrange multipliers to give an alternate solutionThe base of an aquarium with given volume V is made of slate and the sides are made of glass. If slate costs five times as much (per unit area)
Use Lagrange multipliers to give an alternate solutionFind the dimensions of a rectangular box of maximum volume such that the sum of the lengths of its 12 edges is a constant c.
Use Lagrange multipliers to give an alternate solutionFind the dimensions of the rectangular box with largest volume if the total surface area is given as 64 cm2.
Use Lagrange multipliers to give an alternate solutionFind the volume of the largest rectangular box in the first octant with three faces in the coordinate planes and one vertex in the plane x + 2y +
Use Lagrange multipliers to give an alternate solutionFind the dimensions of the box with volume 1000 cm3 that has minimal surface area.
Use Lagrange multipliers to give an alternate solutionFind the maximum volume of a rectangular box that is inscribed in a sphere of radius r.
Use Lagrange multipliers to give an alternate solutionFind three positive numbers whose sum is 12 and the sum of whose squares is as small as possible.
Use Lagrange multipliers to give an alternate solutionFind three positive numbers whose sum is 100 and whose product is a maximum.
Use Lagrange multipliers to give an alternate solutionFind the points on the surface y2 = 9 + xz that are closest to the origin.
Use Lagrange multipliers to give an alternate solutionFind the points on the cone z2 = x2 + y2 that are closest to the point (4, 2, 0).
Use Lagrange multipliers to give an alternate solutionFind the point on the plane x - 2y + 3z = 6 that is closest to the point (0, 1, 1).
Use Lagrange multipliers to give an alternate solutionFind the shortest distance from the point (2, 0, -3) to the plane x + y + z = 1.
Use Lagrange multipliers to prove that the triangle with maximum area that has a given perimeter p is equilateral.Use Heron’s formula for the area:A = √s(s - x)(s - y)(s - z)where s = p/2 and x,
Use Lagrange multipliers to prove that the rectangle with maximum area that has a given perimeter p is a square.
(a) If your computer algebra system plots implicitly defined curves, use it to estimate the minimum and maximum values of f (x, y) = x3 + y3 + 3xy subject to the constraint (x - 3)2 + (y - 3)2 = 9 by
Find the extreme values of f on the region described by the inequality.f(x, y) = e2xy, x2 + 4y2 < 1
Find the extreme values of f on the region described by the inequality.f(x, y) = 2x2 + 3y2 - 4x - 5, x2 + y2 < 16
Find the extreme values of f on the region described by the inequality.f(x, y) = x2 + y2 + 4x - 4y, x2 + y2 < 9
Find the extreme values of f subject to both constraints.f (x, y, z) = x2 + y2 + z2; x - y = 1, y2 - z2 = 1
Find the extreme values of f subject to both constraints.f (x, y, z) = yz + xy; xy = 1, y2 + z2 = 1
Find the extreme values of f subject to both constraints.f (x, y, z) = x + y + z; x2 + z2 = 2, x + y = 1
Find the minimum value of f (x, y, z) = x2 + 2y2 + 3z2 subject to the constraint x + 2y + 3z = 10. Show that f has no maximum value with this constraint.
The method of Lagrange multipliers assumes that the extreme values exist, but that is not always the case. Show that the problem of finding the minimum value of f (x, y) = x2 + y2 subject to the
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.
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
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
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
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
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
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
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
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
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
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
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
(a) Use a graphing calculator or computer to graph the circle x2 + y2 = 1. On the same screen, graph several curves of the form x2 + y = c until you find two that just touch the circle. What is the
Pictured are a contour map of f and a curve with equation g(x, y) = 8. Estimate the maximum and minimum values of f subject to the constraint that g(x, y) = 8. Explain your reasoning. y g(x, y)= 8 40
Find an equation of the plane that passes through the point (1, 2, 3) and cuts off the smallest volume in the first octant.
Three alleles (alternative versions of a gene) A, B, and O determine the four blood types A (AA or AO), B (BB or BO), O (OO), and AB. The Hardy-Weinberg Law states that the proportion of individuals
The Shannon index (sometimes called the Shannon-Wiener index or Shannon-Weaver index) is a measure of diversity in an ecosystem. For the case of three species, it is defined asH = -p1 ln p1 - p2 ln
A model for the yield Y of an agricultural crop as a function of the nitrogen level N and phosphorus level P in the soil (measured in appropriate units) isY(N, P) = kNPe-N-Pwhere k is a positive
If the length of the diagonal of a rectangular box must be L, what is the largest possible volume?
A rectangular building is being designed to minimize heat loss. The east and west walls lose heat at a rate of 10 units/m2 per day, the north and south walls at a rate of 8 units/m2 per day, the
A cardboard box without a lid is to have a volume of 32,000 cm3. Find the dimensions that minimize the amount of cardboard used.
The base of an aquarium with given volume V is made of slate and the sides are made of glass. If slate costs five times as much (per unit area) as glass, find the dimensions of the aquarium that
Find the dimensions of a rectangular box of maximum volume such that the sum of the lengths of its 12 edges is a constant c.
Find the dimensions of the rectangular box with largest volume if the total surface area is given as 64 cm2.
Find the volume of the largest rectangular box in the first octant with three faces in the coordinate planes and one vertex in the plane x + 2y + 3z = 6.
Find the dimensions of the box with volume 1000 cm3 that has minimal surface area.
Find the maximum volume of a rectangular box that is inscribed in a sphere of radius r.
Find three positive numbers whose sum is 12 and the sum of whose squares is as small as possible.

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