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All Textbook Solutions for Calculus (MindTap Course List)
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+...+xn;x12+x22+...+xn2=1The 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 constraint xy=1 can be solved using Lagrange multipliers, but f does not have a maximum value with that constraint.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.17E18E19E20EFind the extreme values of f on the region described by the inequality. f(x,y)=x2+y2+4x4y,x2+y29Find the extreme values of f on the region described by the inequality. f(x,y)=2x2+3y24x5,x2+y216Find the extreme values of f on the region described by the inequality. f(x,y)=exy,x2+4y2124EConsider the problem of minimizing the function f(x,y)=x on the curve y2+x4x3=0 a piriform. a Try using Lagrange multipliers to solve the problem. b Show that the minimum value is f(0,0)=0 but the Lagrange condition f(0,0)g(0,0) is not satisfied for any value of . c Explain why Lagrange multipliers fail to find the minimum value in this case.26EThe total production P of a certain product depends on the amount L of labor used and the amount K of capital investment. In Sections 14.1 and 14.3 we discussed how the Cobb-Douglas model P=bLK1 follows from certain economic assumptions, where b and are positive constants and 1. If the cost of a unit of labor is m and the cost of a unit of capital is n, and the company can spend only p dollars as its total budget, then maximizing the production P is subject to the constraint mL+nK=p. Show that the maximum production occurs when L=pmandK=(1)pn28EUse Lagrange multipliers to prove that the rectangle with maximum area that has a given perimeter p is a square.Use Lagrange multipliers to prove that the triangle with maximum area that has a given perimeter p is equilateral. Hint: Use Herons formula for the area: A=s(sx)(sy)(sz) where s=p/2 and x, y, z are the lengths of the sides.Use Lagrange multipliers to give an alternate solution to the indicated exercise in Section 14.7. Exercise 41 Find the shortest distance from the point (2,0,3) to the plane x+y+z=1.32E33E34EUse Lagrange multipliers to give an alternate solution to the indicated exercise in Section 14.7. Exercise 45 Find three positive numbers whose sum is 100 and whose product is a maximum.36E37E38E39E40E41E42E43EFind the maximum and minimum volumes of a rectangular box whose surface area is 1500 cm2 and whose total edge length is 200 cm.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.The plane 4x3y+8z=5 intersects the cone z2=x2+y2 in an ellipse. aGraph the cone and the plane, and observe the elliptical intersection. bUse Lagrange multipliers to find the highest and lowest points on the ellipse.47EFind 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 only one solution, you may need to use additional commands. f(x,y,z)=x+y+z;x2y2=z,x2+z2=4a Find the maximum value of f(x1,x2,...,xn)=x1x2...xnn given that x1,x2,...,xn are positive numbers and x1+x2+...+xn=c, where c is a constant. b Deduce from part a that if x1,x2,...,xn are positive numbers, then x1x2...xnnx1+x2+...+xnn This inequality says that the geometric mean of n numbers is no larger than the arithmetic mean of the numbers. Under what circumstances are these two means equal?a Maximize i=1nxiyi subject to the constraints i=1nxi2=1 and i=1nyi2=1. b Put xi=aiaj2 and yi=bibj2 to show that aibiaj2bj2 for any numbers a1,...,an,b1,...,bn. This inequality is known as the Cauchy-Schwarz Inequality.1CC2CC3CC4CC5CCWhat does Clairauts Theorem say?7CC8CC9CC10CC11CC12CC13CCaDefine the gradient vector f for a function f of two or three variables. bExpress Duf in terms of f. cExplain the geometric significance of the gradient.15CC16CC17CC18CC19CC1TFQ2TFQ3TFQ4TFQ5TFQ6TFQ7TFQ8TFQ9TFQ10TFQ11TFQ12TFQ1E2E3E4E5E6EMake a rough sketch of a contour map for the function whose graph is shown.The contour map of a function f is shown. a Estimate the value of f(3,2). b Is fx(3,2) positive or negative? Explain. c Which is greater, fy(2,1) or fy(2,2)? Explain.9E10E11E12E13E14E15E16E17E18E19E20E21E22E23E24E25E26E27EFind equations of a the tangent plane and b the normal line to the given surface at the specified point. xy+yz+zx=3,(1,1,1)Find equations of a the tangent plane and b the normal line to the given surface at the specified point. sin(xyz)=x+2y+3z,(2,1,0)30EFind the points on the hyperboloid x2+4y2z2=4 where the tangent plane is parallel to the plane 2x+2y+z=5.32E33EThe two legs of a right triangle are measured as 5 m and 12 m with a possible error in measurement of at most 0.2 cm in each. Use differentials to estimate the maximum error in the calculated value of a the area of the triangle and b the length of the hypotenuse.35E36ESuppose z=f(x,y), where x=g(s,t), y=h(s,t), g(1,2)=3, gs(1,2)=1, gt(1,2)=4, h(1,2)=6, hs(1,2)=5, ht(1,2)=10, fx(3,6)=7, and fy(3,6)=8. Find z/s and z/t when s=1 and t=2.38E39E40E41E42E43E44E45E46E47E48EThe contour map shows wind speed in knots during Hurricane Andrew on August 24, 1992. Use it to estimate the value of the directional derivative of the wind speed at Homestead, Florida, in the direction of the eye of the hurricane.50E51E52E53E54E55E56EUse a graph or level curves or both to estimate the local maximum and minimum values and saddle points of f(x,y)=x33x+y42y2. Then use calculus to find these values precisely.Use a graphing calculator or computer or Newtons method or a computer algebra system to find the critical points of f(x,y)=12+10y2x28xyy4 correct to three decimal places. Then classify the critical points and find the highest point on the graph.59E60E61E62E63EA package in the shape of a rectangular box can be mailed by the US Postal Service if the sum of its length and girth the perimeter of a cross-section perpendicular to the length is at most 108 in. Find the dimensions of the package with largest volume that can be mailed.A pentagon is formed by placing an isosceles triangle on a rectangle, as shown in the figure. If the pentagon has fixed perimeter P, find the lengths of the sides of the pentagon that maximize the area of the pentagon.A rectangle with length L and width W is cut into four smaller rectangles by two lines parallel to the sides. Find the maximum and minimum values of the sum of the squares of the areas of the smaller rectangles.Marine biologists have determined that when a shark detects the presence of blood in the water, it will swim in the direction in which the concentration of the blood increases most rapidly. Based on certain tests, the concentration of blood in parts per million at a point P(x,y) on the surface of seawater is approximated by C(x,y)=e(x2+2y2)/104 where x and y are measured in meters in a rectangular coordinate system with the blood source at the origin. a Identify the level curves of the concentration function and sketch several members of this family together with a path that a shark will follow to the source. b Suppose a shark is at the point (x0,y0) when it first detects the presence of blood in the water. Find an equation of the sharks path by setting up and solving a differential equation.3PFor what values of the number r is the function f(x,y,z)={(x+y+z)rx2+y2+z2if(x,y,z)(0,0,0)0if(x,y,z)=(0,0,0) continuous on 3?5P6PIf the ellipse x2/a2+y2/b2=1 is to enclose the circle x2+y2=2y, what values of a and b minimize the area of the ellipse?Show that the maximum value of the function f(x,y)=(ax+by+c)2x2+y2+1 is a2+b2+c2. Hint: One method for attacking this problem is to use the Cauchy-Schwarz Inequality: |ab||a||b| See Exercise 12.3.61.a Estimate the volume of the solid that lies below the surface z=xy and above the rectangle R={(x,y)|0x6,0y4} Use a Riemann sum with m=3, n=2, and take the sample point to be the upper right corner of each square. b Use the Midpoint Rule to estimate the volume of the solid in part a.If R=[0,4][1,2], use a Riemann sum with m=2, n=3 to estimate the value of R(1xy2)dA. Take the sample points to be a the lower right corners and b the upper left corners of the rectangles.a Use a Riemann sum with m=n=2 to estimate the value of RxexydA, where R=[0,2][0,1]. Take the sample points to be upper right corners. b Use the Midpoint Rule to estimate the integral in part a.4E5EA 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 water in the pool. 0 5 10 15 20 25 30 0 2 3 4 6 7 8 8 5 2 3 4 7 8 10 8 10 2 4 6 8 10 12 10 15 2 3 4 5 6 8 7 20 2 2 2 2 3 4 4A contour map is shown for a function f on the square R=[0,4][0,4]. a Use the Midpoint Rule with m=n=2 to estimate the value of Rf(x,y)dA. b Estimate the average value of f.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 with m=n=4 to estimate the average temperature in Colorado at that time.9E911 Evaluate the double integral by first identifying it as the volume of a solid. R(2x+1)dA,R={(x,y)|0x2,0y4}11EThe integral R9y2dA, where R=[0,4][0,2], represents the volume of a solid. Sketch the solid.13E14E1526 Calculate the iterated integral. 1402(6x2y2x)dydx16E17E1526 Calculate the iterated integral. 0/60/2(sinx+siny)dydx19E20E1526 Calculate the iterated integral. 1412(xy+yx)dydx22E23E24E25E26E2734 Calculate the double integral. Rxsec2ydA,R={(x,y)|0x2,0y/4}28E29E30E31E32E33E34E3536 Sketch the solid whose volume is given by the iterated integral. 0101(4x2y)dxdy36E37E38EFind the volume of the solid that lying under the elliptic paraboloid x2/4+y2/9+z=1 and above the rectangle R=[1,1][2,2].Find the volume of the solid enclosed by the surface z=x2+xy2 and the planes z=0,x=0,x=5, and y=2.41E42E43E44E45E46E4748 Find the average value of f over the given rectangle. f(x,y)=x2y, R has vertices (1,0),(1,5),(1,5),(1,0)48E49EUse symmetry to evaluate the double integral. R(1+x2siny+y2sinx)dA,R=[,][,]51Ea In what way are the theorems of Fubini and Clairaut similar? b If f(x,y) is continuous on [a,b][c,d] and g(x,y)=axcyf(s,t)dtds for axb,cyd, show that gxy=gyx=f(x,y).16 Evaluate the iterated integral. 150x(8x2y)dydx2E3E4E5E16 Evaluate the iterated integral. 010ev1+evdwdv7E8E9E10EDraw an example of a region that is atype I but not type II btype II but not type I12E13E14E15E1516 Set up iterated integrals for both orders of integration. Then evaluate the double integral using the easier order and explain why its easier. Dy2exydA, D is bounded by y=x,y=4,x=017E1722 Evaluate the double integral. D(x2+2y)dA, D is bounded by y=x,y=x3,x01722 Evaluate the double integral. Dy2dA, D is triangular region with vertices (0,1),(1,2),(4,1)1722 Evaluate the double integral. DxydA, D is enclosed by the quarter-circle y=1x2,x0, and the axes1722 Evaluate the double integral. D(2xy)dA, D is bounded by the circle with center the origin and radius 21722 Evaluate the double integral. DydA, D is triangular region with vertices (0,0),(1,1),(4,0), and (4,0)2332 Find the volume of the given solid. Under the plane 3x+2yz=0 and above the region enclosed by the parabolas y=x2 and x=y224E2332 Find the volume of the given solid. Under the surface z=xy and above the triangle with vertices (1,1),(4,1), and (1,2)2332 Find the volume of the given solid. Enclosed by the paraboloid z=x2+y2+1 and the planes x=0,y=0,z=0 and x+y=227E2332 Find the volume of the given solid. Bounded by the planes z=x,y=x,x+y=2, and z=029E2332 Find the volume of the given solid. Bounded by the cylinders y2+z2=4 and the planes x=2y,x=0,z=0 in the first octant31E2332 Find the volume of the given solid. Bounded by the cylinders x2+y2=r2 and y2+z2=r233E34E35E3538 Find the volume of the solid by subtracting two volumes. The solid enclosed by the parabolic cylinders y=x2 and the planes z=3y,z=2+y37E38E39E40E41E4144 Use a computer algebra system to find the exact volume of the solid. Between the paraboloids z=2x2+y2 and z=8x22y2 and inside the cylinder x2+y2=14144 Use a computer algebra system to find the exact volume of the solid. Enclosed by z=1x2y2 and z=044E45E4550 Sketch the region of integration and change the order of integration. 02x24f(x,y)dydx4550 Sketch the region of integration and change the order of integration. 0/20cosxf(x,y)dydx4550 Sketch the region of integration and change the order of integration. 2204y2f(x,y)dxdy49E4550 Sketch the region of integration and change the order of integration. 01arctanx/4f(x,y)dydx5156 Evaluate the integral by reversing the order of integration. 013y3ex2dxdy5156 Evaluate the integral by reversing the order of integration. 01x21ysinydydx5156 Evaluate the integral by reversing the order of integration. 01x1y3+1dydx54E5156 Evaluate the integral by reversing the order of integration. 01arctany/2cosx1+cos2xdxdy56E5758 Express D as a union of regions of type I and type II and evaluate the integral. Dx2dA58E5960 Use Property 11 to estimate the value of the integral. s4x2y2dA,S={(x,y)|x2+y21,x0}60E6162 Find the average value of f over the region D. f(x,y)=xy,D is the triangle with vertices (0,0),(1,0) and (1,3)62E63EIn evaluating a double integral over a region D, a sum of iterated integrals was obtained as follows: Df(x,y)dA=0102yf(x,y)dxdy+1303yf(x,y) dxdy Sketch the region D and express the double integral as an iterated integral with reversed order of integration.6569 Use geometry or symmetry, or both, to evaluate the double integral. D(x+2)dA, D={(x,y)|0y9x2}6569 Use geometry or symmetry, or both, to evaluate the double integral. DR2x2y2dA, D is the disk with center the origin and radius R67E68E69EGraph the solid bounded by the plane x+y+z=1 and the paraboloid z=4x2y2 and find its exact volume. Use your CAS to do the graphing, to find the equations of the boundary curves of the region of integration, and to evaluate the double integral.14 A region R is shown. Decide whether to use polar coordinates or rectangular coordinates and write Rf(x,y)dA as an iterated integral, where f is an arbitrary continuous function on R.14 A region R is shown. Decide whether to use polar coordinates or rectangular coordinates and write Rf(x,y)dA as an iterated integral, where f is an arbitrary continuous function on R.14 A region R is shown. Decide whether to use polar coordinates or rectangular coordinates and write Rf(x,y)dA as an iterated integral, where f is an arbitrary continuous function on R.14 A region R is shown. Decide whether to use polar coordinates or rectangular coordinates and write Rf(x,y)dA as an iterated integral, where f is an arbitrary continuous function on R.5E56 Sketch the region whose area is given by the integral and evaluate the integral. /202sinrdrd714 Evaluate the given integral by changing to polar coordinates. Dx2ydA, where D is the top half of the disk with center the origin and radius 5714 Evaluate the given integral by changing to polar coordinates. R(2xy)dA, where R is the region in the first quadrant enclosed by the circle x2+y2=4 and the lines x=0 and y=x714 Evaluate the given integral by changing to polar coordinates. Rsin(x2+y2)dA, where R is the region in the first quadrant between the circles with centre the origin and radii 1 and 3.714 Evaluate the given integral by changing to polar coordinates. Ry2x2+y2dA, where R is the region that lies between the circles x2+y2=a2 and x2+y2=b2 with 0ab714 Evaluate the given integral by changing to polar coordinates. Dex2y2dA, where D is the region bounded by the semicircle x=4y2 and the y-axis.714 Evaluate the given integral by changing to polar coordinates. Dcosx2+y2dA, where D is the disk with centre the origin and radius 2714 Evaluate the given integral by changing to polar coordinates. Rarctan(y/x)dA, where R={(x,y)|1x2+y24,0yx}14E1518 Use a double integral to find the area of the region. One loop of the rose r=cos31518 Use a double integral to find the area of the region. The region enclosed by both of the cardioids r=1+cos and r=1cos1518 Use a double integral to find the area of the region. The region inside the circle (x1)2+y2=1 and outside the circle x2+y2=11518 Use a double integral to find the area of the region. The region inside the cardioid r=1+cos and outside the circle r=3cos1927 Use polar coordinates to find the volume of the given solid. Under the paraboloid z=x2+y2 and above the disk x2+y2251927 Use polar coordinates to find the volume of the given solid. Below the cone z=x2+y2 and above the ring 1x2+y241927 Use polar coordinates to find the volume of the given solid. Below the plane 2x+y+z=4 and above the disk x2+y211927 Use polar coordinates to find the volume of the given solid. Inside the sphere x2+y2+z2=16 and outside the cylinder x2+y2=423E24E1927 Use polar coordinates to find the volume of the given solid. Above the cone z=x2+y2 and below the sphere x2+y2+z2=11927 Use polar coordinates to find the volume of the given solid. Bounded by the paraboloids z=6x2y2 and z=2x2+2y227E28E2932 Evaluate the iterated integral by converting to polar coordinates. 0204x2ex2y2dydx2932 Evaluate the iterated integral by converting to polar coordinates. 0aa2y2a2y2(2x+y)dxdy2932 Evaluate the iterated integral by converting to polar coordinates. 01/23y1y2xy2dxdy32E33E34EA swimming pool is circular with a 40-ft diameter. The depth is constant along east-west lines and increased linearly from 2 ft at the south end to 7 ft at the north end. Find the volume of water in the pool.36E37E