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All Textbook Solutions for Calculus: Early Transcendentals (3rd Edition)
Optimal box Find the dimensions of the largest rectangular box in the first octant of the xyz-coordinate system that has one vertex at the origin and the opposite vertex on the plane x + 2y + 3z = 6.Absolute maxima and minima Find the absolute maximum and minimum values of the following functions on the given region R. 43.f(x, y)= x2 + y2 2y + 1; R = {(x, y): x2 + y2 4}Absolute maxima and minima Find the absolute maximum and minimum values of the following functions on the given region R. 44.f(x, y) = 2x2 + y2; R = {(x, y): x2 + y2 16Absolute maxima and minima Find the absolute maximum and minimum values of the following functions on the given region R. 45.f(x, y) = 4 + 2x2 + y2; R = {(x, y): 1 x 1, 1 y 1Absolute maxima and minima Find the absolute maximum and minimum values of the following functions on the given region R. 46.f(x, y) = 6 x2 4y2; R = {(x, y): 2 x 2, 1 y 1}Absolute maxima and minima Find the absolute maximum and minimum values of the following functions on the given region R. 47.f(x, y) = 2x2 4x + 3y2 + 2; R = {(x, y): (x 1)2 + y2 1}Absolute maxima and minima Find the absolute maximum and minimum values of the following functions on the given region R. 48.f(x, y) = x2 + y2 2x 2y; R is the closed region bounded by the triangle with vertices (0, 0), (2, 0), and (0, 2).Absolute maxima and minima Find the absolute maximum and minimum values of the following functions on the given region R. 49.f(x, y) = 2x2 + 4x 3y2 3y2 6y 1; R = {(x, y): (x 1)2 + (y + 1)2 1}Absolute maxima and minima Find the absolute maximum and minimum values of the following functions on the given region R. 50.f(x,y)=x2+y22x+2;R={(x,y):x2+y24,y0}55EAbsolute maxima and minima Find the absolute maximum and minimum values of the following functions on the given region R. 52.f(x,y)=x2+y2; R is the closed region bounded by the ellipse x24+y2=1.Pectin Extraction An increase in world production of processed fruit has led to an increase in fruit waste. One way of reducing this waste is to find useful waste byproducts. For example, waste from pineapples is reduced by extracting pectin from pineapple peels (pectin is commonly used as a thickening agent in jam and jellies, and it is also widely used in the pharmaceutical industry). Pectin extraction involves heating and drying the peels, then grinding the peels into a fine powder. The powder is next placed in a solution with a particular pH level H, for 1.5 H 2.5, and heated to a temperature T (in degrees Celsius), for 70 T 90. The percentage of the powder F(H, T) that becomes extracted pectin is F(H,T)=0.042T20.213TH11.219H2+7.327T+58.729H342.684. a. It can be shown that F attains its absolute maximum in the interior of the domain D = {(H, T): 1.5 H 2.5, 70 T 90}. Find the pH level Hand temperature T that together maximize the amount of pectin extracted from the powder. b. What is the maximum percentage of pectin that can be extracted from the powder? Round your answer to the nearest whole number. (Source: Carpathian Journal of Food Science and Technology, Dec 2014)Absolute extrema on open and / or unbounded regions If possible, find the absolute maximum and minimum values of the following functions on the region R. 54.f(x, y) = x + 3y; R = {(x, y): |x| 1, |y| 2}Absolute extrema on open and / or unbounded regions If possible, find the absolute maximum and minimum values of the following functions on the region R. 53.f(x, y) = x2 + y2 4; R = {(x, y): x2 + y2 4}Absolute extrema on open and / or unbounded regions If possible, find the absolute maximum and minimum values of the following functions on the region R. 56.f(x, y) = x2 y2; R = {(x, y); |x| 1, |y| 1}Absolute extrema on open and / or unbounded regions If possible, find the absolute maximum and minimum values of the following functions on the region R. 55.f(x,y)=2exy;R={(x,y):x0,y0}Absolute extrema on open and/or unbounded regions 62. Find the point on the plane x + y + z = 4 nearest the point P(5, 4, 4).Lease distance What point on the plane x y + z = 2 is closest to the point (1, 1, 1)?Absolute extrema on open and/or unbounded regions 64. Find the point on the paraboloid z = x2 + y2 nearest the point P(3, 3, 1).Absolute extrema on open and/or unbounded regions 65. Find the points on the cone z2 = x2 + y2 nearest the point P(6, 8, 0).Absolute extrema on open and / or unbounded regions 60.Rectangular boxes with a volume of 10 m3 are made of two materials. The material for the top and bottom of the box costs 10/m2 and the material for the sides of the box costs 1/m2. What are the dimensions of the box that minimize the cost of the box?Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume that f is differentiable at the points in question. a.The fact that fx(2, 2) = fy(2, 2) = 0 implies that f has a local maximum, local minimum, or saddle point at (2, 2). b.The function f could have a local maximum at (a, b) where fy(a,b)0. c.The function f could have both an absolute maximum and an absolute minimum at two different points that are not critical points. d.The tangent plane is horizontal at a point on a smooth surface corresponding to a critical point.68EExtreme points from contour plots Based on the level curves that are visible in the following graphs, identify the approximate locations of the local maxima, local minima, and saddle points. 63.Optimal box Find the dimensions of the rectangular box with maximum volume in the first octant with one vertex at the origin and the opposite vertex on the ellipsoid 36x2 + 4y2 + 9z2 = 36.Magic triples Let x, y, and z be nonnegative numbers with x + y + z = 200. a.Find the values of x, y, and z that minimize x2 + y2 + z2. b.Find the values of x, y, and z that minimize x2+y2+z2. c.Find the values of x, y, and z that maximize xyz. d.Find the values of x, y, and z that maximize x2y2z2.Maximum/minimum of linear functions Let R be a closed bounded region in 2 and let f(x, y) = ax + by + c, where a, b, and c are real numbers, with a and b not both zero. Give a geometrical argument explaining why the absolute maximum and minimum values of f over R occur on the boundaries of R.73ELeast squares approximation In its many guises, the least squares approximation arises in numerous areas of mathematics and statistics. Suppose you collect data for two variables (for example, height and shoe Size) in the form of pairs (x1, y1), (x2, y2), , (xn, yn). The data may be plotted as a scatterplot in the xy-plane, as shown in the figure. The technique known as linear regression asks the question: What is the equation of the line that best fits the data? The least squares criterion for best fit requires that the sum of the squares of the vertical distances between the line and the data points is a minimum. 70.Let the equation of the best-fit line be y = mx + b, where the slope m and the y-intercept b must be determined using the least squares condition. First assume that there are three data points (1, 2), (3, 5), and (4, 6). Show that the function of m and b that gives the sum of the squares of the vertical distances between the line and the three data points is E(m, b) = ((m + b) 2)2 + ((3m + b) 5)2 + ((4m + b) 6)2. Find the critical points of E and find the values of m and b that minimize E. Graph the three data points and the best-fit line.Least squares approximation In its many guises, least squares approximation arises in numerous areas of mathematics and statistics. Suppose you collect data for two variables (for example, height and shoe size) in the form of pairs (x1, y1), (x2, y2), ., (xn, yn). The data may be plotted as a scatterplot in the xy-plane, as shown in the figure. The technique known as linear regression asks the question: What is the equation of the line that "best fits" the data? The least squares criterion for best fit requires that the sum of the squares of the vertical distances between the line and the data points be a minimum. 75. Generalize the procedure in Exercise 74 by assuming n data points (x1, y1), (x2, y2), , (xn, yn) are given. Write the function E(m, b) (summation notation allows for a more compact calculation). Show that the coefficients of the best-fit line are m=(xk)(yk)nxkyk(xk)2nx2k b=1n(ykmxk) where all sums run from k = 1 to k = n76E77ESecond Derivative Test Suppose the conditions of the Second Derivative Test are satisfied on an open disk containing the point (a, b). Use the test to prove that if (a, b) is a critical point of f at which fx(a, b) = fy(a, b) = 0 and fxx(a, b) 0 fyy(a, b) or fyy(a, b) 0 fxx (a, b), then f has a saddle point at (a, b).Maximum area triangle Among all triangles with a perimeter of 9 units, find the dimensions of the triangle with the maximum area. It may be easiest to use Herons formula, which states that the area of a triangle with side length a, b, and c is A=s(sa)(sb)(sc), where 2s is the perimeter of the triangle.Slicing plane Find an equation of the plane passing through the point (3, 2, 1) that slices off the solid in the first octant with the least volume.Solitary critical points A function of one variable has the property that a local maximum (or minimum) occurring at the only critical point is also the absolute maximum (or minimum) (for example. f(x) = x2). Does the same result hold for a function of two variables? Show that the following functions have the property that they have a single local maximum (or minimum), occurring at the only critical point, but that the local maximum (or minimum) is not an absolute maximum (or minimum) on 2. a. f(x, y) = 3xey x3 e3y b.f(x,y)=(2y2y4)(ex+11+x2)11+x2 This property has the following interpretation. Suppose that a surface has a single local minimum that is not the absolute minimum. Then water can be poured into the basin around the local minimum and the surface never overflows, even though there are points on the surface below the local minimum.Two mountains without a saddle Show that the following two functions have two local maxima but no other extreme points (therefore, there is no saddle or basin between the mountains). a.f(x, y) = (x2 1)2 (x2 ey)2 b.f(x, y) = 4x2ey 2x4 e4yPowers and roots Assume that x + y + z = 1 with x 0, y 0, and z 0. a.Find the maximum and minimum values of (1 + x2)(1 + y2)(1 + z2). b.Find the maximum and minimum values of (1+x)(1+y)(1+z).Ellipsoid inside a tetrahedron (1946 Putnam Exam) Let P be a plane tangent to the ellipsoid x2/a2 + y2/b2 + z2/c2 = 1 at a point in the first octant. Let T be the tetrahedron in the first octant bounded by P and the coordinate planes x = 0, y = 0, and z = 0. Find the minimum volume of T. (The volume of a tetrahedron is one-third the area of the base times the height.)It can be shown that the function f(x, y) = x2 + y2 attains its absolute minimum value on the curve
at the point (1,1). Verify that ∇f(1, 1) and ∇g(1, 1) are parallel, and that both vectors are orthogonal to the line tangent to C at (1, 1), thereby confirming Theorem 15.16.
2QC3QCIn Figure 15.85, explain why, if you move away from the optimal point along the constraint line, the utility decreases.
Explain why, at a point that maximizes or minimizes f subject to a constraint g(x, y) = 0, the gradient of f is parallel to the gradient of g. Use a diagram.Describe the steps used to find the absolute maximum value and absolute minimum value of a differentiable function on a circle centered at the origin of the xy-plane.3E4EGraphical Lagrange multipliers The following figures show the level curves of f and the constraint curve g(x, y) = 0. Estimate the maximum and minimum values of f subject to the constraint. At each point where an extreme value occurs, indicate the direction of f and the direction of g. 50.Graphical Lagrange multipliers The following figures show the level curves of f and the constraint curve g(x, y) = 0. Estimate the maximum and minimum values of f subject to the constraint. At each point where an extreme value occurs, indicate the direction of f and the direction of g. 51.Lagrange multipliers in two variables Use Lagrange multipliers to find the maximum and minimum values of f (when they exist) subject to the given constraint. 5.f(x, y) = x + 2y subject to x2 + y2 = 4Lagrange multipliers in two variables Use Lagrange multipliers to find the maximum and minimum values of f (when they exist) subject to the given constraint. 6.f(x, y) = xy2 subject to x2 + y2 = 1Lagrange multipliers in two variables Use Lagrange multipliers to find the maximum and minimum values of f (when they exist) subject to the given constraint. 7.f(x, y) = x + y subject to x2 xy + y2 = 1Lagrange multipliers in two variables Use Lagrange multipliers to find the maximum and minimum values of f (when they exist) subject to the given constraint. 8.f(x, y) = x2 + y2 subject to 2x2 + 3xy + 2y2 = 7Lagrange multipliers in two variables Use Lagrange multipliers to find the maximum and minimum values of f (when they exist) subject to the given constraint. 9.f(x, y) = xy subject to x2 + y2 xy = 9Lagrange multipliers in two variables Use Lagrange multipliers to find the maximum and minimum values of f (when they exist) subject to the given constraint. 10.f(x, y) = x y subject to x2 + y2 3xy = 20Lagrange multipliers Each function f has an absolute maximum value and absolute minimum value subject to the given constraint. Use Lagrange multipliers to find these values. 13. f(x, y) = e'xy subject to x2 + xy + y2 = 9Lagrange multipliers Each function f has an absolute maximum value and absolute minimum value subject to the given constraint. Use Lagrange multipliers to find these values. 14. f(x, y) = x2y subject to x2 + y2 = 9Lagrange multipliers Each function f has an absolute maximum value and absolute minimum value subject to the given constraint. Use Lagrange multipliers to find these values. 15. f(x, y) = 2x2 + y2 subject to x2 + 2y + y2 = 15Lagrange multipliers Each function f has an absolute maximum value and absolute minimum value subject to the given constraint. Use Lagrange multipliers to find these values. 16. f(x, y) = x2 subject to x2 + xy + y2 = 3Lagrange multipliers in three variables Use Lagrange multipliers to find the maximum and minimum values of f (when they exist) subject to the given constraint. 15.f(x, y, z)= x + 3y z subject to x2 + y2 + z2 = 4Lagrange multipliers in three variables Use Lagrange multipliers to find the maximum and minimum values of f (when they exist) subject to the given constraint. 16.f(x, y, z) = xyz subject to x2 + 2y2 + 4z2 = 9Lagrange multipliers in three variables Use Lagrange multipliers to find the maximum and minimum values of f (when they exist) subject to the given constraint. 17.f(x, y, z) = x subject to x2 + y2 + z2 z = 1Lagrange multipliers in three variables Use Lagrange multipliers to find the maximum and minimum values of f (when they exist) subject to the given constraint. 18.f(x, y, z) = x z subject to x2 + y2 + z2 y = 2Lagrange multipliers Each function f has an absolute maximum value and absolute minimum value subject to the given constraint. Use Lagrange multipliers to find these values. 21. f(x, y, z) = x + y + z subject to x2 + y2 + z2 xy = 5Lagrange multipliers in three variables Use Lagrange multipliers to find the maximum and minimum values of f (when they exist) subject to the given constraint. 20.f(x, y, z) = x + y + z subject to x2 + y2 + z2 2x 2y =1Lagrange multipliers in three variables Use Lagrange multipliers to find the maximum and minimum values of f (when they exist) subject to the given constraint. 21.f(x, y, z) = 2x + z2 subject to x2 + y2 + 2z2 = 25Lagrange multipliers Each function f has an absolute maximum value and absolute minimum value subject to the given constraint. Use Lagrange multipliers to find these values. 24. f(x, y, z) = xy z subject to x2 + y2 + z2 xy = 1Lagrange multipliers Each function f has an absolute maximum value and absolute minimum value subject to the given constraint. Use Lagrange multipliers to find these values. 25. f(x, y, z) = x2 | y | z subject to 2x2 | 2y2 | z2 = 2Lagrange multipliers in three variables Use Lagrange multipliers to find the maximum and minimum values of f (when they exist) subject to the given constraint. 24.f(x, y, z) = (xyz)1/2 subject to x + y + z = 1 with x 0, y 0, z 0Applications of Lagrange multipliers Use Lagrange multipliers in the following problems. When the domain of the objective function is unbounded or open, explain why you have found an absolute maximum or minimum value. 25.Shipping regulations A shipping company requires that the sum of length plus girth of rectangular boxes must not exceed 108 in. Find the dimensions of the box with maximum volume that meets this condition. (The girth is the perimeter of the smallest side of the box.)28E29E30E31E32E33E34E35EApplications of Lagrange multipliers Use Lagrange multipliers in the following problems. When the domain of the objective function is unbounded or open, explain why you have found an absolute maximum or minimum value. 34.Maximum volume cylinder in a sphere Find the dimensions of a right circular cylinder of maximum volume that can be inscribed in a sphere of radius 16.Maximizing utility functions Find the values of l and g with l 0 and g 0 that maximize the following utility functions subject to the given constraints. Give the value of the utility function at the optimal point. 35.U = f(l, g) = 10l1/2 g1/2 subject to 3l + 6g = 18Maximizing utility functions Find the values of l and g with l 0 and g 0 that maximize the following utility functions subject to the given constraints. Give the value of the utility function at the optimal point. 36.U = f(l, g) = 32l2/3 g1/3 subject to 4l + 2g = 12Maximizing utility functions Find the values of l and g with l 0 and g 0 that maximize the following utility functions subject to the given constraints. Give the value of the utility function at the optimal point. 37.U = f(l, g) = 8l4/3 g1/5 subject to 10l + 8g = 40Maximizing utility functions Find the values of l and g with l 0 and g 0 that maximize the following utility functions subject to the given constraints. Give the value of the utility function at the optimal point. 38.U = f(l, g) = l1/6 g5/6 subject to 4l + 5g = 20Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a.Suppose you are standing at the center of a sphere looking at a point P on the surface of the sphere. Your line of sight to P is orthogonal to the plane tangent to the sphere at P. b.At a point that maximizes f on the curve g(x, y) = 0, the dot product fg is zero.42EAlternative method Solve the following problem from section 15.7 using Lagrange multipliers. 43. Exercise 44 44. Cardboard boxes A lidless box is to be made using 2m2 of cardboard. Find the dimensions of the box with the largest possible volume.44E45E46EAlternative method Solve the following problems from Section 15.7 using Lagrange multipliers. 47. Exercise 63 63. Find the point on the plane x y + z = 2 nearest the point P(1, 1, 1).48EAbsolute maximum and minimum values Find the absolute maximum and minimum values of the following functions over the given regions R. Use Lagrange multipliers to check for extreme points on the boundary. 49. f(x, y) = x2 + y 2 2y + 1; R = {(x, y) : x2 + y2 4} (This is Exercise 47, Section 15.7.)50EAbsolute maximum and minimum values Find the absolute maximum and minimum values of the following functions over the given regions R. Use Lagrange multipliers to check for extreme points on the boundary. 51. f(x, y) = 2x2 4x + 3y2 + 2; R = {(x, y): (x 1) 2 + y2 1} (This is Exercise 51, Section 15.71.)Extreme points on flattened spheres The equation x2n + y2n +z2n = 1, where n is a positive integer, describes a flattened sphere. Define the extreme points to be the points on the flattened sphere with a maximum distance from the origin. a.Find all the extreme points on the flattened sphere with n = 2. What is the distance between the extreme points and the origin? b.Find all the extreme points on the flattened sphere for integers n 2. What is the distance between the extreme points and the origin? c.Give the location of the extreme points in the limit as n What is the limiting distance between the extreme points and the origin as n ?Production functions Economists model the output of manufacturing systems using production functions that have many of the same properties as utility functions. The family of Cobb-Douglas production functions has the form P = f(K, L) = CKa L1a, where K represents capital, L represents labor, and C and a are positive real numbers with 0 a 1. If the cost of capital is p dollars per unit, the cost of labor is q dollars per unit, and the total available budget is B, then the constraint takes the form pK + qL = B. Find the values of K and L that maximize the following production functions subject to the given constraint, assuming K 0 and L 0. 53.P = f(K, L) = K1/2 L1/2 for 20K + 30L = 300Production functions Economists model the output of manufacturing systems using production functions that have many of the same properties as utility functions. The family of Cobb-Douglas production functions has the form P = f(K, L) = CKa L1a, where K represents capital, L represents labor, and C and a are positive real numbers with 0 a 1. If the cost of capital is p dollars per unit, the cost of labor is q dollars per unit, and the total available budget is B, then the constraint takes the form pK + qL = B. Find the values of K and L that maximize the following production functions subject to the given constraint, assuming K 0 and L 0. 54.P = f(K, L) = 10K1/3 L2/3 for 30K + 60L = 360Production functions Economists model the output of manufacturing systems using production functions that have many of the same properties as utility functions. The family of Cobb-Douglas production functions has the form P = f(K, L) = CKa L1a, where K represents capital, L represents labor, and C and a are positive real numbers with 0 a 1. If the cost of capital is p dollars per unit, the cost of labor is q dollars per unit, and the total available budget is B, then the constraint takes the form pK + qL = B. Find the values of K and L that maximize the following production functions subject to the given constraint, assuming K 0 and L 0. 55.Given the production function P = f(K, L) = KaL1a and the budget constraint pK + qL = B, where a, p, q and B are given, show that P is maximized when K = aB/p and L = (1 a)B/q.Temperature of an elliptical plate The temperature of points on an elliptical plate x2 + y2 + xy 1 is given by T(x, y) = 25(x2 + y2). Find the hottest and coldest temperatures on the edge of the elliptical plate.Maximizing a sum 57.Find the maximum value of x1 + x2 + x3 + x4 subject to the condition that x12+x22+x32+x42=16.58E59EGeometric and arithmetic means Given positive numbers x1, , xn, prove that the geometric mean (x1x2 xn)1/n is no greater than the arithmetic mean (x1+ + xn)/n in the following cases. a.Find the maximum value of xyz, subject to x + y + z = k, where k is a real number and x 0, y 0, and z 0. Use the result to prove that (xyz)1/3x+y+z3. b.Generalize part (a) and show that (x1x2xn)1/nx1++xnn.Problems with two constraints Given a differentiable function w = f(x; y, z), the goal is to find its maximum and minimum values subject to the constraints g(x, y, z) = 0 and h(x, y, z) = 0, where g and h are also differentiable. a.Imagine a level surface of the function f and the constraint surfaces g(x, y, z) = 0 and h(x, y, z) = 0. Note that g and h intersect (in general) in a curve C on which maximum and minimum values of f must be found. Explain why g and h are orthogonal to their respective surfaces. b.Explain why f lies in the plane formed by g and h at a point of C where f has a maximum or minimum value. c.Explain why part (b) implies that f = g + h at a point of C where f has a maximum or minimum value, where and . (the Lagrange multipliers) are real numbers. d.Conclude from part (c) that the equations that must be solved for maximum or minimum values of f subject to two constraints are f = g + h, g(x, y, z) = 0 and h(x, y, z) = 0.62ETwo-constraint problems Use the result of Exercise 61 to solve the following problems. 63.Find the maximum and minimum values of f(x, y, z) = xyz subject to the conditions that x2 + y2 = 4 and x + y + z = 1.Two-constraint problems Use the result of Exercise 61 to solve the following problems. 65.Find the maximum and minimum values of f(x, y, z) = x2 + y2 + z2 on the curve on which the cone z2 = 4x2: + 4y2: and the plane 2x + 4z = 5 intersect.Check assumptions Consider the function f(x, y) = xy + x + y + 100 subject to the constraint xy = 4.
Use the method of Lagrange multipliers to write a system of three equations with three variables x, y, and λ.
Solve the system in part (a) to verity that (x, y) = (−2, −2) and (x, y) = (2, 2) are solutions.
Let the curve C1 be the branch of the constraint curve corresponding to x > 0. Calculate f(2, 2) and determine whether this value is an absolute maximum or minimum value of f over C1.(Hint: Let h1 (x), for x > 0, equal the values of f over the curve C1 and determine whether h1 attains an absolute maximum or minimum value at x = 2.)
Let the curve C2 be the branch of the constraint curve corresponding to x < 0. Calculate f(−2, −2) and determine whether this value is an absolute maximum or minimum value of f over C2. (Hint: Let h2(x), for x < 0, equal the values of f over the curve C2 and determine whether h2 attains an absolute maximum or minimum value at x = −2.)
Show that the method of Lagrange multipliers falls to find the absolute maximum and minimum values of f over the constraint curve xy = 4. Reconcile your explanation with the method of Lagrange multipliers.
1RE2RE3RE4RE5REGraphs Describe the graph of the following functions, and state the domain and range of the function. 6. f(x, y) = x2+y2Graphs Describe the graph of the following functions, and state the domain and range of the function. 7. g(x, y) = x2+y21Level curves Make a sketch of several level curves of the following functions. Label at least two level curves with their z-values. 28.f(x, y) = x2 yLevel curves Make a sketch of several level curves of the following functions. Label at least two level curves with their z-values. 9. f(x, y) = x2 + 4y2Matching level curves with surfaces Match level curve plots ad with surfaces AD.11RE12RE13RE14RE15RE16RE17RE18RE19RE20RE21RE22RE23RE24RE25RE26RE27RE28RE29RELaplaces equation Verify that the following functions satisfy Laplaces equation 2ux2+2uy2=0. 48.u(x, y) = ln (x2 + y2)31REChain Rule Use the Chain Rule to evaluate the following derivatives. 32. w'(t), where w = x cos yz, x = t2 + 1, y = t, and z = t3Chain Rule Use the Chain Rule to evaluate the following derivatives 33. w'(t), where w = z ln (x2 | y2), x = 3ct, y = 4ct, and z = tChain Rule Use the Chain Rule to evaluate the following derivatives. 52.ws and wt, where w = xyz, x = 2st, y = st2, and z = s2t35REImplicit differentiation Find dy/dx for the following implicit relations. 54.2x2 + 3xy 3y4 = 2Implicit differentiation Find dy/dx for the following implicit relations. 55.y ln (x2 + y2) = 4Walking on a surface Consider the following surfaces and parameterized curves C in the xy-plane. a.In each case, find z' (t) on C. b.Imagine that you are walking on the surface directly above C consistent with the positive orientation of C. Find the values of t for which you are walking uphill. 56.z = 4x2 + y2 2; C: x = cos t, y = sin t, for 0 t 2Walking on a surface Consider the following surfaces and parameterized curves C in the xy-plane. a.In each case, find z'(t) on C. b.Imagine that you are walking on the surface directly above C consistent with the positive orientation of C. Find the values of t for which you are walking uphill. 57.z = x2 2y2 + 4; C: x = 2 cos t, y = 2 sin t, for 0 t 2Constant volume cones Suppose the radius of a right circular cone increases as r(t) = ta and the height decreases as h(t) = tb, for t 1, where a and b are positive constants. What is the relationship between a and b such that the volume of the cone remains constant (that is, V(t) = 0, where V = (/3)r2h)?Directional derivatives Consider the function f(x, y) = 2x2 4y2 + 10, whose graph is shown in the figure. a.Fill in the table showing the value of the directional derivative at points (a, b) in the direction given by the unit vectors u, v, and w. b.Interpret each of the directional derivatives computed in part (a) at the point (2, 0).Computing gradients Compute the gradient of the following functions, evaluate it at the given point P, and evaluate the directional derivative at that point in the given direction. 60.f(x,y)=x2;P(1,2);u=12,12Computing gradients Compute the gradient of the following functions, evaluate it at the given point P, and evaluate the directional derivative at that point in the given direction. 61.g(x,y)=x2y3;P(1,1);u=513,1213Computing gradients Compute the gradient of the following functions, evaluate it at the given point P, and evaluate the directional derivative at that point in the given direction. 62.f(x,y)=xy2;P(0,3);u=32,12Computing gradients Compute the gradient of the following functions, evaluate it at the given point P, and evaluate the directional derivative at that point in the given direction. 63.h(x,y)=2+x2+2y2;P(2,1);u=35,45Computing directional derivatives Compute the gradient of the following functions, evaluate it at the given point P, and evaluate the directional derivative at that point in the direction of the given vector. 46. f(x, y, z) = xe1+y2+z2; P(0, 1, 2), u= 49,19,89Computing directional derivatives Compute the gradient of the following functions, evaluate it at the given point P, and evaluate the directional derivative at that point in the direction of the given vector. 47. f(x, y, z) = sin xy + cos z; P (1, , 0); u = 27,37,67Direction of steepest ascent and descent a.Find the unit vectors that give the direction of steepest ascent and steepest descent at P. b.Find a unit vector that points in a direction of no change. 66.f(x, y) = 1n (1 + xy); P(2, 3)49RELevel curves Let f(x, y) = 8 2x2 y2. For the following level curves f(x, y) = C and points (a, b), compute the slope of the line tangent to the level curve at (a, b) and verify that the tangent line is orthogonal to the gradient at that point. 68.f(x, y) = 5; (a, b) = (1, 1)Level curves Let f(x, y) = 8 2x2 y2. For the following level curves f(x, y) = C and points (a, b), compute the slope of the line tangent to the level curve at (a, b) and verify that the tangent line is orthogonal to the gradient at that point. 69.f(x, y) = 0, (a, b) = (2, 0)52RE53RETangent planes Find an equation of the plane tangent to the following surfaces at the given points. 72.z = 2x2 + y2; (1, 1, 3) and (0, 2, 4)Tangent planes Find an equation of the plane tangent to the following surfaces at the given points. 73.x2+y24z29=1;(0,2,0)and(1,1,32)56RETangent planes Find an equation of the plane tangent to the following surfaces at the given points. 57. exy2z31=1 = 1; (1, 1, 1) and (1, 1, 1)Tangent planes Find an equation of the plane tangent to the following surfaces at the given points. 58. z tan-1xy = 0; ( 1, 1, /4) and (1, 3, /3)Tangent planes Find an equation of the plane tangent to the following surfaces at the given points. 59. x+yz=1; (2, 2, 4) and (10, 1, 9)Linear approximation a.Find the linear approximation to the function f at the point (a, b). b.Use part (a) to estimate the given function value. 78.f(x,y)=4cos(2xy);(a,b)=(4,4);estimatef(0.8,0.8).Linear approximation a.Find the linear approximation to the function f at the point (a, b). b.Use part (a) to estimate the given function value. 79.f(x, y) = (x + y)exy; (a, b) = (2, 0); estimate f(1.95, 0.05).Changes in a function Estimate the change in the function f(x, y) = 2y2 + 3x2 + xy when (x, y) changes from (1, 2) to (1.05, 1.9).Volume of a cylinder The volume of a cylinder with radius r and height h is V = r2h. Find the approximate percentage change in the volume when the radius decreases by 3% and the height increases by 2%.Volume of an ellipsoid The volume of an ellipsoid with axes of length 2a, 2b, and 2c is V = abc. Find the percentage change in the volume when a increases by 2%, b increases by 1.5%, and c decreases by 2.5%.Water-level changes A hemispherical tank with a radius of 1.50 m is filled with water to a depth of 1.00 m. Water is released from the tank and the water level drops by 0.05 m (from 1.00 m to 0.95 m). a.Approximate the change in the volume of water in the tank. The volume of a spherical cap is V = h2(3r h)/3, where r is the radius of the sphere and h is the thickness of the cap (in this case, the depth of the water). b.Approximate the change in the surface area of the water in the tank.66REAnalyzing critical points Identify the critical points of the following functions. Then determine whether each critical point corresponds to a local maximum, local minimum, or saddle point. State when your analysis is inconclusive. Confirm your results using a graphing utility. 85.f(x, y) = x3/3 y3/3 + 2xyAnalyzing critical points Identify the critical points of the following functions. Then determine whether each critical point corresponds to a local maximum, local minimum, or saddle point. State when your analysis is inconclusive. Confirm your results using a graphing utility. 86.f(x, y) = xy (2 + x)(y 3)Analyzing critical points Identify the critical points of the following functions. Then determine whether each critical point corresponds to a local maximum, local minimum, or saddle point. State when your analysis is inconclusive. Confirm your results using a graphing utility. 87.f(x, y) = 10 x3 y3 3x2 + 3y2Absolute maxima and minima Find the absolute maximum and minimum values of the following functions on the specified region R. 88.f(x, y) = x3/3 y3/3 + 2xy on the rectangle R = {(x, y):0 x 3, 1 y 1}Absolute maxima and minima Find the absolute maximum and minimum values of the following functions on the specified region R. 89.f(x, y) = x4 + y4 4xy + 1 on the square R = {(x, y): 2 x 2, 2 y 2}72REAbsolute maxima and minima Find the absolute maximum and minimum values of the following functions on the specified region R. 91.f(x, y) = xy on the semicircular disk R={(x,y):1x1,0y1x2}74RELagrange multipliers Use Lagrange multipliers to find the maximum and minimum values of f (when they exist) subject to the given constraint. 93.f(x,y)=2x+y+10 subject to 2(x1)2+4(y1)2=176RELagrange multipliers Use Lagrange multipliers to find the maximum and minimum values of f (when they exist) subject to the given constraint. 95.f(x,y)=x+2yz subject to x2+y2+z2=1Lagrange multipliers Use Lagrange multipliers to find the maximum and minimum values of f (when they exist) subject to the given constraint. 96.f(x,y,z)=x2y2z subject to 2x2+y2+z2=25Maximum perimeter rectangle Use Lagrange multipliers to find the dimensions of the rectangle with the maximum perimeter that can be inscribed with sides parallel to the coordinate axes in the ellipse x2/a2 + y2/b2 = 1.Minimum surface area cylinder Use Lagrange multipliers to find the dimensions of the right circular cylinder of minimum surface area (including the circular ends) with a volume of 32 in3.Minimum distance to a cone Find the point(s) on the cone z2x2y2=0 that are closest to the point (1, 3, 1). Give an argument showing you have found an absolute minimum of the distance function.82RE83REExplain why the sum for the volume is an approximation. How can the approximation be improved?Consider the integral 3412f(x,y)dxdy. Give the limits of integration and the variable of integration for the first (inner) integral and the second (outer) integral. Sketch the region of integration.3QC1. Write an iterated integral that gives the volume of the solid bounded by the surface f(x, y) = xy over the square {(x,y):0x2,1y3}.Write an iterated integral that gives the volume of a box with height 10 and base R {(x,y):0x5,2y4}.Write two iterated integrals that equal Rf(x,y)dA, where R={(x,y):2x4,1y5}.Consider the integral 1311(2y2+xy)dydx. State the variable of integration in the first (inner) integral and the limits of integration. State the variable of integration in the second (outer) integral and the limits of integration.Region R = {(x, y) : 0 ≤ x ≤ 4, 0 ≤ y ≤ 6} is partitioned into six equal subregions (see figure, which also shows the level curves of a function f continuous on the region R). Estimate the value of by evaluating the Riemann sum , where (xk*, yk*) is the center of the kth subregion, for k = 1, …, 6.
Pictures of solids Draw the solid whose volume is given by the following iterated integrals. Then find the volume of the solid. 41.061210dydxIterated integrals Evaluate the following iterated integrals. 5.02014xydxdyIterated integrals Evaluate the following iterated integrals. 6.0201(3x2+4y3)dydxIterated integrals Evaluate the following iterated integrals. 7.1302x2ydxdyIterated integrals Evaluate the following iterated integrals. 8.0321(2x+3y)dxdyIterated integrals Evaluate the following iterated integrals. 9.130/2xsinydydxIterated integrals Evaluate the following iterated integrals. 10.1312(y2+y)dxdyIterated integrals Evaluate the following iterated integrals. 11.1404uvdudvIterated integrals Evaluate the following iterated integrals. 16.0/403rsecdrdIterated integrals Evaluate the following iterated integrals. 15.1ln50ln3ex+ydxdyIterated integrals Evaluate the following iterated integrals. 16. 0/201uv cos (u2v) du dvIterated integrals Evaluate the following iterated integrals. 17.0101 t2est ds dtIterated integrals Evaluate the following iterated integrals. 18. 02018xy1+x4dxdyIterated integrals Evaluate the following iterated integrals. 19. 1e014(p+q) ln q dp dqIterated integrals Evaluate the following iterated integrals. 20. 010y2cos xy dx dyMore integration practice Evaluate the following iterated integrals. 43.1212xx+ydydxMore integration practice Evaluate the following iterated integrals. 44.0201x5y2ex3y3dydxMore integration practice Evaluate the following iterated integrals. 45.01143yx+y2dxdyIterated integrals Evaluate the following iterated integrals. 24. 0101x2y2ex3ydx dyDouble integrals Evaluate each double integral over the region R by converting it to an iterated integral. 17.R(x+2y)dA;R={(x,y):0x3,1y4}Double integrals Evaluate each double integral over the region R by converting it to an iterated integral. 18.R(x2+xy)dA;R={(x,y):1x2,1y1}Double integrals Evaluate each double integral over the region R by converting it to an iterated integral. 27.Rs2t sin (st2) dA; R = {(s, t) : 0 s , 0 t 1}Double integrals Evaluate each double integral over the region R by converting it to an iterated integral. 28.Rx1+xy dA; R ={(x, y) : 0 x 1, 0 y 1Double integrals Evaluate each double integral over the region R by converting it to an iterated integral. 21.RxydA;R={(x,y):0x1,1y4}Double integrals Evaluate each double integral over the region R by converting it to an iterated integral. 22.Rxysinx2dA;R={(x,y):0x/2,0y1}Double integrals Evaluate each double integral over the region R by converting it to an iterated integral. 23.Rex+2dA;R={(x,y):0xln2,1yln4}Double integrals Evaluate each double integral over the region R by converting it to an iterated integral. 24.R(x2+y2)dA;R={(x,y):1x2,0y1}Double integrals Evaluate each double integral over the region R by converting it to an iterated integral. 25.R(x5+y5)dA;R={(x,y):0x1,1y1}Double integrals Evaluate each double integral over the region R by converting it to an iterated integral. 34.Rcos (xy) dA; R = {(x, y) : 0 x 1, 0 y 2/4}Double integrals Evaluate each double integral over the region R by converting it to an iterated integral 35.Rx3y cos (x2 y2) dA; R = {(x, y) : 0 x /2, 0 y 1}Volumes of solids Find the volume of the following solids. 47.The solid beneath the cylinder f(x,y)=ex and above the region R={(x,y):0xln4,2y2}Volumes of solids Find the volume of the following solids. 49.The solid beneath the cylinder f(x,y)=243x4y and above the region R={(x,y):1x3,0y2}Volumes of solids Find the volume of the following solids 38. The solid in the first octant bounded above by the surface z = 9xy1x24+y2 and below by the xy-plane.Volumes of solids Find the volume of the following solids 39. The solid in the first octant bounded by the surface z = xy21x2 and the planes z = 0 and y = 3Choose a convenient order When convened to an iterated integral, the following double integrals are easier to evaluate in one order than the other. Find the best order and evaluate the integral. 26.RycosxydA;R={(x,y):0x1,0y/3}Choose a convenient order When convened to an iterated integral, the following double integrals are easier to evaluate in one order than the other. Find the best order and evaluate the integral. 27.R(y+1)ex(y+1)dA;R={(x,y):0x1,1y1}Choose a convenient order When convened to an iterated integral, the following double integrals are easier to evaluate in one order than the other. Find the best order and evaluate the integral. 28.Rxsec2xydA;R={(x,y):0x/3,0y1}Choose a convenient order When convened to an iterated integral, the following double integrals are easier to evaluate in one order than the other. Find the best order and evaluate the integral. 29.R6x5ex3ydA;R={(x,y):0x2,0y2}Choose a convenient order When convened to an iterated integral, the following double integrals are easier to evaluate in one order than the other. Find the best order and evaluate the integral. 30.Ry3sinxy2dA;R={(x,y):0x2,0y/2}Choose a convenient order When convened to an iterated integral, the following double integrals are easier to evaluate in one order than the other. Find the best order and evaluate the integral. 31.Rx(1+xy)2dA;R={(x,y):0x4,0y2}Average value Compute the average value of the following functions over the region R. 32.f(x,y)=4xy;R={(x,y):0x2,0y2}Average value Compute the average value of the following functions over the region R. 33.f(x,y)=ey;R={(x,y):0x6,0yln2}Average value Compute the average value of the following functions over the region R. 34.f(x,y)=sinxsiny;R={(x,y):0x,0y}Average value 35.Find the average squared distance between the points of R={(x,y):2x2,0y2} and the origin.Average value 35.Find the average squared distance between the points of R={(x,y):0x3,0y3} and the origin.Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a.The region of integration for 46134dxdy is a square. b.If f is continuous on 2, then 4613f(x,y)dxdy=4613f(x,y)dydx. c.If f is continuous on 2, then 4613f(x,y)dxdy=1346f(x,y)dydx.Symmetry Evaluate the following integrals using symmetry arguments. Let R={(x,y):=axa,byb}, where a and b are positive real numbers. a.Rxye(x2+y2) b.Rsin(xy)x2+y2+1dAComputing populations The population densities in nine districts of a rectangular county are shown in the figure. a.Use the fact that population = (population density) (area) to estimate the population of the county. b.Explain how the calculation of part (a) is related to Riemann sums and double integrals.54ECylinders Let S be the solid in 3between the cylinder z = f(x) and the region, where f(x) 0 on R. Explain why cdabf(x)dxdy equals the area of the constant cross section of S multiplied by (d c), which is the volume of S.Product of integrals Suppose f(x, y) = g(x)h(y), where g and h are continuous functions for all real values of x: and y. a.Show that cdabf(x,y)dxdy=(abg(x)dx)(cdh(y)dy) Interpret this result geometrically. b.Write (abg(x)dx)2 as an iterated integral. c.Use the result of part (a) to evaluate 021030e4y2cosxdydx.Solving for a parameter Let R={x,y}:{0x,0ya}. For what values of a, with 0a, is Rsin(x+y)dA equal to 1?58EZero average value Find the value of a 0 such that the average value of the following functions over R={(x,y):0xa,0ya} is zero. 53.f(x,y)=4x2y260EDensity and mass Suppose a thin rectangular plate, represented by a region R in the xy-plane, has a density given by the function (x, y); this function gives the area density in units such as grams per square centimeter (g/cm2). The mass of the plate is R(x,y)dA. Assume that R={(x,y):0x/2,0y} and find the mass of the plates with the following density functions. a.(x, y) = 1 + sin x b.(x, y) = 1 + sin y c.(x, y) = 1 + sin x sin yApproximating volume Propose a method based on Riemann sums to approximate the volume of the shed shown in the figure (the peak of the roof is directly above the rear comer of the shed). Carry out the method and provide an estimate of the volume.63EA region R is bounded by the x- and y-axes and the line x + y = 2. Suppose you integrate first with respect to y. Give the limits of the iterated integral over R.Could the integral in Example 2 be evaluated by integrating first (inner integral) with respect to y?
Example 2 Computing a volume
Find the volume of the solid below the surface and above the region R in the xy-plane bounded by the lines y = x, y = 8–x, and y = 1. Notice that f(x, y) > 0 on R.
Change the order of integration of the integral 010y f(x, y) dx dy4QCDescribe and sketch a region that is bounded above and below by two curves.Describe and a sketch a region that is bounded on the left and on the right by two curves.Which order of integration is preferable to integrate f(x, y) = xy over R={(x,y):y1x1y,0y1}Which order of integration would you use to find the area of the region bounded by the x-axis and the lines y = 2x + 3 and y = 3x 4 using a double integral?Change the order of integration in the integral 01y2yf(x,y)dxdy.Sketch the region of integration for 22x24exydydxSketch the region of integration for 0202x dy dx and use geometry to evaluate the iterated integral.Describe a solid whose volume equals 4416z216z2 10 dy dx and evaluate this iterated integral using geometry.Regions of integration Consider the regions R shown in the figures and write an iterated integral of a continuous function f over R. 7.Regions of integration Consider the regions R shown in the figures and write an iterated integral of a continuous function f over R. 8.Evaluating integrals Evaluate the following integrals as they are written. 17.01x16ydydxEvaluating integrals Evaluate the following integrals as they are written. 18.0102x15xy2dydxEvaluating integrals Evaluate the following integrals as they are written. 19.02x22xxydydxEvaluating integrals Evaluate the following integrals as they are written. 21./4/4sinxcosxdydxEvaluating integrals Evaluate the following integrals as they are written. 23.22x28x2xdydxEvaluating integrals Evaluate the following integrals as they are written. 24.0ln2ex2dydxEvaluating integrals Evaluate the following integrals as they are written. 25.010x2ex2dydxEvaluating integrals Evaluate the following integrals as they are written. 26.0/230xycosx3dydxEvaluating integrals Evaluate the following integrals as they are written. 43.0ln2ey2yxdxdyEvaluating integrals Evaluate the following integrals as they are written. 44.04y2yxydxdyEvaluating integrals Evaluate the following integrals as they are written. 45.0/2y/26sin(2x3y)dxdyEvaluating integrals Evaluate the following integrals as they are written. 46.0/20cosyesinydxdyEvaluating integrals Evaluate the following integrals. 23.0/20ycosy dx dyEvaluating integrals Evaluate the following integrals. 24.01tan1x/4 2x dy dxEvaluating integrals Evaluate the following integrals as they are written. 41.0416y216y22xydxdyEvaluating integrals Evaluate the following integrals. 26.010x 2exdy dxEvaluating integrals Evaluate the following integrals 27./20y2 cos xy dx dyRegions of integration Sketch each region and write an iterated integral of a continuous function f over the region. Use order dy dx. 10.R={(x,y):0x2,3x2y6x+24}Regions of integration Sketch each region and write an iterated integral of a continuous function f over the region. Use order dy dx. 11.R={(x,y):1x2,x+1y2x+4}Regions of integration Sketch each region and write an iterated integral of a continuous function f over the region. Use order dy dx. 12.R={(x,y):0x4,x2y8x}Regions of integration Sketch each region and write an iterated integral of a continuous function f over the region. Use order dy dx. 13.R is the triangular region with vertices (0, 0), (0, 2), and (1, 0).Regions of integration Sketch each region and write an iterated integral of a continuous function f over the region. Use order dy dx. 14.R is the triangular region with vertices (0,0), (0,2), and (1,1).Regions of integration Sketch each region and write an iterated integral of a continuous function f over the region. Use order dy dx. 15.R is the region in the first quadrant bounded by a circle of radius 1 centered at the origin.Regions of integration Sketch each region and write an iterated integral of a continuous function f over the region. Use order dy dx. 16.R is the region in the first quadrant bounded by the y-axis and the parabolas y = x2 and y = 1 x2Regions of integration Write an iterated integral of a continuous function f over the region R shown in the figure. 31.Regions of integration Write an iterated integral of a continuous function f over the region R shown in the figure. 32.Regions of integration Write an iterated integral of a continuous function f over the region R. Use the order dy dx. Start by sketching the region of integration if it is not supplied.
37. R is the region bounded by y = 4 – x, y = 1, and x = 0.
Regions of integration Write an iterated integral of a continuous function f over the region R. Use the order dy dx. Start by sketching the region of integration if it is not supplied.
38. R = {(x, y): 0 ≤ x ≤ y(1–y)}.
Regions of integration Write an iterated integral of a continuous function f over the region R. Use the order dy dx. Start by sketching the region of integration if it is not supplied.
39. R is the region bounded by y = 2x + 3, y = 3x – 7, and y = 0.