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mathematics
calculus with applications

Calculus With Applications 12th Edition Margaret L. Lial - Solutions

In Exercises, find fx(x, y) and fy(x, y). Then find fx (2, -1) and fy (-4, 3). Leave the answers in terms of e in Exercises. f(x, y) || 3x et t2 ху
The production function for one country is z = x0.65y0.35, where x stands for units of labor and y for units of capital. At present, 50 units of labor and 29 units of capital are available. Use differentials to estimate the change in production if the number of units of labor is increased to 52 and
Graph the first-octant portion of each plane.5x + 2y = 10
Describe the procedure for finding critical points of a function in two independent variables.
Show that the function ƒ(x, y) = 8x2y in Exercise 8, subject to 3x - y = 9, does not have an absolute minimum or maximum.Find the relative maxima or minima in Exercises.Maximum of ƒ(x, y) = 8x2y, subject to 3x - y = 9
Evaluate each iterated integral. 42 LL ( + ) dx dy 3
Discuss how a function of three variables in the form w = ƒ(x, y, z) might be graphed.
The production function for another country is z = x0.8y0.2, where x stands for units of labor and y for units of capital. At present, 20 units of labor and 18 units of capital are being provided. Use differentials to estimate the change in production if an additional unit of labor is provided and
In Exercise 60 of Section 9.2, we found that the rate of heat loss (in watts) in harbor seal pups could be approximated bywhere m is the body mass of the pup (in kg), and T and A are the body core temperature and ambient water temperature, respectively (in°C). Suppose m is 25 kg, T is 36.0°, and
Graph the first-octant portion of each plane.4x + 3z = 12
Find each double integral over the rectangular region R with the given boundaries. ff (3.x² (3x² + 4y) dx dy; 0≤x≤ 3,1 ≤ y ≤ 4 R
How are second-order partial derivatives used in finding extrema?
Figures (a)–(f) on the next page show the graphs of the functions defined in Exercises. Find all relative extrema for each function, and then match the equation to its graph.(a)(b)(c)(d)(e)(f) z = −3xy + x² - y² + N -100 8
In Exercises, find fx(x, y) and fy(x, y). Then find fx(2, -1) and fy(-4, 3). Leave the answers in terms of e in Exercises.ƒ(x, y) = (7ex+2y + 4)(ex2 + y2 + 2)
Show that the three equations in Step 4 of the box “Using Lagrange Multipliers” are equivalent to the three equations ƒx(x, y) = λgx(x, y), ƒy(x, y) = λgy(x, y), g(x, y) = 0.
Suppose the graph of a plane ax + by + cz = d has a portion in the first octant. What can be said about a, b, c, and d?
A piece of bone in the shape of a right circular cylinder is 7 cm long and has a radius of 1.4 cm. It is coated with a layer of preservative 0.09 cm thick. Estimate the volume of preservative used.
Graph the first-octant portion of each plane.x = 3
Find each double integral over the rectangular region R with the given boundaries. ff (x² (x² + 4y³) dy dx; 1 ≤ x ≤ 2,0 ≤ y ≤ 3 R
Consider the problem of minimizing ƒ(x, y) = x2 + 2x + 9y2 + 4y + 8xy subject to x + y = 1.(a) Find the solution using the method of Lagrange multipliers.(b) Suppose you erroneously applied the method of finding the discriminant D from the previous section to determine whether the point found in
Figures (a)–(f) on the next page show the graphs of the functions defined in Exercises. Find all relative extrema for each function, and then match the equation to its graph.(a)(b)(c)(d)(e)(f) 3 1 s=-x+ - x + 2- Z 1 16
Find all second-order partial derivatives for the following.ƒ(x, y) = 4x2y2 - 16x2 + 4y
In Exercise 62 of Section 9.2, we found that the number of liters of blood pumped through the lungs in one minute is given bySuppose a = 160, b = 200, and v = 125. Estimate the change in C if a becomes 145, b becomes 190, and v changes to 130.In Exercise 62 of Section 9.2:According to the Fick
The vertical line test was presented, which tells whether a graph is the graph of a function. Does this test apply to functions of two variables? Explain.
A portion of a blood vessel is measured as having length 7.9 cm and radius 0.8 cm. If each measurement could be off by as much as 0.15 cm, estimate the maximum possible error in calculating the volume of the vessel.
Graph the first-octant portion of each plane.y = 4
Find each double integral over the rectangular region R with the given boundaries. R √x + y dy dx; 1 ≤ x ≤ 3,0 ≤ y ≤1
Figures (a)–(f) on the next page show the graphs of the functions defined in Exercises. Find all relative extrema for each function, and then match the equation to its graph.(a)(b)(c)(d)(e)(f) z = y² 2y² + x² 17 16
Discuss the advantages and disadvantages of the method of Lagrange multipliers compared with solving the equation g(x, y) = 0 for y (or x), substituting that expression into ƒ and then minimizing or maximizing ƒ as a function of one variable. You might want to try some examples both ways and
Find all second-order partial derivatives for the following.g(x, y) = 5x4y2 + 12y3 - 9x
A graph that was not shown in this section is the hyperboloid of one sheet, described by the equation x2 + y2 - z2 = 1. Describe it as completely as you can.
Maximize each of the following utility functions, with the cost of each commodity and total amount available to spend given.ƒ(x, y) = xy2, cost of a unit of x is $1, cost of a unit of y is $2, and $60 is available.
Let z = ƒ(x, y) = 3x3 + 4x2y - 2y2. Find the following.(a) ∂z/dx(b) ∂z/dy(-1, 4)(c) ƒxy(2, -1)
Find each double integral over the rectangular region R with the given boundaries. [[ ³√P³ + 2 R + 2y dx dy; 0 ≤ x ≤ 2,0 ≤ y ≤ 3
LetFind the following.(a) ∂z/dy(b) ∂z/dx(0, 2)(c) ƒxx(-1, 0) z = f(x, y): x + y² x - y².
Figures (a)–(f) on the next page show the graphs of the functions defined in Exercises. Find all relative extrema for each function, and then match the equation to its graph.(a)(b)(c)(d)(e)(f) z = −2x³ - 3y4 + 6xy² + 1 16
Find all second-order partial derivatives for the following.R(x, y) = 4x2 - 5xy3 + 12y2x2
Match each equation in Exercises with its graph in (a)–(f) below and in the next column.z = x2 + y2
Match each equation in Exercises with its graph in (a)–(f) below and in the next column.z2 - y2 - x2 = 1(a)(b)(c)(d)(e)(f) N Z У
Maximize each of the following utility functions, with the cost of each commodity and total amount available to spend given.ƒ(x, y) = x2y3, cost of a unit of x is $2, cost of a unit of y is $1, and $80 is available.
Figures (a)–(f) on the next page show the graphs of the functions defined in Exercises. Find all relative extrema for each function, and then match the equation to its graph.(a)(b)(c)(d)(e)(f) 1 Z = −x² + y² + 2x²2 - 2y² + 16
Match each equation in Exercises with its graph in (a)–(f) below and in the next column.x2 - y2 = z(a)(b)(c)(d)(e)(f) N Z У
Find all second-order partial derivatives for the following.h(x, y) = 30y + 5x2y + 12xy2
Find each double integral over the rectangular region R with the given boundaries. R 3 (x + y)² در dy dx; 2 ≤x≤ 4,1 ≤ y ≤ 6
Find all second-order partial derivatives for the following. r(x, y) = бу x + y
As we saw in Exercise 66 of Section 9.2, researchers have estimated the maximum life span (in years) for various species of mammals according to the formula L(E, P) = 23E0.6P-0.267, where E is the average brain mass and P is the average body mass (both in g). Consider humans, with E = 14,100 g and
The volume of the horns from bighorn sheep was estimated by researchers using the equationwhere h is the length of a horn segment (in centimeters) and r1 and r2 are the radii of the two ends of the horn segment (in centimeters).(a) Determine the volume of a segment of horn that is 40 cm long with
Find fx(x, y) and fy(x, y).ƒ(x, y) = 6x2y3 - 4y
Maximize each of the following utility functions, with the cost of each commodity and total amount available to spend given.ƒ(x, y) = x4y2, cost of a unit of x is $2, cost of a unit of y is $4, and $60 is available.
Figures (a)–(f) on the next page show the graphs of the functions defined in Exercises. Find all relative extrema for each function, and then match the equation to its graph.(a)(b)(c)(d)(e)(f) 1 z = y² + 4xy - 2x² +. 16
Find each double integral over the rectangular region R with the given boundaries. R y √2x + 5y² dx dy; 0≤x≤ 2,1 ≤ y ≤ 3
Find all second-order partial derivatives for the following. k(x, y) = -7x 2x + 3y
Match each equation in Exercises with its graph in (a)–(f) below and in the next column.z = y2 - x2(a)(b)(c)(d)(e)(f) N Z У
Find fx(x, y) and fy(x, y).ƒ(x, y) = 5x4y3 - 6x5y
Ring shake, which is the separation of the wood between growth rings, is a serious problem in hemlock trees. Researchers have developed the following function that estimates the probability P that a given hemlock tree has ring shake.where A is the age of the tree (yr), B is 1 if bird pecking is
Find fx(x, y) and fy(x, y). f(x, y) = √4x² + y²
Maximize each of the following utility functions, with the cost of each commodity and total amount available to spend given.ƒ(x, y) = x3y4, cost of a unit of x is $3, cost of a unit of y is $3, and $42 is available.
Because of terrain difficulties, two sides of a fence can be built for $6 per ft, while the other two sides cost $4 per ft. (See the sketch.) Find the field of maximum area that can be enclosed for $1200. $4 per ft $6 per ft $4 per ft $6 per ft
Find each double integral over the rectangular region R with the given boundaries. [[yerty² yexy dx dy; 2 ≤ x ≤ 3,0 ≤ y ≤ 2 R
Match each equation in Exercises with its graph in (a)–(f) below and in the next column.(a)(b)(c)(d)(e)(f) x 16 + 25 ม 4 || 1
A model that estimates the concentration of urea in the body for a particular dialysis patient, following a dialysis session, is given bywhere t represents the number of minutes of the dialysis session and g represents the rate at which the body generates urea in mg per minute.(a) Find C(180,
Show that ƒ(x, y) = 1 - x4 - y4 has a relative maximum, even though D in the theorem is 0.
Find fx(x, y) and fy(x, y). f(x, y) 2x + 5y² 3x² + y²
Find all second-order partial derivatives for the following.z = 9yex
The amount of time in seconds it takes for a swimmer to hear a single, hand-held, starting signal is given by the formulawhere (x, y) is the location of the starter (in meters), (0, p) is the location of the swimmer (in meters), and C is the air temperature (in degrees Celsius). Assume that the
Find each double integral over the rectangular region R with the given boundaries. [[x²e²³+23 d PR 3+2y dx dy; 1 ≤ x ≤ 2, 1 ≤ y ≤ 3
Match each equation in Exercises with its graph in (a)–(f) below and in the next column.z = 5(x2 + y2)-1/2(a)(b)(c)(d)(e)(f) N Z У
Show that D = 0 for ƒ(x, y) = x3 + (x - y)2 and that the function has no relative extrema.
To enclose a yard, a fence is built against a large building, so that fencing material is used only on three sides. Material for the ends costs $15 per ft; material for the side opposite the building costs $25 per ft. Find the dimensions of the yard of maximum area that can be enclosed for $2400.
Find all second-order partial derivatives for the following.z = -6xey
Find fx(x, y) and fy(x, y).ƒ(x, y) = x3e3y
A friend taking calculus is puzzled. She remembers that for a function of one variable, if the first derivative is zero at a point and the second derivative is positive, then there must be a relative minimum at the point. She doesn’t understand why that isn’t true for a function of two
The total cost to produce x large jewelry-making kits and y small ones is given by C(x, y) = 2x2 + 6y2 + 4xy + 10. If a total of ten kits must be made, how should production be allocated so that total cost is minimized?
Find fxx(x, y) and fxy(x, y). f(x, y) = 3x 2х - у
In Exercises, find all points where the functions defined below have any relative extrema. Find any saddle points.z = 2x2 - 3y2 + 12y
In Exercises, find all points where the functions defined below have any relative extrema. Find any saddle points.ƒ(x, y) = x2 + 3xy - 7x + 5y2 - 16y
In Exercises, find all points where the functions defined below have any relative extrema. Find any saddle points.z = x3 - 8y2 + 6xy + 4
A model of airline competition defines ƒ(x, y) as the probability that a customer will buy a ticket from Airline 1 rather than Airline 2 if the ticket price is x from Airline 1 and y from Airline 2. A similar function g(x, y) gives the probability that a customer will buy a ticket from Airline 2
The average energy expended for an animal to walk or run 1 km can be estimated by the functionwhere ƒ(m, v) is the energy used (in kcal per hour), m is the mass (in g), and v is the speed of movement (in km per hour) of the animal.(a) Find ƒ(300, 10).(b) Find fm(300, 10) and interpret. f(m, v) =
The idea of the average value of a function, discussed earlier for functions of the form y = f (x), can be extended to functions of more than one independent variable. For a function z = f (x, y), he average value of f over a region R is defined aswhere A is the area of the region R. Find the
Use the region R, with boundaries as indicated, to evaluate the given double integral. R (2x + 3y) dx dy; 0 ≤ y ≤ 1, y ≤ x ≤ 2-y
A sphere of radius 2 ft is to receive an insulating coating 1 in. thick. Approximate the volume of the coating needed.
The height of a sample cone from a production line is measured as 11.4 cm, while the radius is measured as 2.9 cm. Each of these measurements could be off by 0.2 cm. Approximate the maximum possible error in the volume of the cone.
The total profit from 1 acre of a certain crop depends on the amount spent on fertilizer, x, and on hybrid seed, y, according to the model P(x, y) = 0.01(-x2 + 3xy + 160x - 5y2 + 200y + 2600). The budget for fertilizer and seed is limited to $280.(a) Use the budget constraint to express one
A production function is given by P(x, y) = 400x0.3y0.7, where x is the number of units of labor and y is the number of units of capital. Find the average production level if x varies from 40 to 50 units of labor and y varies from 20 to 40 units of capital.
The cost (in dollars) for producing x units of one product and y units of a second product is C(x, y) = 0.03x2 + 6y + 2xy + 10. The weekly production of the first product varies from 100 units to 150 units, and the weekly production of the second product varies from 50 to 100 units. Estimate the
Accurate prediction of total body water is critical in determining adequate dialysis doses for patients with renal disease. For African American males, total body water can be estimated by the function T(A, M, S) = -18.37 - 0.09A + 0.34M + 0.25S, where T is the total body water (in liters), A is
The following figure shows survival curves (percent surviving as a function of age) for people in the United States in 1900 and 2000. Let ƒ(x, y) give the proportion surviving at age x in year y. Use the graph to estimate the following. Interpret each answer in words.(a) ƒ(60, 1900) (b)
The bottom of a planter is to be made in the shape of an isosceles triangle, with the two equal sides 3 ft long and the third side 2 ft long. The area of an isosceles triangle with two equal sides of length a and third side of length b is(a) Find the area of the bottom of the planter.(b) The
A length of blood vessel is measured as 2.7 cm, with the radius measured as 0.7 cm. If each of these measurements could be off by 0.1 cm, estimate the maximum possible error in the volume of the vessel.
Researchers from New Zealand have determined that the length of a brown trout depends on both its mass and age and that the length can be estimated by L(m, t) = (0.00082t + 0.0955)e(ln m+10.492/2.84), where L(m, t) is the length of the trout (in centimeters), m is the mass of the trout (in grams),
A closed box with square ends must have a volume of 125 in3. Use Lagrange multipliers to find the dimensions of such a box that has minimum surface area.
Use Lagrange multipliers to find the maximum rectangular area that can be enclosed with 400 ft of fencing, if no fencing is needed along one side.
For the functions defined as follows, find all values of x and y such that both fx(x, y) = 0 and fy(x, y) = 0.ƒ(x, y) = 6x2 + 6y2 + 6xy + 36x - 5
Find the volume under the given surface z = f(x, y) and above the rectangle with the given boundaries. z = yxVx² + y²; 0 ≤ x ≤ 4,0 ≤ y ≤ 1
Find the level curve at a production of 500 for the production functions in Exercises. Graph each level curve in the xy-plane.In their original paper, Cobb and Douglas estimated the production function for the United States to be z = 1.01x3/4y1/4, where x represents the amount of labor and y the
A hose has a radius of approximately 0.5 in. and a length of approximately 20 ft. By what factor does a change in the radius affect the volume compared with a change in the length?
Find fxx(x, y) and fxy(x, y).ƒ(x, y) = -3x2y3 + x3y
Find the area of the largest rectangular field that can be enclosed with 600 m of fencing. Assume that no fencing is needed along one side of the field.
Suppose that the profit (in hundreds of dollars) of a certain firm is approximated by P(x, y) = 1500 + 36x - 1.5x2 + 120y - 2y2, where x is the cost of a unit of labor and y is the cost of a unit of goods. Find values of x and y that maximize profit. Find the maximum profit.

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