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mathematics
calculus with applications
Calculus With Applications 12th Edition Margaret L. Lial - Solutions
LetFind the following.(a) ƒ(10, 2) (b) ƒ(100, 1)(c) ƒ(1000, 0) (d) f(x, y) = V9x + 5y log x
Let h(x, y) = √x2 + y2. Find the following.(a) h(5, 3) (b) h(2, 4)(c) h(-1, -3) (d) h(-3, -1)
A function of two variables may have both a relative maximum and an absolute maximum at the same point.Determine whether each of the following statements is true or false, and explain why.
Evaluate dw using the given information. 5x² + y² z+1 dy = -0.03, dz = 0.02 W = ; x = -2, y = 1, z = 1, dx = 0.02,
In Exercises, find all points where the functions have any relative extrema. Identify any saddle points.ƒ(x, y) = x2 + xy + y2 - 6x - 3
Determine whether each of the following statements is true or false, and explain why. -45 [T, C³x. 2 -45 (3x + 4y) dy dx = (3x 2 (3x + 4y) dx dy
Find the relative maxima or minima in Exercises.Maximum of ƒ(x, y) = 8x2y, subject to 3x - y = 9
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) = 9x2y2 - 4y2
Evaluate dz using the given information.z = ln(x2 + y2); x = 2, y = 3, dx = 0.02, dy = -0.03
The method of Lagrange multipliers tells us whether a point identified by the method is a maximum or minimum.Determine whether each of the following statements is true or false, and explain why.
In Exercises, find all points where the functions have any relative extrema. Identify any saddle points.ƒ(x, y) = xy - x2 - y2 - 2x - 2y - 6
Find the relative maxima or minima in Exercises.Minimum of ƒ(x, y) = x2 + 2y2 - xy, subject to x + y = 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) = 5x2y3
Let ƒ(x, y) = ex + ln (x + y). Find the following.(a) ƒ(1, 0) (b) ƒ(2, -1)(c) ƒ(0, e) (d) ƒ(0, e2)
Evaluate dw using the given information. w = dx = x ln(yz) — y ln; x = 2, y = 1, z = 4, Z 0.03, dy = 0.02, dz = -0.01
Determine whether each of the following statements is true or false, and explain why. 2 [Lxe dy dx = [[' xe -2 -2J0 xe" dx dy
In Exercises, find all points where the functions have any relative extrema. Identify any saddle points.ƒ(x, y) = 2x2 + 3xy + 2y2 - 5x + 5y
Find the relative maxima or minima in Exercises.Minimum of ƒ(x, y) = 3x2 + 4y2 - xy - 2, subject to 2x + y = 21
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) = -3x4y3 + 10
Let ƒ(x, y) = xex+y. Find the following.(a) ƒ(1, 0) (b) ƒ(2, -2)(c) ƒ(3, 2) (d) ƒ(-1, 4)
Determine whether each of the following statements is true or false, and explain why. (x + xy²) dx dy 1 0 (x + x), JJ = P & P ( x + x)]]] xp J.J
In Exercises, find all points where the functions have any relative extrema. Identify any saddle points.ƒ(x, y) = x2 + 3xy + 3y2 - 6x + 3y
Find the relative maxima or minima in Exercises.Maximum of ƒ(x, y) = x2 - 10y2, subject to x - y = 18
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) = ex+y
Graph the first-octant portion of each plane.x + y + z = 9
Use the total differential to approximate each quantity. Then use a calculator to approximate the quantity, and give the absolute value of the difference in the two results to 4 decimal places.
In Exercises, find all points where the functions have any relative extrema. Identify any saddle points.ƒ(x, y) = 5xy - 7x2 - y2 + 3x - 6y - 4
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) = 4e3x+2y
Graph the first-octant portion of each plane.x + y + z = 15
Use the total differential to approximate each quantity. Then use a calculator to approximate the quantity, and give the absolute value of the difference in the two results to 4 decimal places. V4.96² +12.06²
Describe in words how to take a partial derivative.
In Exercises, find all points where the functions have any relative extrema. Identify any saddle points.ƒ(x, y) = 4xy - 10x2 - 4y2 + 8x + 8y + 9
Find the relative maxima or minima in Exercises.Maximum of ƒ(x, y, z) = xyz2, subject to x + y + z = 6
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) = -6e4x-3y
Graph the first-octant portion of each plane.2x + 3y + 4z = 12
Use the total differential to approximate each quantity. Then use a calculator to approximate the quantity, and give the absolute value of the difference in the two results to 4 decimal places.(1.922 + 2.12)1/3
Describe what a partial derivative means geometrically.
In Exercises, find all points where the functions have any relative extrema. Identify any saddle points.ƒ(x, y) = 4y2 + 2xy + 6x + 4y - 8
Find the relative maxima or minima in Exercises.Minimum of ƒ(x, y, z) = xy + 2xz + 2yz, subject to xyz = 32
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) = 8e7x-y
Graph the first-octant portion of each plane.4x + 2y + 3z = 24
Use the total differential to approximate each quantity. Then use a calculator to approximate the quantity, and give the absolute value of the difference in the two results to 4 decimal places.(2.932 - 0.942)1/3
Evaluate each iterated integral. -25 < + J J 0 (x+y + y) dx dy
Describe what a total differential is and how it is useful.
In Exercises, find all points where the functions have any relative extrema. Identify any saddle points.ƒ(x, y) = x2 + xy - 2x - 2y + 2
Find positive numbers x and y such that x + y = 24 and 3xy2 is maximized.
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) = x² + y² x³ - y²
Suppose you are walking through the region of New York state shown in the topographical map in Figure 11 in the first section of this chapter. Assume you are heading north, toward the top of the map, over the western side of the mountain at the left, but not directly over the peak. Explain why you
Graph the first-octant portion of each plane.x + y = 4
Use the total differential to approximate each quantity. Then use a calculator to approximate the quantity, and give the absolute value of the difference in the two results to 4 decimal places.1.03e0.04
In Exercises, find all points where the functions have any relative extrema. Identify any saddle points.ƒ(x, y) = x2 + xy + y2 - 3x - 5
Find positive numbers x and y such that x + y = 48 and 5x2y + 10 is maximized.
Evaluate each iterated integral. 3-2 [ (xy³ - x) dy dx
Evaluate each iterated integral. 3-2 [ (xy³ - x) dy dx
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) = 3x²y³ 2 x² + y²
Graph the first-octant portion of each plane.y + z = 5
Use the total differential to approximate each quantity. Then use a calculator to approximate the quantity, and give the absolute value of the difference in the two results to 4 decimal places.0.98e-0.04
Evaluate each iterated integral. -1.6 [LIVE √x² + 3y dx dy 0 √3
Find f(-1, 2) and f(6,-3) for the following.ƒ(x, y) = -4x2 + 6xy - 3
In Exercises, find all points where the functions have any relative extrema. Identify any saddle points.ƒ(x, y) = 3x2 + 2y3 - 18xy + 42
Find three positive numbers whose sum is 90 and whose product is a maximum.
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) = ln |1 + 5x3y2|
Evaluate each iterated integral. -3 -5 LLAVE + xVx² + 3y dy dx
Find f(-1, 2) and f (6,-3) for the following. f(x, y) = x - 2y x + 5y
Graph the first-octant portion of each plane.x = 5
Use the total differential to approximate each quantity. Then use a calculator to approximate the quantity, and give the absolute value of the difference in the two results to 4 decimal places.0.99 ln 0.98
Find f(-1, 2) and f(6,-3) for the following.ƒ(x, y) = 2x2y2 - 7x + 4y
In Exercises, find all points where the functions have any relative extrema. Identify any saddle points.ƒ(x, y) = 7x3 + 3y2 - 126xy - 63
Find three positive numbers whose sum is 240 and whose product is a maximum.
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) = ln |4x4 - 2x2y2|
Evaluate each iterated integral. -2 1 J4 3 + 5y Vx dx dy
Graph the first-octant portion of each plane.z = 4
Use the total differential to approximate each quantity. Then use a calculator to approximate the quantity, and give the absolute value of the difference in the two results to 4 decimal places.2.03 ln 1.02
In Exercises, find all points where the functions have any relative extrema. Identify any saddle points.ƒ(x, y) = 6xy - x2 - 4y3 + 1
Find f(-1, 2) and f(6,-3) for the following. f(x, y) 2 √x² + y² x - y
Find the maximum and minimum values of ƒ(x, y) = x3 + 2xy + 4y2 subject to x + 2y = 12. Be sure to use the method at the end of Example 1 to determine whether each solution is a maximum or a minimum.
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) = xex2y
Evaluate each iterated integral. -25 -7 16 J2 3 + 5y √x dy dx
Graph the level curves in the first quadrant of the xy-plane for the following functions at heights of z = 0, z = 2, and z = 4.3x + 2y + z = 24
Approximate the volume of aluminum needed for a beverage can of radius 2.5 cm and height 14 cm. Assume the walls of the can are 0.08 cm thick.
In Exercises, find all points where the functions have any relative extrema. Identify any saddle points.ƒ(x, y) = 3x2 + 7y3 - 42xy + 5
Explain the difference between the two methods we used in Sections 3 and 4 to solve extrema problems.
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) = y2ex+3y
Evaluate each iterated integral. -33 1 -1 ху dy dx
Graph the level curves in the first quadrant of the xy-plane for the following functions at heights of z = 0, z = 2, and z = 4.3x + y + 2z = 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. f(x, y) = √x² + 3xy + y² + 10
Approximate the volume of material needed to make a water tumbler of diameter 3 cm and height 9 cm. Assume the walls of the tumbler are 0.2 cm thick.
Graph the first-octant portion of each plane.x + y + z = 4
In Exercises, find all points where the functions have any relative extrema. Identify any saddle points.ƒ(x, y) = ex(y+1)
Evaluate each iterated integral. 4 1 - dx dy -5 TY √2 y
Why is it unnecessary to find the value of λ when using the method explained in this section?
Graph the level curves in the first quadrant of the xy-plane for the following functions at heights of z = 0, z = 2, and z = 4.y2 - x = -z
An industrial coating 0.1 in. thick is applied to all sides of a box of dimensions 10 in. by 9 in. by 18 in. Estimate the volume of the coating used.
Graph the level curves in the first quadrant of the xy-plane for the following functions at heights of z = 0, z = 2, and z = 4. 2y X 3 Z
Graph the first-octant portion of each plane.x + 2y + 6z = 6
In Exercises, find all points where the functions have any relative extrema. Identify any saddle points.ƒ(x, y) = y2 + 2ex
Evaluate each iterated integral. 5 IL ( + ³) dx 3, 2 J3 dx dy
Show that the function ƒ(x, y) = xy2 in Exercise 7, subject to x + 2y = 15, does not have an absolute minimum or maximum.Exercise 7Find the relative maxima or minima in Exercises.Maximum of ƒ(x, y) = xy2, subject to x + 2y = 15
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) = (7x2 + 18xy2 + y3)1/3
The manufacturing cost of a smartphone is approximated by M(x, y) = 45x2 + 40y2 - 20xy + 50, where x is the cost of the parts and y is the cost of labor. Right now, the company spends $8 on parts and $14 on labor. Use differentials to approximate the change in cost if the company spends $8.25 on
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