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mathematics
calculus 6th edition

Calculus 6th Edition James Stewart - Solutions

Use the weather map in Figure 1 to estimate the value of the directional derivative of the temperature function at Reno in the southeasterly direction.Figure 1 1997 USA Today do 60 50 Reno San Francisco 70 0 50 100 150 200 (Distance in miles) 60 (Las Vegás 70 80 •Los Angeles
Find the extreme values of f(x, y) = y2 – x2.
A rectangular box without a lid is to be made from 12 m3 of cardboard. Find the maximum volume of such a box.
Find the directional derivative Duf(x, y) if f(x, y) = x3 – 3xy + 4y2 and u is the unit vector given by angle θ = π/6. What is Duf(1, 2)?
Find the local maximum and minimum values and saddle points of f(x, y) = x4 + y4 – 4xy + 1.
Find and classify the critical points of the functionAlso find the highest point on the graph of f. f(x, y) = 10x²y - 5x² - 4y² - x - 2y4
Find the directional derivative of the function f(x, y) = x2y2 – 4y at the point (2, –1) in the direction of the vector v = 2i + 5j.
Find the extreme values of f(x, y) = x2 + 2y2 on the disk x2 + y2 ≤ 1.
If f(x, y, z) = x sin yz, (a) Find the gradient of f (b) Find the directional derivative of f at (1, 3,0) in the direction of v = i + 2j k.
Find the shortest distance from the point (1, 0, –2) to the plane x + 2y + z = 4.
Find the points on the sphere x2 + y2 + z2 = 4 that are closest to and farthest from the point (3, 1, –1).
(a) If f(x, y) = xey, find the rate of change of f at the point P(2,0) in the direction from P to Q(1/2, 2). (b) In what direction does f have the maximum rate of change? What is this maximum rate of change?
A rectangular box without a lid is to be made from 12 m3 of cardboard. Find the maximum volume of such a box.
Find the maximum value of the function f(x, y, z) = x + 2y + 3z on the curve of intersection of the plane x – y + z = 1 and the cylinder x2 + y2 = 1.
Suppose that the temperature at a point (x, y, z) in space is given by T(x, y, z) = 80/(1+ x2 + 2y2 + 3z2), where T is measured in degrees Celsius and x, y, z in meters. In which direction does the temperature increase fastest at the point (1, 1, –2)? What is the maximum rate of increase?
Find the equations of the tangent plane and normal line at the point (–2, 1, –3) to the ellipsoid 시 4 + y2 + || 3
Find the absolute maximum and minimum values of the function f(x, y) = x2 – 2xy + 2y on the rectangle D = {(x, y) | 0 ≤ x ≤ 3,0 ≤ y ≤ 2}.
Estimate the volume of the solid that lies above the square R= [0, 2] × [0, 2] and below the elliptic paraboloid z = 16 – x2 –2y2. Divide R into four equal squares and choose the sample point to be the upper right corner of each square Rij. Sketch the solid and the approximating rectangular
Evaluate ∫∫D(x + 2y) dA, where D is the region bounded by the parabolas y = 2x2 and y = 1 + x2.
Charge is distributed over the triangular region D in Figure 3 so that the charge density at (x, y) is σ(x, y) = xy, measured in coulombs per square meter (C/m2). Find the total charge.Figure 3 YA 1 0 y=1 D y=1-x (1, 1) X
Evaluate ∫∫R(3x + 4y2) dA, where R is the region in the upper half-plane bounded by the circles x2+ y2 = 1 and x2 + y2 = 4.
Evaluate the triple integral ∫∫∫Bxyz2 dV, where B is the rectangular box given by B = {(x, y, z) | 0 ≤ x ≤ 1, − 1 ≤ y ≤ 2, 0≤ z
(a) Plot the point with cylindrical coordinates (2, 2π/3, 1) and find its rectangular coordinates. (b) Find cylindrical coordinates of the point with rectangular coordinates (3, –3, –7).
A transformation is defined by the equationsFind the image of the square S = {(u, v) |0 ≤ u ≤ 1, 0 ≤ v ≤ 1}. x=u² - v² y = 2uv
The point (2, π/4, π/3) is given in spherical coordinates. Plot the point and find its rectangular coordinates.
If R = {(x, y) –1 ≤ x ≤ 1, –2 ≤ y ≤ 2}, evaluate the integral SS √₁ - R √1 - x² dA
Evaluate the double integral ∫∫R (x – 3y2) dA, where R = {(x, y) |0 ≤ x ≤ 2,1 ≤ y ≤ 2}.
Find the volume of the solid that lies under the paraboloid z = x2 + y2 and above the region D in the xy-plane bounded by the line y = 2x and the parabola y = x2.
Find the volume of the solid bounded by the plane z = 0 and the paraboloid z = 1 – x2 – y2.
Find the mass and center of mass of a triangular lamina with vertices (0, 0), (1, 0), and (0, 2) if the density function is ρ(x, y) = 1 + 3x + y.
Evaluate ∫∫∫E z dV, where E is the solid tetrahedron bounded by the four planes x = 0, y = 0, z = 0, and x + y + z = 1.
Describe the surface whose equation in cylindrical coordinates is z = r.
Use the Midpoint Rule with m = n = 2 to estimate the value of the integral SSR (x − 3y²) dA, where R = {(x, y) |0 ≤ x ≤ 2,1 ≤ y ≤ 2}. -
The point (0,2√3, –2) is given in rectangular coordinates. Find spherical coordinates for this point.
Use the change of variables x = u2 – v2, y = 2uv to evaluate the integral ∫∫R y dA, where R is the region bounded by the x-axis and the parabolas y2 = 4 – 4x and y2 = 4 + 4x, y ≥ 0.
Evaluate ∫∫Ry sin(xy) dA, where R = [1, 2] × [0, π].
Evaluate ∫∫Dxy dA, where D is the region bounded by the line y = x – 1 and the parabola y2 = 2x + 6.
Use a double integral to find the area enclosed by one loop of the four- leaved rose r = cos 2θ.
A solid E lies within the cylinder x2 + y2 = 1, below the plane z = 4, and above the paraboloid z = 1 – x2 – y2. (See Figure 8.) The density at any point is proportional to its distance from the axis of the cylinder. Find the mass of E.Figure 8 z = 4 XX N (0,0,4) I T T T (0, 0,
Evaluatewhere E is the region bounded by the paraboloid y = x2 + z2 and the plane y = 4. SSS √x² + z² dv. JJJE
The density at any point on a semicircular lamina is proportional to the distance from the center of the circle. Find the center of mass of the lamina.
Evaluate where B is the unit ball: SSS e(x² + y² +2²³/² dv.
Evaluate the integral where R is the trapezoidal region with vertices (1, 0), (2, 0), (0, –2), and (0, –1). JJR (x+y)/(x−y) dA,
The contour map in Figure 12 shows the snowfall, in inches, that fell on the state of Colorado on December 20 and 21, 2006. (The state is in the shape of a rectangle that measures 388 mi west to east and 276 mi south to north.) Use the contour map to estimate the average snowfall for the entire
Find the volume of the solid S that is bounded by the elliptic paraboloid x2 + 2y2 + z = 16, the planes x = 2 and y = 2, and the three coordinate planes.
Find the volume of the tetrahedron bounded by the planes x + 2y + z = 2, x = 2y, x = 0, and z = 0.
Find the volume of the solid that lies under the paraboloid z = x2 + y2, above the xy-plane, and inside the cylinder x2 + y2 = 2x.
Use spherical coordinates to find the volume of the solid that lies above the cone z = √x2 + y2 and below the sphere x2 + y2 + z2 = z. (See Figure 9.)Figure 9 X* ZA (0,0,1) x² + y² +2²= z = √√x² + y² y
Find the moments of inertia Ix, Iy, and I0 of a homogeneous disk D with density ρ(x, y) = ρ, center the origin, and radius a.
Use a triple integral to find the volume of the tetrahedron T bounded by the planes x + 2y + z = 2x = 2y, x = 0, and z = 0.
Use Formula 13 to derive the formula for triple integration in spherical coordinates.Formula 13 13 fff f(x, y, z) dv = fff ƒ(x(u, v, w), y(u, v, w), z(u, v), w)) dV= R S a(x, y, z) a(u, v, w) du dv dw
Evaluate the iterated integral Soft sin(y²) dy dx.
Find the center of mass of a solid of constant density that is bounded by the parabolic cylinder x = y2 and the planes x = z, z = 0, and x = 1.
Use Property 11 to estimate the integral ∫∫Desin x cos y dA, where D is the disk with center the origin and radius 2. Data from 11 If m≤ f(x, y) ≤ M for all (x, y) in D, then ff D mA(D) ≤ f(x, y) dA ≤ MA(D)
If the joint density function for X and Y is given by find the value of the constant C. Then find P(X ≤ 7, Y ≥ 2). f(x, y) = [C(x + 2y) if 0≤x≤ 10, 0 ≤ y ≤ 10 otherwise
The manager of a movie theater determines that the average time moviegoers wait in line to buy a ticket for this week’s film is 10 minutes and the average time they wait to buy popcorn is 5 minutes. Assuming that the waiting times are independent, find the probability that a moviegoer waits a
A factory produces (cylindrically shaped) roller bearings that are sold as having diameter 4.0 cm and length 6.0 cm. In fact, the diameters X are normally distributed with mean 4.0 cm and standard deviation 0.01 cm while the lengths Y are normally distributed with mean 6.0 cm and standard deviation
Compute the surface integral ∫∫S x2 ds, where S is the unit sphere x2 + y2 + z2 = 1.
Evaluate ∫C (2 + x2y) ds, where C is the upper half of the unit circle x2 + y2 = 1.
Evaluate ∫C F · dr, where F(x, y, z) = –y2i + xj + z2k and C is the curve of intersection of the plane y + z = 2 and the cylinder x2 + y2 = 1.
Identify and sketch the surface with vector equation r(u, v) = 2 cos ui + v j + 2 sin u k
Find the work done by the gravitational field in moving a particle with mass m from the point (3, 4, 12) to the point (2, 2, 0) along a piecewise-smooth curve C. F(x) mMG |x| X
Find the flux of the vector field F(x, y, z) = zi + yj + xk over the unit sphere x2 + y2 + z2 = 1.
Evaluate ∫Cx4dx + xy dy, where C is the triangular curve consisting of the line segments from (0, 0) to (1, 0), from (1, 0) to (0, 1), and from (0, 1) to (0, 0).
Show that the vector field F(x, y, z) = xz i + xyz j – y2k is not conservative.
Evaluate ∫∫S y ds, where S is the surface z = x + y2, 0 ≤ x ≤ 1,0 ≤ y ≤ 2. (See Figure 2.)Figure 2 N X
Sketch the vector field on R3 given by F(x, y, z) = z k.
Use Stokes' Theorem to compute the integral ∫∫S curl F · dS, where F(x, y, z) = xz i + yz j+ xy k and S is the part of the sphere x2 + y2 + z2 = 4 that lies inside the cylinder x2 + y2 = 1 and above the xy-plane. (See Figure 4.)Figure 4Stokes' Theorem xx N 0 S C x² + y² + z² = 4 y x² +
Evaluate ∫C 2x ds, where C consists of the arc C1 of the parabola y = x2 from (0, 0) to (1, 1) followed by the vertical line segment C2 from (1, 1) to (1, 2).
Use a computer algebra system to graph the surfaceWhich grid curves have u constant? Which have v constant? r(u, v) = ((2 + sin v) cos u, (2 + sin v) sin u, u + cos v)
Evaluate ∫∫S F · ds, whereand S is the surface of the region E bounded by the parabolic cylinder z = 1 – x3 and the planes z = 0, y = 0, and y + z = 2. (See Figure 2.)Figure 2 F(x, y, z)=xy i+ (y² + e*²²)j + sin(xy) k
Evaluate where C is the circle x2 + y2 = 9. fc (3y - esinx) dx + (7x + √y+ + 1) dy,
Determine whether or not the vector field F(x, y) = (x – y)i + (x – 2) j is conservative.
(a) Show thatis a conservative vector field. (b) Find a function f such that F = ∇f. F(x, y, z) = y²z³i + 2xyz³j + 3xy²z² k
Evaluate ∫∫S z dS, where S is the surface whose sides S1 are given by the cylinder x2 + y2 = 1, whose bottom S2 is the disk x2 + y2 ≤ 1 in the plane z = 0, and whose top S3 is the part of the plane z = 1 + x that lies above S2.
Find the gradient vector field of f(x, y) = x2y – y3. Plot the gradient vector field together with a contour map of f. How are they related?
A wire takes the shape of the semicircle x2 + y2 = 1, y ≥ 0, and is thicker near its base than near the top. Find the center of mass of the wire if the linear density at any point is proportional to its distance from the line y = 1.
Find a vector function that represents the plane that passes through the point P0 with position vector r0 and that contains two nonparallel vectors a and b.
Determine whether or not the vector field F(x, y) = (3 + 2xy) i + (x2 – 3y2) j is conservative.
Evaluate where C is the boundary of the semi annular region D in the upper half-plane between the circles x2 + y2 = 1 and x2 + y2 = 4. $y² dx + 3xy dy,
Find the area enclosed by the ellipse x2/a2 + y2/b2 = 1.
Evaluate ∫C y2dx + x dy, where (a) C = C1 is the line segment from (–5, –3) to (0, 2)(b) C = C2 is the arc of the parabola x = 4 – y2 from (–5, –3) to (0, 2). (See Figure 7.)Figure 7 (-5,-3) C₁ y 0 (0, 2) C₂ 4 x=4-y² X
Find the flux of the vector field F(x, y, z) = zi + y j + x k across the unit sphere x2 + y2 + z2 = 1.
If F(x, y, z) = xzi + xyz j – y2 k, find div F.
(a) If F(x, y) = (3 + 2xy) i + (x2 – 3y2) j, find a function f such that F = ∇f. (b) Evaluate the line integral ∫CF · dr, where C is the curve given by r(t) = e' sin ti + e' costj 0≤1≤T
Find a parametric representation of the sphere x2 + y2 + z2 = a2
If F(x, y) = (–yi + xj)/(x2 + y2), show that ∫C F · dr = 2π for every positively oriented simple closed path that encloses the origin.
Evaluate ∫C y sin z ds, where C is the circular helix given by the equations x = cos t, y = sin t, z = t, 0 ≤ t ≤ 2π. (See Figure 9.)Figure 9 2 6 4 2 у 0 11 0 C x
Evaluate ∫∫SF · ds, where F(x, y, z) = yi + x j + zk and S is the boundary of the solid region E enclosed by the paraboloid z = 1 – x2 – y2 and the plane z = 0.
Show that the vector field F(x, y, z) = xz i + xyz j - y2k can't be written as the curl of another vector field, that is, F ≠ curl G.
Find a parametric representation for the cylinderx2 + y2 = 4 0 ≤ z ≤ 1
If F(x, y, z) = y2 i + (2xy + e3z)j + 3ye3zk, find a function of such that ∇f = F.
The temperature u in a metal ball is proportional to the square of the distance from the center of the ball. Find the rate of heat flow across a sphere S of radius a with center at the center of the ball.
Evaluate ∫C y dx + z dy + x dz, where C consists of the line segment C1 from (2, 0, 0) to (3, 4, 5), followed by the vertical line segment C2 from (3, 4, 5) to (3, 4, 0).
Find a vector function that represents the elliptic paraboloid z = x2 + 2y2.
Find the work done by the force field F(x, y) = x2 i – xy j in moving a particle along the quarter-circle r(t) = cos t i+ sin t j, 0 ≤ t ≤ π/2.
Find a parametric representation for the surface z = 2√x2 + y2, that is, the top half of the cone z2 = 4x2 + 4y2.
Evaluate ∫CF · dr, where F(x, y, z) = xyi + yz j + zx k and C is the twisted cubic given by .
Find parametric equations for the surface generated by rotating the curve y = sin x, 0 ≤ x ≤ 2π, about the x-axis. Use these equations to graph the surface of revolution.
Find the tangent plane to the surface with parametric equations x = u2, y = v2, z = u + 2v at the point (1, 1, 3).

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