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mathematics
calculus 6th edition

Calculus 6th Edition James Stewart - Solutions

Show that f(x, y) = xexy is differentiable at (1, 0) and find its linearization there. Then use it to approximate (1.1, –0.1).
If f(x, y) = 4 – x2 – 2y2, find fx(1, 1) and fy(1, 1) and interpret these numbers as slopes.
The pressure P (in kilopascals), volume V (in liters), and temperature T (in kelvins) of a mole of an ideal gas are related by the equation PV = 8.317. Find the rate at which the pressure is changing when the temperature is 300 K and increasing at a rate of 0.1 K/s and the volume is 100 L and
Use the Chain Rule to find dz/dt or dw/dt.z = cos(x + 4y), x = 5t4, y = 1/t
Find the domain and range of g(x, y) = √9 − x² − y².
Find an equation of the tangent plane to the given surface at the specified point.z = 3(x – 1)2 + 2(y + 3)2 + 7, (2, –2, 12)
If does exist? f(x, y) xy² x² + y² 42
At the beginning of Section 14.3 we discussed the heat index (perceived temperature) as a function of the actual temperature and the relative humidity and gave the following table of values from the National Weather ServiceFind a linear approximation for the heat index I = f(T, H) when I is near
If calculate f(x, y) = sin X 1 + y
Find if it exists. 3x²y lim (x,y) → (0,0) x² + y² 2
Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.fxy = ∂2f/∂x ∂y
If z = ex sin y, where x = st2 and y = s2t, find ∂z/∂s and ∂z/∂t.
Sketch the graph of the function f(x, y) = 6 – 3x – 2y.
(a) If z = f(x, y) = x2 + 3xy – y2, find the differential dz. (b) If x changes from 2 to 2.05 and y changes from 3 to 2.96, compare the values of Δz and dz.
Sketch the graph of g(x, y)=√√9x² - y².
Find ∂z/∂x and ∂z/∂y if z is defined implicitly as a function of x and y by the equationx3 + y3 + z3 + 6xyz = 1
Write out the Chain Rule for the case where w = f(x, y, z, t) and x = x(u, v), y = y(u, v), z = z(u, v), and t = t(u, v).
Evaluate lim (x²y³ x³y² + 3x + 2y). (x,y)→→(1,2)
The base radius and height of a right circular cone are measured as 10 cm and 25 cm, respectively, with a possible error in measurement of as much as 0.1 cm in each. Use differentials to estimate the maximum error in the calculated volume of the cone.
Find fx, fy, and fz if f(x, y, z) = exy ln z.
If u = x4y + y2z3, where x = rset, y =rs2e–t, and z = r2s sin t, find the value of ∂u/∂s when r = 2, s = 1,t = 0.
Use a computer to draw the graph of the Cobb-Douglas production function P(L, K) = 1.01L0.75K0.25.
Find the second partial derivatives off(x, y) = x3 + x2y3 – 2y2
Sketch the level curves of the function g(x, y) = √√9x² - y² for k= 0, 1, 2, 3
Find ∂z/∂x and ∂z/∂y if x3 + y3 + z3 + 6xyz = 1.
Find y' if x3 + y3 = 6xy.
Where is the function continuous? x2 - 2 f(x,y) = - + y2 X
The dimensions of a rectangular box are measured to be 75 cm, 60 cm, and 40 cm, and each measurement is correct to within 0.2 cm. Use differentials to estimate the largest possible error when the volume of the box is calculated from these measurements.
If g(s, t) = f(s2 – t2, t2 – s2) and is differentiable, show that g satisfies the equation Т ag as +s ag at = 0
Find the domain and range and sketch the graph of h(x, y) = 4x2 + y2.
Calculate fxxyz if f(x, y, z) = sin(3x + yz).
If z = f(x, y) has continuous second-order partial derivatives and x = r2 + s2 and y = 2rs, find (a) ∂z/∂r (b) ∂2z/∂r2.
A contour map for a function f is shown in Figure 14. Use it to estimate the values of f(1, 3) and f(4, 5).Figure 14 y 5 4 3 2 0 80 70 60 50 50 80 70 60 1 2 3 4 5 *
Show that the function u(x, y) = ex sin y is a solution of Laplace's equation.
Sketch the level curves of the function f(x, y) = 6 – 3x – 2y for the values k = –6, 0, 6, 12.
Verify that the function u(x, t) = sin(x – at) satisfies the wave equation.
Where is the function h(x, y) = arctan(y/x) continuous?
Find the domain of f if f(x, y, z) = n(z - y) + xy sin z
Sketch some level curves of the function h(x, y) = 4x2 + y2.
Find the level surfaces of the function 2 f(x, y, z) = x + y +z
Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function.kf(x, y) = xy + 1/x + 1/y
Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function.f(x, y) = ey(y2 – x2)
Computewhere D is the disk x2 + y2 ≤ 1, by first identifying the integral as the volume of a solid. 2 SSD √1-x² - y² JD
Sketch the vector field F by drawing a diagram like Figure 5 or Figure 9.F(x, y) = 1/2(i + j)Figure 5Figure 9 y F (0, 3) 0 F (2,2) F (1,0) X
Use spherical coordinates.Find the centroid of the solid in Exercise 25.Data from Exercises 25Evaluate ∫∫∫E x2dV, where E is bounded by the xz-plane and the hemispheres y = √9 – x2 – z2 and y = √16 – x2 – z2
Find (a) The curl (b) The divergence of the vector field.F(x, y, z) = xyz i – x2yk
Evaluate the line integral by two methods:(a) Directly (b) Using Green's Theorem. C is the circle with center the origin and radius 2Data from Green's TheoremLet C be a positively oriented, piecewise-smooth, simple closed curve in the plane and let D be the region bounded by C. If P and Q have
Evaluate the line integral, where C is the given curve.∫c y3ds, C: x = t3, y = t, 0 ≤ t ≤ 2
Sketch the vector field F by drawing a diagram like Figure 5 or Figure 9.F(x, y) = i + xjFigure 5Figure 9 y F (0, 3) 0 F (2,2) F (1,0) X
Determine whether the points P and Q lie on the given surface.r(u, v) = 〈2u + 3v, 1 + 5u – v, 2 + u + v〉 P(7, 10, 4), Q(5, 22, 5)
Use Stokes Theorem to evaluate ∫∫S curl F · ds.kF(x, y, z) = 2y cos zi + exsin zj + xey k, S is the hemisphere x2 + y2 + z2 = 9, z ≥ 0, oriented upwardData from Stokes TheoremLet S be an oriented piecewise-smooth surface that is bounded by a simple, closed, piecewise-smooth boundary curve C
Find (a) The curl (b) The divergence of the vector field.F(x, y, z) = x2yz i + xy2zj + xyz2k
Evaluate the line integral by two methods:(a) Directly (b) Using Green's Theorem. C is the rectangle with vertices (0, 0), (3, 0), (3, 1), and (0, 1)Data from Green's TheoremLet C be a positively oriented, piecewise-smooth, simple closed curve in the plane and let D be the region bounded by C. If
Evaluate the line integral, where C is the given curve.∫c xy ds, C: x = t2, y = 2t, 0 ≤ t ≤ 1
(a) What is a conservative vector field?(b) What is a potential function?
Determine whether the points P and Q lie on the given surface.r(u, v) = 〈u + v, u2 –v, u + v2〉 P(3, –1, 5), Q(–1, 3, 4)
Verify that the Divergence Theorem is true for the vector field F on the region E.F(x, y, z) = x2 i + xy j + zk, E is the solid bounded by the paraboloid z = 4 – x2 – y2 and the xy-planeData from the Divergence TheoremLet E be a simple solid region and let S be the boundary surface of E, given
Sketch the vector field F by drawing a diagram like Figure 5 or Figure 9.F(x, y) = yi + 1/2jFigure 5Figure 9 y F (0, 3) 0 F (2,2) F (1,0) X
Determine whether or not F is a conservative vector field. If it is, find a function f such that F = ∇f. F(x, y) = (2x – 3y)i + (–3x + 4y – 8)j
Use Stokes Theorem to evaluate ∫∫S curl F · ds.F(x, y, z) = x2z2 i + y2z2j + xyz k, S is the part of the paraboloid z = x2 + y2 that lies inside the cylinder x2 + y2 = 4, oriented upwardData from Stokes TheoremLet S be an oriented piecewise-smooth surface that is bounded by a simple, closed,
Find (a) The curl (b) The divergence of the vector field.F(x, y, z)i + (x + yz)j + (xy – √z)k
Verify that the Divergence Theorem is true for the vector field F on the region E.F(x, y, z) = xyi + yz j + zx k, E is the solid cylinder x2 + y2 ≤ 1, 0 ≤ z ≤ 1Data from the Divergence TheoremLet E be a simple solid region and let S be the boundary surface of E, given with positive (outward)
Sketch the vector field F by drawing a diagram like Figure 5 or Figure 9.F(x, y) = (x – y)i + xjFigure 5Figure 9 y F (0, 3) 0 F (2,2) F (1,0) X
Determine whether or not F is a conservative vector field. If it is, find a function f such that F = ∇f.F(x, y) = ex cos y i + ex sin y j
Find (a) The curl (b) The divergence of the vector field.F(x, y, z) = cos xz j – sin xy k
Use Stokes Theorem to evaluate ∫∫S curl F · ds.F(x, y, z) = x2y3zi + sin(xyz)j + xyz k, S is the part of the cone y2 = x2 + z2 that lies between the planes y = 0 and y = 3, oriented in the direction of the positive y-axisData from Stokes TheoremLet S be an oriented piecewise-smooth surface
Evaluate the line integral, where C is the given curve.∫c x sin y ds, C is the line segment from (0, 3) to (4, 6)
Identify the surface with the given vector equation.r(u, v) = 2 sin ui + 3 cos uj + vk, 0 ≤ v ≤ 2
Determine whether or not F is a conservative vector field. If it is, find a function f such that F = ∇f.F(x, y) = ex sin yi + ex cos y j
Find (a) The curl (b) The divergence of the vector field.F(x, y, z) = 1/√x2 + y2 + z2 (xi + yj + zk)
Evaluate the line integral, where C is the given curve.∫c (x2y3 – √x) dy, C is the arc of the curve y = √x from (1, 1) to (4, 2)
Use Green’s Theorem to evaluate the line integral along the given positively oriented curve.∫c xy2 dx + 2x2y dy, C is the triangle with vertices (0, 0). (2, 2), and (2, 4)Data from Green's TheoremLet C be a positively oriented, piecewise-smooth, simple closed curve in the plane and let D be
Identify the surface with the given vector equation.r(s, t) = 〈s, t, t2 – s2〉
Use the Divergence Theorem to calculate the surface integral ∫∫S F · ds; that is, calculate the flux of F across S.F(x, y, z) = ex sin yi + ex cos yj + yz2k, S is the surface of the box bounded by the planes x = 0, x = 1, y = 0, y = 1, z = 0, and z = 2Data from the Divergence TheoremLet E be a
Determine whether or not F is a conservative vector field. If it is, find a function f such that F = ∇f.F(x, y) = (3x2 – 2y2)i + (4xy + 3) j
Find (a) The curl (b) The divergence of the vector field.F(x, y, z)= exy sin z j + y tan–1 (x/z) k
Use Stokes Theorem to evaluate ∫∫S curl F · ds.F(x, y, z)= exy cos zi + x2zj + xyk, S is the hemisphere x = √1 – y2 – z2, oriented in the direction of the positive x-axis Data from Stokes TheoremLet S be an oriented piecewise-smooth surface that is bounded by a simple, closed,
Evaluate the line integral, where C is the given curve.∫cxey dx, C is the arc of the curve x = ey from (1, 0) to (e, 1)
Identify the surface with the given vector equation.r(s, t) = 〈s sin 2t, s2, s cos 2t〉
Use Green’s Theorem to evaluate the line integral along the given positively oriented curve.∫c cos y dx + x2 sin y dy, C is the rectangle with vertices (0, 0), (5, 0), (5, 2), and (0, 2)Data from Green's TheoremLet C be a positively oriented, piecewise-smooth, simple closed curve in the plane
Evaluate the surface integral.∫∫S xy ds, S is the triangular region with vertices (1, 0, 0), (0, 2, 0), and (0, 0, 2)
Sketch the vector field F by drawing a diagram like Figure 5 or Figure 9.F(x, y, z) = kFigure 5Figure 9 y F (0, 3) 0 F (2,2) F (1,0) X
Use the Divergence Theorem to calculate the surface integral ∫∫S F · ds; that is, calculate the flux of F across S.F(x, y, z) = x2z3 i + 2xyz3j + xz4k, S is the surface of the box with vertices (±1, ±2, ±3)Data from the Divergence TheoremLet E be a simple solid region and let S be the
Find (a) The curl (b) The divergence of the vector field.F(x, y, z)= 〈ln x, ln(xy), ln(xyz)〉
Evaluate the line integral, where C is the given curve.∫c xy dx + (x – y) dy, C consists of line segments from (0, 0) to (2, 0) and from (2, 0) to (3, 2)
Use a computer to graph the parametric surface. Get a printout and indicate on it which grid curves have u constant and which have v constant.r(u, v) = 〈u2 + 1, v3 + 1, u + v〉, –1 ≤ u ≤ 1, –1 ≤ v ≤ 1
Sketch the vector field F by drawing a diagram like Figure 5 or Figure 9.F(x, y, z) = –ykFigure 5Figure 9 y F (0, 3) 0 F (2,2) F (1,0) X
Determine whether or not F is a conservative vector field. If it is, find a function f such that F = ∇f.F(x, y) = (xy cos xy + sin xy)i + (x2 cos xy) j
Use a computer to draw the curve with vector equation r(t) = 〈t, t2, t3〉.This curve is called a twisted cubic.
Evaluate the line integral, where C is the given curve.∫c sin x dx + cos y dy, C consists of the top half of the circle x2 + y2 = 1 from (1, 0) to (–1,0) and the line segment from (–1, 0) to (–2, 3)
Find (a) The curl (b) The divergence of the vector field.F(x, y, z) = (ex, exy, exyz)
Use a computer to graph the parametric surface. Get a printout and indicate on it which grid curves have u constant and which have v constant.r(u, v) = 〈u cos v, u sin v, u5〉, –1 ≤ u ≤ 1, 0 ≤ v ≤ 2π
Use Green’s Theorem to evaluate the line integral along the given positively oriented curve.∫c sin y dx + x cos y dy, C is the ellipse x2 + xy + y2 = 1Data from Green's TheoremLet C be a positively oriented, piecewise-smooth, simple closed curve in the plane and let D be the region bounded by
Use the Divergence Theorem to calculate the surface integral ∫∫S F · ds; that is, calculate the flux of F across S.F(x, y, z) = x2yi + xy2j + 2xyzk, S is the surface of the tetrahedron bounded by the planes x = 0, y = 0, z = 0, and x + 2y + z = 2Data from the Divergence TheoremLet E be a
Use the Divergence Theorem to calculate the surface integral ∫∫S F · ds; that is, calculate the flux of F across S.F(x, y, z) = (cos z + xy2)i + xe–zj + (sin y + x2z)k, S is the surface of the solid bounded by the paraboloid z = x2 + y2 and the plane z = 4Data from the Divergence TheoremLet
Use Green’s Theorem to evaluate ʃC F · dr. (Check the orientation of the curve before applying the theorem.)F(x, y) = (√x + y3, x2 + √y), C consists of the arc of the curve y = sin x from (0, 0) to (π, 0) and the line segment from (π, 0) to (0, 0)Data from Green's TheoremLet C be a
Match the vector fields F with the plots labeled I–IV. Give reasons for your choices.kF(x, y) = 〈y, x〉 -3 -3 برا • " ✔ //\\\/ 1/ //^^\/ // \ 5 III 3 IV 5 دل ✔ ند ✔ 5 I 3 II 5 $-3 -3 " // \ \\/ 1/ //^^\/ // \ 5 III 3 IV 5 5 I 3 II 5$
Use Green’s Theorem to evaluate ʃC F · dr. (Check the orientation of the curve before applying the theorem.)F(x, y) = 〈y2 cos x, x2 + 2y sin x〉, C is the triangle from (0, 0) to (2, 6) to (2, 0) to (0, 0)Data from Green's TheoremLet C be a positively oriented, piecewise-smooth, simple
Verify that Stokes Theorem is true for the given vector field F and surface S.F(x, y, z) = xi + yj + xyz k, S is the part of the plane 2x + y + z = 2 that lies in the first octant, oriented upwardData from Stokes TheoremLet S be an oriented piecewise-smooth surface that is bounded by a simple,
Use a calculator or CAS to evaluate the line integral correct to four decimal places.∫c F · dr, where F(x, y, z) = y sin z i + z sin xj + x sin yk and r(t) = cos ti + sin t j + sin 5t k, 0 ≤ t ≤ π

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