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mathematics
calculus 6th edition

Calculus 6th Edition James Stewart - Solutions

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.lf(x, y) = 9 – 2x + 4y – x2 – 4y2
Find the directional derivative of f at the given point in the direction indicated by the angle θ.f(x, y) = ye–x, (0.4), θ = 2π/3
Find an equation of the tangent plane to the given surface at the specified point.z = y cos(x – y), (2, 2, 2)
Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint(s).f(x,y) = x2y; x2 + 2y2 = 6
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) = –x3y + 12x2 – 8y
Find the directional derivative of f at the given point in the direction indicated by the angle θ.f(x, y) = x sin(xy), (2, 0), θ = π/3
Find an equation of the tangent plane to the given surface at the specified point.z = ex2–y2, (1, –1, 1)
Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint(s).f(x, y) = exy; x3 + y3 = 16
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) = x4 + y4 – 4xy + 2
Use the Chain Rule to find ∂z/∂s and ∂z/∂t.z = x2y3, x = s cos t, y = s sin t
Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint(s).f(x, y, z) = 2x + 6y + 10z; x2 + y2 + z2 = 35
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) = e4y–x2–y2
Among all planes that are tangent to the surface xy2z2 = 1, find the ones that are farthest from the origin.
(a) Find the gradient of f. (b) Evaluate the gradient at the point P. (c) Find the rate of change of f at P in the direction of the vector u.f(x, y) = y2/x, P(1,2), u = 1/3(2i + √5j)
Use the Chain Rule to find ∂z/∂s and ∂z/∂t.z = arcsin(x – y), x = s2 + t2, y = 1 – 2st
Graph the surface and the tangent plane at the given point. (Choose the domain and viewpoint so that you get a good view of both the surface and the tangent plane.) Then zoom in until the surface and the tangent plane become indistinguishable.z = arctan(xy2), (1, 1, π/4)
Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint(s).f(x, y, z) = 8x – 4z; x2 + 10y2 + z2 = 5
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.(x, y) = (1 + xy)(x + y)
(a) Find the gradient of f. (b) Evaluate the gradient at the point P. (c) Find the rate of change of f at P in the direction of the vector u.f(x, y, z)= xe2yz, P(3, 0, 2), u = 〈2/3, –2/3, 1/3〉
Use the Chain Rule to find ∂z/∂s and ∂z/∂t.z = sin θ cos Φ , θ = st2, Φ = s2t
Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint(s).f(x, y, z) = xyz; x2 + 2y2 + 3z2 = 6
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) = 2x3 + xy2 + 5x2 + y2
(a) Find the gradient of f. (b) Evaluate the gradient at the point P. (c) Find the rate of change of f at P in the direction of the vector u.f(x, y, z) = √x + yz, P(1, 3, 1), u = 〈2/7, 3/7, 6/7〉
Use the Chain Rule to find ∂z/∂s and ∂z/∂t.z = ex + 2y, x = s/t, y = t/s
Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint(s).kf(x, y, z) = x2y2z2; x2 + y2 + z2 = 1
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) = x3 – 12xy + 8y3
Find the directional derivative of the function at the given point in the direction of the vector v.f(x, y) = 1 + 2x√y, (3, 4), v = 〈4, –3〉
Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint(s).f(x, y, z) = x2 + y2 + z2; x4 + y4 + z4 = 1
Find the directional derivative of the function at the given point in the direction of the vector v.f(x, y) = ln(x2 + y2), (2, 1), v = 〈–1, 2〉
Explain why the function is differentiable at the given point. Then find the linearization L(x, y) of the function at that point.f(x, y) = x3y4, (1, 1)
Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint(s).f(x, y, z) = x4 + y4 + z4; x2 + y2 + z2 = 1
If z = f(x, y), where f is differentiable, andfind dz/dt when t = 3. x = g(t) g(3) = 2 g'(3) = 5 fx(2,7) = 6 y = h(t) h(3) = 7 h'(3) fy(2,7) = -8 = -4
Find the first partial derivatives.f(x, y) = √2x + y2
Find the directional derivative of the function at the given point in the direction of the vector v.g(p, q) = p4 – p2q3, (2, 1), v = i + 3j
Explain why the function is differentiable at the given point. Then find the linearization L(x, y) of the function at that point.f(x, y) = x/x + y, (2, 1)
Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint(s).f(x, y, z, t) = x + y + z + t; x2 + y2 + z2 + t2 = 1
Find the first partial derivatives.u = e–r sin 2θ
Let W(s, t) = F(u(s, t), v(s, t)), where F, u, and v are differentiable, andFind Ws (1, 0) and Wt(1,0). u(1,0) = 2 us(1,0) = -2 u,(1,0) = 6 F(2, 3) = -1 v(1,0) = 3 vs (1,0) = 5 v,(1,0) = 4 F,(2, 3) 10 =
Find the directional derivative of the function at the given point in the direction of the vector v.g(r, s) = tan–1(rs). (1, 2), v = 5i + 10j
Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint(s). f(x1,x2,. f(x₁, x2₁, X₂) = x₁ + x₂ + x² + x² + + x = 1 + xni
Explain why the function is differentiable at the given point. Then find the linearization L(x, y) of the function at that point.f(x, y) = √x + e4y, (3,0)
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) = (x2 + y2)ey2 –x2
Find the first partial derivatives.g(u, v) = u tan–1 v
Find the directional derivative of the function at the given point in the direction of the vector v.f(x, y, z) = xey + yez + zex, (0, 0, 0), v = 〈5, 1, –2〉
Explain why the function is differentiable at the given point. Then find the linearization L(x, y) of the function at that point.f(x, y) = e–xycos y, (7,0)
Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint(s).f(x, y, z)= x + 2y; x + y+ z = 1, y2 + z2 = 4
Find the first partial derivatives.w = x/y – z
Find the directional derivative of the function at the given point in the direction of the vector v.f(x, y, z) = √xyz, (3, 2, 6), v = 〈–1, –2, 2〉
The dimensions of a closed rectangular box are measured as 80 cm, 60 cm, and 50 cm, respectively, with a possible error of 0.2 cm in each dimension. Use differentials to estimate the maximum error in calculating the surface area of the box.
Find the first partial derivatives of the function.u = xy/z
For the given contour map draw the curves of steepest ascent starting at P and at Q. P -60 50 40 30 20
Find the first partial derivatives of the function.f(x, y, z, t) = xyz2 tan(yt)
Find the first partial derivatives of the function.f(x, y, z, t) = xy2/t + 2z
A boundary stripe 3 in. wide is painted around a rectangle whose dimensions are 100 ft by 200 ft. Use differentials to approximate the number of square feet of paint in the stripe.
Find the first partial derivatives of the function.u = √x12 + x22 + · · · · · + xn2
Find the indicated partial derivatives.f(x, y) = ln(x + √x2 + y2); fx(3, 4)
Find the indicated partial derivatives.f(x, y) = arctan(y/x); fx(2.3)
Four positive numbers, each less than 50, are rounded to the first decimal place and then multiplied together. Use differentials to estimate the maximum possible error in the computed product that might result from the rounding.
Find equations of (a) The tangent plane (b) The normal line to the given surface at the specified point.y = x2 – z2, (4, 7, 3)
Find the indicated partial derivatives.f(x, y, z) = y/x + y + z ; fy(2, 1, –1)
Find equations of (a) The tangent plane (b) The normal line to the given surface at the specified point.x2 – 2y2 + z2 + yz = 2, (2, 1, –1)
Car A is traveling north on Highway 16 and car B is traveling west on Highway 83. Each car is approaching the intersection of these highways. At a certain moment, car A is 0.3 km from the intersection and traveling at 90 km/h while car B is 0.4 km from the intersection and traveling at 80 km/h. How
Find the indicated partial derivatives.f(x, y, z) = √sin2x + sin2y + sin2z; fz(0, 0, π/4)
If yz4 + x2z3 = exyz, find ∂z/∂x and ∂z/∂y.
Find equations of (a) The tangent plane (b) The normal line to the given surface at the specified point.x – z = 4 arctan(yz), (1 + π, 1, 1)
Find the gradient of the functionf(x, y, z) = z2ex√y.
Find equations of (a) The tangent plane (b) The normal line to the given surface at the specified point.z + 1 = xey cos z, (1, 0, 0)
Find equations of (a) The tangent plane (b) The normal line to the given surface at the specified point.yz = ln(x + z), (0, 0, 1)
Use implicit differentiation to find ∂z/∂x and ∂z/∂y.x2 + y2 + z2 = 3xyz
Find the directional derivative of f at the given point in the indicated direction.f(x, y) = 2√x – y2, (1, 5), in the direction toward the point (4, 1)
Use implicit differentiation to find ∂z/∂x and ∂z/∂y.yz = ln(x + z)
Use implicit differentiation to find ∂z/∂x and ∂z/∂y.x – z = arctan(yz)
Use implicit differentiation to find ∂z/∂x and ∂z/∂y.sin(xyz) = x + 2y + 3z
Suppose that a scientist has reason to believe that two quantities x and y are related linearly, that is, y = mx + b, at least approximately, for some values of m and b. The scientist performs an experiment and collects data in the form of points (x1, y2), (x2. y2) · · · · (xn, yn), and then
Find all the second partial derivatives.f(x, y) = x3y5 + 2x4y
Find all the second partial derivatives.f(x, y) = sin2(mx + ny)
Find all the second partial derivatives.w = √u2 + v2
Find all the second partial derivatives.v = xy/x – y
Find all the second partial derivatives.z = arctan x + y/1 – xy
Find all the second partial derivatives.v = exey
Use Lagrange multipliers to find the maximum and minimum values of f subject to the given constraint(s). f(x, y) = X y 1 1 y²
Verify that the conclusion of Clairaut's Theorem holds, that is, uxy = uyx.u = ln√x2 + y2
Verify that the conclusion of Clairaut's Theorem holds, that is, uxy = uyx.u = xyey
Find the indicated partial derivative.f(x, y) = 3xy4 + x3y2; fxxy: fyyy
Find the indicated partial derivative.f(x, t) = x2e–et ; fttt, ftxx
Find the indicated partial derivative.f(x, y, z) = cos(4x + 3y + 2z); fxyz, fyzz
Find the indicated partial derivative.f(r, s, t) = r ln(rs2t3); frss: frst
Ifshow that U= е ea₁x₁ + a₂x₂++ax, where a1 + a2 + An Xn . + a = 1,
Find the indicated partial derivative.u = erθ sin θ; ∂3u/∂r2 ∂θ
Find the indicated partial derivative.z = u√v - w; ∂3z/∂u∂v∂w
Verify that the function z = In(ex + ey) is a solution of the differential equations and дz ax дz + = 1 ду
A particle of mass m moves on the surface z = f(x, y). Let x = x(t) and y = y(t) be the x- and y-coordinates of the particle at time t.(a) Find the velocity vector v and the kinetic energy K = 1/2m|v|2 of the particle. (b) Determine the acceleration vector a. (c) Let z = x2 + y2 and x(t) = t cos
Show that the Cobb-Douglas production function P = bLαKβ satisfies the equation L ар ƏL + K ар ak (a + B)P
Evaluate the iterated integral. ff xy² dx dy 10
If f(x, y) = 3√x3 + y3, find fx(0, 0).
Plot the point whose cylindrical coordinates are given. Then find the rectangular coordinates of the point.(a) (2, π/4, 1) (b) (4, –π/3, 5)
Find the Jacobian of the transformation.x = 5u – v, y = u + 3v
Plot the point whose spherical coordinates are given. Then find the rectangular coordinates of the point.(a) (1, 0, 0) (b) (2, π/3, π/4)
Evaluate the integral ∫∫∫E (xz – y3) dV, where using three different orders of integration. E = {(x, y, z) | − 1 ≤ x ≤ 1,0 ≤ y ≤ 2,0 ≤ z
Find ∫05f(x, y) dx and ∫01f(x, y) dy.f(x, y) = 12x2y3

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