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study help
mathematics
precalculus
Calculus Early Transcendentals 8th edition James Stewart - Solutions
Find the distance between the points.(1, 1), (4, 5)
(a) Is the sum of two irrational numbers always an irrational number?(b) Is the product of two irrational numbers always an irrational number?
Show that the sum, difference, and product of rational numbers are rational numbers.
Prove that |x - y | > |x| - |y|.the Triangle Inequality with a = x - y and b = y.
Show that if 0 < a < b, then a2 < b2.
Prove that |a/b| = |a|/|b|.
Prove that |ab| = |a||b|.
Use Rule 3 to prove Rule 5 of (2).
Show that if a < b, then a < a + b/2 < b.
Show that if |x + 3| < 1/2, then |4x + 13 | < 3.
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.If y1 and y2 are solutions of y0 + y = 0, then y1 + y2 is also a solution of the equation.
As in Exercise 9, consider a spring with mass m, spring con-stant k, and damping constant c = 0, and let w = √k/m.If an external force F(t) = F0 cos wt is applied (the applied frequency equals the natural frequency), use the method of undetermined coefficients to show that the motion of the mass
Verify that the Divergence Theorem is true for the vector field F(x, y, z) = x i + y j + z k, where E is the unit ball x2 + y2 + z2 < 1.
Find an equation of the tangent plane to the given parametric surface at the specified point.r(u, v) = sin u i + cos u sin v j + sin v k; u = π/6, v = π/6
The surface with parametric equationsx = 2 cos θ + r cos(θ/2)y = 2 sin θ + r cos(θ/2)z = r sin(θ/2)where -1/2 < r < 1/2 and 0 < θ < 2π, is called a Möbius strip. Graph this surface with several viewpoints. What is unusual about it?
Find parametric equations for the surface obtained by rotating the curve x = 1/y, y > 1, about the y-axis and use them to graph the surface.
Find parametric equations for the surface obtained by rotating the curve y = 1/(1 + x2), -2 < x < 2, about the x-axis and use them to graph the surface.
Use a graphing device to produce a graph that looks like the given one. 0- -1
Use a graphing device to produce a graph that looks like the given one. 3- -3 -3 y
Find a parametric representation for the surface.The part of the plane z = x + 3 that lies inside the cylinder x2 + y2 = 1
Find a parametric representation for the surface.The part of the sphere x2 + y2 + z2 = 36 that lies between the planes z = 0 and z = 3√3
Find a parametric representation for the surface.The part of the cylinder x2 + z2 = 9 that lies above the xy-plane and between the planes y = 24 and y = 4
Find a parametric representation for the surface.The part of the sphere x2 + y2 + z2 = 4 that lies above the cone z = √x2 + y2
Find a parametric representation for the surface.The part of the ellipsoid x2 + 2y2 + 3z2 = 1 that lies to the left of the xz-plane
Find a parametric representation for the surface.The part of the hyperboloid 4x2 - 4y2 - z2 = 4 that lies in front of the yz-plane
Find a parametric representation for the surface.The plane that passes through the point (0, -1, 5) and contains the vectors (2, 1, 4) and (-3, 2, 5)
Find a parametric representation for the surface.The plane through the origin that contains the vectors i - j and j - k
Match the equations with the graphs labeled I–VI and give reasons for your answers. Determine which families of grid curves have u constant and which have v constant.x = sin u, y = cos u sin v, z = sin v П ZA х Ш IV ZA х х ZA VI х х
Match the equations with the graphs labeled I–VI and give reasons for your answers. Determine which families of grid curves have u constant and which have v constant.x = cos3u cos3v, y = sin3u cos3v, z = sin3v П ZA х Ш IV ZA х х ZA VI х х
Match the equations with the graphs labeled I–VI and give reasons for your answers. Determine which families of grid curves have u constant and which have v constant.x = (1 - u)(3 + cos v) cos 4πu,y = (1 - u)(3 + cos v) sin 4πu,z = 3u + (1 - u) sin v П ZA х Ш IV ZA х х ZA VI х х
Match the equations with the graphs labeled I–VI and give reasons for your answers. Determine which families of grid curves have u constant and which have v constant.r(u, v) = (u3 - u) i + v - j + u2 k П ZA х Ш IV ZA х х ZA VI х х
Match the equations with the graphs labeled I–VI and give reasons for your answers. Determine which families of grid curves have u constant and which have v constant.r(u, v) = uv - i + u2v j + (u2 - v2) k П ZA х Ш IV ZA х х ZA VI х х
Match the equations with the graphs labeled I–VI and give reasons for your answers. Determine which families of grid curves have u constant and which have v constant.r(u, v) = u cos v i + u sin v j + v k П ZA х Ш IV ZA х х ZA VI х х
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.x = cos u, y = sin u sin v, z = cos v, 0 < u < 2π, 0 < v < 2π
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.x = sin v, y = cos u sin 4v, z = sin 2u sin 4v, 0 < u < 2π, -π/2 < v < π/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) = (u, sin(u + v), sin v), -π < u < π, -π < v < π
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) = (u3, u sin v, u cos v), -1 < u < 1, 0 < v < 2π
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, v3, -v), -2 < u < 2, -2 < v < 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, v2, u + v), -1 < u < 1, -1 < v < 1
Identify the surface with the given vector equation.r(s, t) = (3 cos t, s, sin t), -1 < s < 1
Identify the surface with the given vector equation.r(s, t) = (s cos t, s sin t, s)
Identify the surface with the given vector equation.r(u, v) = u2 i + u cos v j + u sin v k
Identify the surface with the given vector equation.r(u, v) = (u + v) i + (3 - v) j + (1 + 4u + 5v) k
Determine whether the points P and Q lie on the given surface.r(u, v) = (1 + u - v, u + v2, u2 - v2) P(1, 2, 1), Q(2, 3, 3)
Determine whether the points P and Q lie on the given surface.r(u, v) = (u + v, u - 2v, 3 + u - v) P(4, -5, 1), Q(0, 4, 6)
We have seen that all vector fields of the form F = ∇g satisfy the equation curl F = 0 and that all vector fields of the form F = curl G satisfy the equation div F = 0 (assuming continuity of the appropriate partial derivatives). This suggests the question: are there any equations that all
This exercise demonstrates a connection between the curl vector and rotations. Let B be a rigid body rotating about the z-axis. The rotation can be described by the vector w = k, where is the angular speed of B, that is, the tangential speed of any point P in B divided by the distance d from
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. 9. x² sin(x – y) dx dy = x* sin(x – y) dy dx
Use Green’s first identity to show that if f is harmonic on D, and if f (x, y) = 0 on the boundary curve C, then(Assume the same hypotheses as inExercise 33.) |Vf l° dA = 0. D
Recall from Section 14.3 that a function g is called harmonic on D if it satisfies Laplace’s equation, that is, ∇2g = 0 on D. Use Green’s first identity (with the same hypotheses as in Exercise 33) to show that if g is harmonic on D, then ∮CDngds = 0. Here Dng is the normal derivative of t
Use Green’s first identity (Exercise 33) to prove Green’s second identity:where D and C satisfy the hypotheses of Green’s Theorem and the appropriate partial derivatives of f and g exist and are continuous. S (SV'g – gV°f) dA – f.(SVg – 9Vf) • n ds
Use Green’s Theorem in the form of Equation 13 to prove Green’s first identity:where D and C satisfy the hypotheses of Green’s Theorem and the appropriate partial derivatives of f and g exist and are continuous. (The quantity ∇g • n = Dn g occurs in the line integral. This is the
Let r = x i + y j + z k and r = |r|.If F = r/rp, find div F. Is there a value of p for which div F = 0?
Let r = x i + y j + z k and r = |r|.(a) ∇r = r/r(b) ∇ x r = 0(c) ∇ x (1/r) = - r/r3(d) ∇ In r = r/r2
Let r = x i + y j + z k and r = |r|.Verify each identity.(a) ∇ • r = 3 (b) ∇ • (r r) = 4r(c) ∇2r3 = 12r
Prove the identity, assuming that the appropriate partial derivatives exist and are continuous. If f is a scalar field and F, G are vector fields, then f F, F • G, and F x G are defined by(f F)(x, y, z) = f (x, y, z) F(x, y, z)(F • G)(x, y, z) = F(x, y, z) • G(x, y, z)(F x G)(x, y, z) = F(x,
Prove the identity, assuming that the appropriate partial derivatives exist and are continuous. If f is a scalar field and F, G are vector fields, then f F, F • G, and F x G are defined by(f F)(x, y, z) = f (x, y, z) F(x, y, z)(F • G)(x, y, z) = F(x, y, z) • G(x, y, z)(F x G)(x, y, z) = F(x,
Prove the identity, assuming that the appropriate partial derivatives exist and are continuous. If f is a scalar field and F, G are vector fields, then f F, F • G, and F x G are defined by(f F)(x, y, z) = f (x, y, z) F(x, y, z)(F • G)(x, y, z) = F(x, y, z) • G(x, y, z)(F x G)(x, y, z) = F(x,
Prove the identity, assuming that the appropriate partial derivatives exist and are continuous. If f is a scalar field and F, G are vector fields, then f F, F • G, and F x G are defined by(f F)(x, y, z) = f (x, y, z) F(x, y, z)(F • G)(x, y, z) = F(x, y, z) • G(x, y, z)(F x G)(x, y, z) = F(x,
Prove the identity, assuming that the appropriate partial derivatives exist and are continuous. If f is a scalar field and F, G are vector fields, then f F, F • G, and F x G are defined by(f F)(x, y, z) = f (x, y, z) F(x, y, z)(F • G)(x, y, z) = F(x, y, z) • G(x, y, z)(F x G)(x, y, z) = F(x,
Prove the identity, assuming that the appropriate partial derivatives exist and are continuous. If f is a scalar field and F, G are vector fields, then f F, F • G, and F x G are defined by(f F)(x, y, z) = f (x, y, z) F(x, y, z)(F • G)(x, y, z) = F(x, y, z) • G(x, y, z)(F x G)(x, y, z) = F(x,
Prove the identity, assuming that the appropriate partial derivatives exist and are continuous. If f is a scalar field and F, G are vector fields, then f F, F • G, and F x G are defined by(f F)(x, y, z) = f (x, y, z) F(x, y, z)(F • G)(x, y, z) = F(x, y, z) • G(x, y, z)(F x G)(x, y, z) = F(x,
If F is the vector field of Example 5, show that ʃC F • dr = 0 for every simple closed path that does not pass through or enclose the origin.
Use the method of Example 5 to calculate ʃC F • dr, whereand C is any positively oriented simple closed curve that encloses the origin. 2хyi + (у? — х?) j (x² + y?)? F(x, y) = - y³)?
A plane lamina with constant density p(x, y) = p occupies a region in the xy-plane bounded by a simple closed path C. Show that its moments of inertia about the axes are x' dy уз dx 3 Jc .3 х I, I, 3 Jc
Use Green’s Theorem to evaluate ʃC F • dr. (Check the orientation of the curve before applying the theorem.)F(x, y) = (√x2 + 1, tan-1 x), C is the triangle from (0, 0) to (1, 1) to (0, 1) to (0, 0)
Show that if the vector field F = P i + Q j + R k is conservative and P, Q, R have continuous first-order partial derivatives, then де дz ӘR ӘР до ӘР ƏR дz ду Әх дх ду
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.The integralrepresents the volume enclosed by the cone z = √x2 + y2 and the plane z = 2. 2т
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. CI (² + Vy) sin(x?y?)dx dy < 9 11
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.If f is continuous on [0, 1], then 12 SCT )S(9) dy dx = | () dx f(x) dx
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. ex?+y° sin y dx dy = 0 -1 Jo
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. re dy dx = fx* dx f, e' dy 13 13
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. Vx + y² dy dx = /0 J0 Vx + y? dx dy
Find the mass and center of mass of the lamina that occupies the region D and has the given density function p.D is the triangular region enclosed by the lines y = 0, y = 2x, and x + 2y = 1; (x, y) = x
Find the mass and center of mass of the lamina that occupies the region D and has the given density function p.D is the triangular region with vertices (0, 0), (2, 1), (0, 3); (x, y) = x + y
Show that the maximum value of the functionf (x, y) = (ax + by + c)2/x2 + y2 + 1is a2 + b2 + c2.One method for attacking this problem is to use the Cauchy-Schwarz Inequality:|a • b | < |a||b|
If the ellipse x2/a2 + y2/b2 = 1 is to enclose the circle x2 + y2 = 2y, what values of a and b minimize the area of the ellipse?
Suppose f is a differentiable function of one variable. Show that all tangent planes to the surface z = x f(y/x) intersect in a common point.
For what values of the number r is the function
A long piece of galvanized sheet metal with width w is to be bent into a symmetric form with three straight sides to make a rain gutter. A cross-section is shown in the figure. (a) Determine the dimensions that allow the maximum possible flow; that is, find the dimensions that give the maximum
Marine biologists have determined that when a shark detects the presence of blood in the water, it will swim in the direction in which the concentration of the blood increases most rapidly. Based on certain tests, the concentration of blood (in parts per million) at a point P(x, y) on the surface
A rectangle with length L and width W is cut into four smaller rectangles by two lines parallel to the sides. Find the maximum and minimum values of the sum of the squares of the areas of the smaller rectangles.
A pentagon is formed by placing an isosceles triangle on a rectangle, as shown in the figure. If the pentagon has fixed perimeter P, find the lengths of the sides of the pentagon that maximize the area of the pentagon.
A package in the shape of a rectangular box can be mailed by the US Postal Service if the sum of its length and girth (the perimeter of a cross-section perpendicular to the length) is at most 108 in. Find the dimensions of the package with largest volume that can be mailed.
Find the points on the surface xy2z3 = 2 that are closest to the origin.
Use Lagrange multipliers to find the maximum and minimum values of f subject to the given constraint(s).f (x, y, z) = x2 + 2y2 + 3z2;x + y + z = 1, x - y + 2z = 2
Use Lagrange multipliers to find the maximum and minimum values of f subject to the given constraint(s).f(x, y, z) = xyz; x2 + y2 + z2 = 3
Use Lagrange multipliers to find the maximum and minimum values of f subject to the given constraint(s).f (x, y) = 1/x + 1/y; 1/x2 + 1/y2 = 1
Use Lagrange multipliers to find the maximum and minimum values of f subject to the given constraint(s).f (x, y) = x2y; x2 + y2 = 1
Use a graphing calculator or computer (or Newton’s method or a computer algebra system) to find the critical points of f (x, y) = 12 + 10y - 2x2 - 8xy - y4 correct to three decimal places. Then classify the critical points and find the highest point on the graph.
Use a graph or level curves or both to estimate the local maximum and minimum values and saddle points of f (x, y) = x3 - 3x + y4 - 2y2. Then use calculus to find these values precisely.
Find the absolute maximum and minimum values of f on the set D.f (x, y) = e-x 2-y2(x2 + 2y2); D is the disk x2 + y2 < 4
Find the absolute maximum and minimum values of f on the set D.f (x, y) = 4xy2 - x2y2 - xy3; D is the closed triangular region in the xy-plane with vertices (0, 0), (0, 6), and (6, 0)
Find the local maximum and minimum values and saddle points 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 + y)ey/2
Find the local maximum and minimum values and saddle points 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) = 3xy - x2y - xy2
Find the local maximum and minimum values and saddle points 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 - 6xy + 8y3
Find the local maximum and minimum values and saddle points 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 - xy + y2 + 9x - 6y + 10
Find parametric equations of the tangent line at the point (-2, 2, 4) to the curve of intersection of the surface z = 2x2 - y2 and the plane z = 4.
The contour map shows wind speed in knots during Hurricane Andrew on August 24, 1992. Use it to estimate the value of the directional derivative of the wind speed at Homestead, Florida, in the direction of the eye of the hurricane. 70 60 70 80 Homestead 65 55 65 \ 75 60 55 50 45- -40 35 30 *Key
Find the direction in which f (x, y, z) = zexy increases most rapidly at the point (0, 1, 2). What is the maximum rate of increase?
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