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mathematics
calculus 10th edition
Calculus 10th Edition Ron Larson, Bruce H. Edwards - Solutions
In Exercises convert the point from cylindrical coordinates to spherical coordinates. (-4₁) این 4
In Exercises determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.If u and v are orthogonal to w, then u + v is orthogonal to w.
In Exercises find a unit vector(a) parallel to (b) perpendicular to the graph of ƒ at the given point. Then sketch the graph of ƒ and sketch the vectors at the given point. f(x) = x² + 5, (1,4) -
In Exercises convert the point from cylindrical coordinates to spherical coordinates. (12, TT, 5)
In Exercises sketch a graph of the plane and label any intercepts.x + z = 6
In Exercises(a) Find all points of intersection of the graphs of the two equations(b) Find the unit tangent vectors to each curve at their points of intersection(c) Find the angles (0° ≤ θ ≤ 90°) between the curves at their points of intersection. y = x3, y = x1/3
In Exercises use vectors to show that the points form the vertices of a parallelogram. (2, 9, 1), (3, 11, 4), (0, 10, 2), (1, 12, 5)
In Exercises sketch a graph of the plane and label any intercepts.2x + y = 8
Prove that ||u − v||² = ||u||² + ||v||² - 2u • v.
In Exercises sketch a graph of the plane and label any intercepts.x = 5
In Exercises use vectors to find the point that lies two-thirds of the way from P to Q. P(1, 2, 5), Q(6, 8, 2)
In Exercises(a) Find all points of intersection of the graphs of the two equations(b) Find the unit tangent vectors to each curve at their points of intersection(c) Find the angles (0° ≤ θ ≤ 90°) between the curves at their points of intersection. y = 1x², y = x² - 1
In Exercises find inequalities that describe the solid, and state the coordinate system used. Position the solid on the coordinate system such that the inequalities are as simple as possible. The solid between the spheres x² + y² + z² = 4 and x² + y² + z² = 9, and inside the cone z² = x² +
In Exercises find inequalities that describe the solid, and state the coordinate system used. Position the solid on the coordinate system such that the inequalities are as simple as possible. The solid inside both x² + y² + z² = 9 and (x - 2)² + y² = = 014
Identify the curve of intersection of the surfaces (in spherical coordinates) p = 2 sec Ø and p = 4.
Identify the curve of intersection of the surfaces (in cylindrical coordinates) z = sin θ and r = 1.
In Exercises determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.The spherical coordinates of a point (x, y, z) are unique.
Suppose the length of each cable has a fixed length L = a, and the radius of each disc is r0 inches. Make a conjecture about the limit and give a reason for your answer. lim T Toa
In Exercises sketch the solid that has the given description in spherical coordinates. 0 ≤ 0 ≤ π, 0 ≤ φ = π/2, 1 < p = 3
In Exercises sketch the solid that has the given description in spherical coordinates. 0 ≤ 0 ≤ π/2, 0 ≤ φ = π/2, 0 < p = 2
In Exercises determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.The cylindrical coordinates of a point (x, y, z) are unique.
Consider the vector u = (x, y). Describe the set of all points (x, y) such that |u|| = 5.
In Exercises find inequalities that describe the solid, and state the coordinate system used. Position the solid on the coordinate system such that the inequalities are as simple as possible.The solid that remains after a hole 1 inch in diameter is drilled through the center of a sphere 6 inches in
In Exercises verify that the lines are parallel, and find the distance between them. L₁: x = 3 + 6t, y = -2 +9t, z = 1 - 12t L₂: x= -1 + 4t, y = 3 + 6t, z = - 8t
In Exercises sketch the solid that has the given description in spherical coordinates. 0 ≤ 0 ≤ 2π, π/4 = φ = π/2, 0 < p = 1
In Exercises find inequalities that describe the solid, and state the coordinate system used. Position the solid on the coordinate system such that the inequalities are as simple as possible.A spherical shell with inside and outside radii of 4 inches and 6 inches, respectively
In Exercises find inequalities that describe the solid, and state the coordinate system used. Position the solid on the coordinate system such that the inequalities are as simple as possible.A cylindrical shell 8 meters long with an inside diameter of 0.75 meter and an outside diameter of 1.25
In Exercises sketch the solid that has the given description in cylindrical coordinates. 0 ≤ 0 ≤ 27, 2 ≤ r ≤ 4, z² ≤ - r² + 6r - 8
In Exercises sketch the solid that has the given description in spherical coordinates. 0 ≤ 0 ≤ 2π, 0 ≤ φ = 7/6, 0 < p = a sec φ α
In Exercises find inequalities that describe the solid, and state the coordinate system used. Position the solid on the coordinate system such that the inequalities are as simple as possible.A cube with each edge 10 centimeters long
In Exercises verify that the lines are parallel, and find the distance between them. L₁: x=2 t, y = 3 + 2t, z = 4 + t L₂: x = 3t, y = 16t, z = 4 - 3t
Prove that the vectorbisects the angle between u and v. w = || u ||v + ||v||u V
In Exercises sketch the solid that has the given description in cylindrical coordinates. 0 ≤ 0 ≤ 2T, 0 ≤ r ≤ a, r ≤ z ≤ a
The initial and terminal points of the vector v are (x₁, y₁, z₁) and (x, y, z). Describe the set of all points (x, y, z) such that ||v|| = 4.
Give the parametric equations and the symmetric equations of a line in space. Describe what is required to find these equations.
In Exercises sketch the solid that has the given description in cylindrical coordinates. - π/2 ≤ 0 ≤ π/2,0 ≤ r ≤ 3,0 ≤ z ≤rcos (
In Exercises find the distance between the point and the line given by the set of parametric equations. (-2, 1, 3); x = 1-t, y = 2+t, z = −2t
In Exercises find the distance between the point and the line given by the set of parametric equations. (4, -1,5); x = 3, y = 1 + 3t, z = 1+ t
Give the formula for the distance between the points (x₁, y₁, Z₁) and (x₂, y2, z₂).
A point in the three- dimensional coordinate system has coordinates (x0, Y0, Z0).Describe what each coordinate measures.
Prove thatare unit vectors for any angle θ. u = (cos ) i (sin )j and V= (sin )i + (cos 0) j
In Exercises sketch the solid that has the given description in cylindrical coordinates. 0 ≤ 0 ≤ T/2,0 ≤ r ≤ 2,0 ≤z ≤ 4
Using vectors, prove that the diagonals of a parallelogram bisect each other.
Using vectors, prove that the line segment joining the midpoints of two sides of a triangle is parallel to, and one-half the length of, the third side.
In Exercises find the distance between the point and the line given by the set of parametric equations. (1, -2, 4); x = 2t, y = t - 3, z = 2t + 2
In Exercises determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If a b, then || ai + bj|| = √√√2a.
Let u = i + j, v = j + k, and w = au + bv.(a) Sketch u and v.(b) If w = 0, show that a and b must both be zero.(c) Find a and b such that w = i + 2j + k.(d) Show that no choice of a and b yields w = i + 2j + 3k.
In Exercises verify that the two planes are parallel, and find the distance between the planes. 2x - 4z = 4 2x - 4z = 10
In Exercises find the distance between the point and the line given by the set of parametric equations. (1,5,-2); x = 4t2, y = 3, z = -t + 1
In Exercises use vectors to find the point that lies two-thirds of the way from P to Q. P(4, 3, 0), Q(1, -3, 3)
In Exercises determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.If u and v have the same magnitude but opposite directions, then u + v = 0.
In Exercises convert the rectangular equation to an equation in(a) Cylindrical coordinates (b) Spherical coordinatesy = 4
In Exercises verify that the two planes are parallel, and find the distance between the planes. -3x + 6y + 72 = 1 = 6x - 12y - 142 = 25
In Exercises verify that the two planes are parallel, and find the distance between the planes. 4x - 4y + 9z = 7 4x4y + 9z = 18
In Exercises convert the rectangular equation to an equation in(a) Cylindrical coordinates (b) Spherical coordinatesx2 - y2 = 9
In Exercises determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.If u is a unit vector in the direction of v, then v = ||v||u.
In Exercises sketch the vector v and write its component form.v lies in the xz-plane, has magnitude 5, and makes an angle of 45° with the positive z-axis.
In Exercises determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.If v = ai + bj = 0, then a = -b.
In Exercises find the vector with the given magnitude and the same direction as u. Magnitude ||v|| = 7 Direction u = (-4, 6, 2)
In Exercises convert the rectangular equation to an equation in(a) Cylindrical coordinates (b) Spherical coordinatesx² + y² = 36
In Exercises find the vector with the given magnitude and the same direction as u. Magnitude ||v|| = 2/1 Direction u = (2, -2, 1)
In Exercises sketch the vector v and write its component form.v lies in the yz-plane, has magnitude 2, and makes an angle of 30° with the positive y-axis.
In Exercises determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.If u = ai + bj is a unit vector, then a² + b² = 1.
In Exercises verify that the two planes are parallel, and find the distance between the planes. x - 3y + 4z = 10 x - 3y + 4z = 6
In Exercises find the distance between the point and the plane. (1,3,-1) 3x - 4y + 5z = 6
In Exercises convert the rectangular equation to an equation in(a) Cylindrical coordinates (b) Spherical coordinatesx² + y² = 4y
In Exercises find the vector with the given magnitude and the same direction as u. Magnitude ||v|| = 3 Direction u = (1, 1, 1)
In Exercises find the distance between the point and the plane. (2,8, 4) 2x + y + z = 5
In Exercises convert the rectangular equation to an equation in(a) Cylindrical coordinates (b) Spherical coordinatesx2 + y2 = z
A plane flies at a constant groundspeed of 400 miles per hour due east and encounters a 50-mile-per-hour wind from the northwest. Find the airspeed and compass direction that will allow the plane to maintain its groundspeed and eastward direction.
In Exercises determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.If u and y have the same magnitude and direction, then u and v are equivalent.
In Exercises find the distance between the point and the plane. (0, 0, 0) 5x +y - z = 9
In Exercises convert the rectangular equation to an equation in(a) Cylindrical coordinates (b) Spherical coordinatesx² + y² + z² = 2z = 0
In Exercises find the vector with the given magnitude and the same direction as u. Magnitude ||v|| = 10 Direction u = (0, 3, 3)
A plane is flying with a bearing of 302°. Its speed with respect to the air is 900 kilometers per hour. The wind at the plane's altitude is from the southwest at 100 kilometers per hour (see figure). What is the true direction of the plane, and what is its speed with respect to the ground? 900
Identify the surface graphed and match the graph with its rectangular equation. Then find an equation in cylindrical coordinates for the equation given in rectangular coordinates.(a)(b)(i)(ii) 3 Z 2
In Exercises find the distance between the point and the plane. (0, 0, 0) 2x + 3y + z = 12
In Exercises convert the rectangular equation to an equation in(a) Cylindrical coordinates (b) Spherical coordinates4(x² + y²) = z²
Determine (x, y, z) for each figure. Then find the component form of the vector from the point on the x-axis to the point (x, y, z).(a)(b) X Z (x, y, z) (3, 0, 0) (0, 3, 3) (0,3,0)
To carry a 100-pound cylindrical weight, two workers lift on the ends of short ropes tied to an eyelet on the top center of the cylinder. One rope makes a 20° angle away from the vertical and the other makes a 30° angle (see figure).(a) Find each rope's tension when the resultant force is
In Exercises convert the rectangular equation to an equation in(a) Cylindrical coordinates (b) Spherical coordinatesx² + y² + z² = 25
In Exercises find the point(s) of intersection (if any) of the plane and the line. Also, determine whether the line lies in the plane. 5x + 3y = 17, x-4 2 y + 1 -3 Z+2 5
In Exercises find a unit vector(a) In the direction of v (b) In the direction opposite of v v = 5i + 3j - k
Determine the tension in each cable supporting the given load for each figure.(a)(b) A 50⁰ 3000 lb 30° B
Consider the two nonzero vectors u and v, and let s and t be real numbers. Describe the geometric figure generated by the terminal points of the three vectors tv, u + tv, and su + tv.
A gun with a muzzle velocity of 1200 feet per second is fired at an angle of 6° above the horizontal. Find the vertical and horizontal components of the velocity.
In Exercises find the point(s) of intersection (if any) of the plane and the line. Also, determine whether the line lies in the plane. 2x + 3y = -5, x-1 4 y 2 z-3 6
In Exercises find the point(s) of intersection (if any) of the plane and the line. Also, determine whether the line lies in the plane. 2x + 3y = 10, x - 1 3 لیا y + 1 -2 =Z-3
In Exercises find a unit vector(a) In the direction of v (b) In the direction opposite of v v = 4i - 5j + 3k
Give the equations for the coordinate conversion from rectangular to spherical coordinates and vice versa.
In Exercises find the point(s) of intersection (if any) of the plane and the line. Also, determine whether the line lies in the plane. 2x - 2y + z = 12, x I 2 y + (3/2) -1 z + 1 2
In Exercises find a unit vector(a) In the direction of v (b) In the direction opposite of v v = (6, 0, 8)
Explain why in spherical coordinates the graph of θ = c is a half-plane and not anentire plane.
In Exercises convert the point from spherical coordinates to cylindrical coordinates. 7, п 3п 4' 4
Prove Theorem 11.6.Data from in Theorem 11.6 THEOREM 11.6 Projection Using the Dot Product If u and v are nonzero vectors, then the projection of u onto v is proj, u = u. V ||v||² V.
Consider two forces of equal magnitude acting on a point.(a) When the magnitude of the resultant is the sum of the magnitudes of the two forces, make a conjecture about the angle between the forces.(b) When the resultant of the forces is 0, make a conjecture about the angle between the forces.(c)
Give the equations for the coordinate conversion from rectangular to cylindrical coordinates and vice versa.
In Exercises find a unit vector(a) In the direction of v (b) In the direction opposite of v V v (2, 1, 2) = -
In Exercises convert the point from spherical coordinates to cylindrical coordinates. 8. 7 п 9.9
In Exercises convert the point from spherical coordinates to cylindrical coordinates. 5, 5 п 6 П
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