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mathematics
calculus 4th
Calculus 4th Edition Jon Rogawski, Colin Adams, Robert Franzosa - Solutions
Suppose that vectors u, v, and w are mutually orthogonal—that is, u ⊥ v, u ⊥ w, and v ⊥ w. Prove that (u × v) × w = 0 and u × (v × w) = 0.
Show that three points P, Q, R are collinear (lie on a line) if and only if POX PR = 0.
State whether the given equation defines an ellipsoid or hyperboloid, and if a hyperboloid, whether it is of one or two sheets. () + ( ) ( ) = 1
State whether the given equation defines an ellipsoid or hyperboloid, and if a hyperboloid, whether it is of one or two sheets. 2 + + = 1
State whether the given equation defines an ellipsoid or hyperboloid, and if a hyperboloid, whether it is of one or two sheets. 2 2 + = 1
State whether the given equation defines an ellipsoid or hyperboloid, and if a hyperboloid, whether it is of one or two sheets.x2 + 3y2 + 9z2 = 1
State whether the given equation defines an ellipsoid or hyperboloid, and if a hyperboloid, whether it is of one or two sheets.x2 − 3y2 + 9z2 = 1
State whether the given equation defines an ellipsoid or hyperboloid, and if a hyperboloid, whether it is of one or two sheets.x2 − 3y2 − 9z2 = 1
State whether the given equation defines an ellipsoid or hyperboloid, and if a hyperboloid, whether it is of one or two sheets.x2 + y2 = 4 − 4z2
State whether the given equation defines an elliptic paraboloid, a hyperbolic paraboloid, or an elliptic cone. 22 = ( 2
State whether the given equation defines an elliptic paraboloid, a hyperbolic paraboloid, or an elliptic cone. z= X 4. 2 + 3.
State whether the given equation defines an elliptic paraboloid, a hyperbolic paraboloid, or an elliptic cone. 2 2 X y = (-) - (/) 12, Z:
State whether the given equation defines an ellipsoid or hyperboloid, and if a hyperboloid, whether it is of one or two sheets.x2 + 3y2 = 9 + z2
State whether the given equation defines an elliptic paraboloid, a hyperbolic paraboloid, or an elliptic cone.4z = 9x2 + 5y2
State whether the given equation defines an elliptic paraboloid, a hyperbolic paraboloid, or an elliptic cone.3x2 − 7y2 = z
State whether the given equation defines an elliptic paraboloid, a hyperbolic paraboloid, or an elliptic cone.3x2 + 7y2 = 14z2
State whether the given equation defines an elliptic paraboloid, a hyperbolic paraboloid, or an elliptic cone.y2 = 5x2 − 4z2
State the type of the quadric surface and describe the trace obtained by intersecting with the given plane. +2 + ()+ + 2 = 1,_y = 0 +2=
State whether the given equation defines an elliptic paraboloid, a hyperbolic paraboloid, or an elliptic cone.y = 3x2 − 4z2
State the type of the quadric surface and describe the trace obtained by intersecting with the given plane. x + +2= 1, y = 5
State the type of the quadric surface and describe the trace obtained by intersecting with the given plane. x + () + z = 1, z || 1 + 4
State the type of the quadric surface and describe the trace obtained by intersecting with the given plane. ( ) + ( ) 5z = 1, x = 0
State the type of the quadric surface and describe the trace obtained by intersecting with the given plane. 3/ - 5z = 1, y = 1 -
Match each of the ellipsoids in Figure 14 with the correct equation: (a) x + 4y + 4z = 16 (c) 4x + 4y + z = 16 N (A) (B) (b) 4x + y + 4z = 16 (C)
State the type of the quadric surface and describe the trace obtained by intersecting with the given plane. 4x + (3)-2 = -1. -1, z=1
State the type of the quadric surface and describe the trace obtained by intersecting with the given plane.y = 3x2, z = 27
What is the equation of the surface obtained when the elliptic paraboloid is rotated about the x-axis by 90°? Refer to Figure 15. Z = 2 () + ()
State the type of the quadric surface and describe the trace obtained by intersecting with the given plane.y = 3x2, y = 27
Describe the surface that is obtained when, in the equation ±8x2 ± 3y2 ± z2 = 1, we choose (a) All plus signs, (b) One minus sign, (c) Two minus signs.
Sketch the given surface. x + y z = 1
Describe the intersection of the horizontal plane z = h and the hyperboloid −x2 − 4y2 + 4z2 = 1. For which values of h is the intersection empty?
Sketch the given surface. 2 (7) + ( ) + ( ) = 1
Sketch the given surface. N = () + 4, z = 8,
Sketch the given surface. >100 = 2 N
Sketch the given surface. lo +) =
Sketch the given surface. -x - y +9z = 9
Sketch the given surface.y = −x2
Sketch the given surface.x2 + 36y2 = 1
Sketch the given surface.xy = 1
Sketch the given surface.x = 2y2 − z2
Sketch the given surface.x = 1 + y2 + z2
Sketch the given surface.x2 − 4y2 = z
Find the equation of the ellipsoid passing through the points marked in Figure 16(A). -4 N Z 6 -6 (A) 4 2 2 (B)
Sketch the given surface.x2 + 9y2 + 4z2 = 36
Sketch the given surface.y2 − 4x2 − z2 = 4
Find the equation of the ellipsoid passing through the points marked in Figure 16(A). X -4 6 2 Y (A)
Find the equation of the hyperboloid shown in Figure 17(A). x 4 8 (A) 12 6 9
Find the equation of the quadric surface shown in Figure 17(B). N 8 6 (B) 5
Determine the vertical traces of elliptic and parabolic cylinders in standard form.
What is the equation of a hyperboloid of one or two sheets in standard form if every horizontal trace is a circle?
The eccentricity of a conic section is defined in Section 11.5. Show that the horizontal traces of the ellipsoidare ellipses of the same eccentricity (apart from the traces at height h = ±c, which reduce to a single point). Find the eccentricity. () - + b. 2 + () = 1
Let C be an ellipse in a horizonal plane lying above the xy-plane. Which type of quadric surface is made up of all lines passing through the origin and a point on C?
Let C be a curve in R3 not passing through the origin. The cone on C is the surface consisting of all lines passing through the origin and a point on C [Figure 19(A)]. Cone on ellipse C N Cone on parabola C (half of cone shown)
Let S be the hyperboloid x2 + y2 = z2 + 1 and let P = (α, β, 0) be a point on S in the (x, y)-plane. Show that there are precisely two lines through P entirely contained in S (Figure 18). Consider the line r(t) = (α + at, β + bt, t) through P. Show that r(t) is contained in S if (a, b) is one
Let C be a curve in R3 not passing through the origin. The cone on C is the surface consisting of all lines passing through the origin and a point on C [Figure 19(A)].Let a and c be nonzero constants and let C be the parabola at height c consisting of all points (x, ax2, c) [Figure 19(B)]. Let S be
Write the equation of the plane with normal vector n passing through the given point in the scalar form ax + by + cz = d.n = (1, 3, 2), (4, −1, 1)
Write the equation of the plane with normal vector n passing through the given point in the scalar form ax + by + cz = d.n = (−1, 2, 1), (3, 1, 9)
Write the equation of the plane with normal vector n passing through the given point in the scalar form ax + by + cz = d.n = (−1, 2, 1), (4, 1, 5)
Write the equation of the plane with normal vector n passing through the given point in the scalar form ax + by + cz = d.n = (2, −4, 1), (1/3, 2/3, 1)
Write the equation of the plane with normal vector n passing through the given point in the scalar form ax + by + cz = d.n = i, (3, 1, −9)
Write the equation of the plane with normal vector n passing through the given point in the scalar form ax + by + cz = d.n = j, (−5, 1/2, 1/2)
Write the equation of the plane with normal vector n passing through the given point in the scalar form ax + by + cz = d.n = k, (6, 7, 2)
Write the equation of the plane with normal vector n passing through the given point in the scalar form ax + by + cz = d.n = i − k, (4, 2, −8)
Write the equation of any plane through the origin.
Write the equations of any two distinct planes with normal vector n = (3, 2, 1) that do not pass through the origin.
Find a normal vector n and an equation for the planes in Figures 8(A)–(C). N (A) -3 (B) (C)
Which of the following statements are true of a plane that is parallel to the yz-plane?(a) n = (0, 0, 1) is a normal vector.(b) n = (1, 0, 0) is a normal vector.(c) The equation has the form ay + bz = d.(d) The equation has the form x = d.
Find a vector normal to the plane with the given equation.9x − 4y − 11z = 2
Find a vector normal to the plane with the given equation.x − z = 0
Find a vector normal to the plane with the given equation.3(x − 4) − 8(y − 1) + 11z = 0
Find a vector normal to the plane with the given equation.x = 1
Find the equation of the plane with the given description.Passes through O and is parallel to 4x − 9y + z = 3
Find the equation of the plane with the given description.Passes through (4, 1, 9) and is parallel to x + y + z = 3
Find the equation of the plane with the given description.Passes through (4, 1, 9) and is parallel to x = 3
Find an equation of the plane passing through the three points given. P= (2,-1,4), Q=(1,1,1), R=(3, 1,-2)
Find the equation of the plane with the given description.Passes through P = (3, 5, −9) and is parallel to the xz-plane
Find an equation of the plane passing through the three points given.P = (5, 1, 1), Q = (1, 1, 2), R = (2, 1, 1)
Find an equation of the plane passing through the three points given.P = (1, 0, 0), Q = (0, 1, 1), R = (2, 0, 1)
Find an equation of the plane passing through the three points given.P = (2, 0, 0), Q = (0, 4, 0), R = (0, 0, 2)
In each case, describe how to find a normal vector to the plane:(a) Three noncollinear points are given. The plane contains all three points.(b) Two lines are given that intersect in a point. The plane contains the lines.
In each case, determine whether or not the lines have a single point of intersection. If they do, give an equation of a plane containing them. (a) r(t) = (t, 2t - 1,t-3) and r(t) =(4,2t - 1,-1) (b) r(t) = (3t, 2t + 1, t-5) and r(t) = (4t, 4t -3,-1)
In each case, describe how to find a normal vector to the plane:(a) A line and a point that is not on the line are given. The plane contains the line and the point.(b) Two lines are given that are parallel and distinct. The plane contains the lines.
In each case, determine whether or not the lines have a single point of intersection. If they do, give an equation of a plane containing them. (a) r(t) = (5t, 2t - 1, 2t - 2) and r(t) = (t -5, -t +4, t - 7) (b) r(t) = (3t, -2t + 1,t-3) and r(t) = (2t - 1,-t, -t - 1)
In each case, determine whether or not the point lies on the line. If it does not, give an equation of a plane containing the point and the line. (a) (2,2,-1) and r(t) = (4t, 6t - 1,-1) (b) (3,-3,2) and r(t) = (4t + 3,4t - 3,t + 1)
In each case, determine whether or not the point lies on the line. If it does not, give an equation of a plane containing the point and the line. (a) (-7, 10,-3) and r(t) = (1-4t, 6t - 5, t-5) (b) (-1,5,9) and r(t) = (4t + 3,t + 6,5-4t)
In each case, determine whether or not the lines are distinct parallel lines. If they are, give an equation of a plane containing them. (a) r(t) = (t, 2t - 1, t - 3) and r(t) = (3t 3, 6t - 1,3t - 1) (b) ri(t) = (3t, 2t + 1, t-5) and r(t) = (-6t, 1 - 4t, 2t-3) -
In each case, determine whether or not the lines are distinct parallel lines. If they are, give an equation of a plane containing them. (a) r(t) = (2t + 1, -2t - 1, 3t-7) and r(t) = (7-6t, 6t - 7,2 - 9t) (b) r(t) = (-4t, 2t + 1, 8t + 5) and r(t) = (2t - 2, -t + 4,5-4t)
Draw the plane given by the equation.x + y + z = 4
Draw the plane given by the equation.3x + 2y − 6z = 12
Draw the plane given by the equation.12x − 6y + 4z = 6
Draw the plane given by the equation.x + 2y = 6
Let a, b, c be constants. Which two of the following equations define the plane passing through (a, 0, 0), (0, b, 0), (0, 0, c)? (a) ax + by + cz = 1 (c) bx + cy + az = 1 (b) bcx + acy+abz = abc X y Z + + - b (d) - a C = 1
Draw the plane given by the equation.x + y + z = 0
Find an equation of the plane P in Figure 9. X 3 N 5 2
Verify that the plane x − y + 5z = 10 and the line r(t) = (1, 0, 1) + t (−2, 1, 1) intersect at P = (−3, 2, 3).
Find the intersection of the line and the plane.x + y + z = 14, r(t) = (1, 1, 0) + t (0, 2, 4)
Find the intersection of the line and the plane.2x + y = 3, r(t) = (2, −1, −1) + t(1, 2, −4)
Find the intersection of the line and the plane.z = 12, r(t) = t (−6, 9, 36)
Find the intersection of the line and the plane.x − z = 6, r(t) = (1, 0, −1) + t (4, 9, 2)
Find the trace of the plane in the given coordinate plane.3x − 9y + 4z = 5, yz
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