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mathematics
calculus 4th

Calculus 4th Edition Jon Rogawski, Colin Adams, Robert Franzosa - Solutions

Find parametric equations for the lines with the given description.Passes through (1, −1, 0) and (0, −1, 2)
What are the coordinates a and b in the parallelogram in Figure 26(B)? (-1, b) (-3, 2) У (2, 3) (a, 1) (B) Х
Let where A, B,C are three distinct points in the plane. Match (a)–(d) with (i)–(iv). v = AB and w = AC,
Find a parametrization of the line through P = (4, 9, 8) perpendicular to the yz-plane.
Show that r1(t) and r2(t) define the same line, whereShow that r2(t) passes through (3, −1, 4) and that the direction vectors for r1(t) and r2(t) are parallel. r₁(t)=(3,-1, 4) + t (8, 12,-6) r₂(t) = (11, 11, -2) + (4,6,-3)
Show that r1(t) and r2(t) define the same line, where r₁(t) = t (2, 1, 3), r₂(t) = (-6, -3, -9) + (8, 4, 12)
Find the components and length of the following vectors:(a) 4i + 3j (b) 2i − 3j (c) i +j (d) i − 3j
Find two different vector parametrizations of the line through P = (5, 5, 2) with direction vector v = (0, −2, 1).
Calculate the linear combination. -3i+5(¹j-i)
Calculate the linear combination.3j + (9i + 4j)
Find the point of intersection of the lines r(t) = (1, 0, 0) + t (-3, 1, 0) and s(t) = (0, 1, 1) + t (2,0, 1).
Calculate the linear combination.(3i + j) − 6j + 2(j − 4i)
Show that the lines r1(t) = (−1, 2, 2) + t(4, −2, 1) and r2(t) = (0, 1, 1) + t(2, 0, 1) do not intersect.
Determine whether the lines r1(t) = (2, 1, 1) + t(−4, 0, 1) and r2(s) = (−4, 1, 5) + s(2, 1, −2) intersect, and if so, find the point of intersection.
For each of the position vectors u with endpoints A, B, and C in Figure 27, indicate with a diagram the multiples rv and sw such that u = rv + sw. A sample is shown for u = OQ. u=
Calculate the linear combination.3(3i − 4j) + 5(i + 4j)
Determine whether the lines r1(t) =(0, 1, 1) + t(1, 1, 2) and r2(s) = (2, 0, 3) + s(1, 4, 4) intersect, and if so, find the point of intersection.
Sketch the parallelogram spanned by v = (1, 4) and w = (5, 2). Add the vector u = (2, 3) to the sketch and express u as a linear combination of v and w.
Express u as a linear combination u = rv + sw. Then sketch u, v,w, and the parallelogram formed by rv and sw. u = (3,-1); v = (2, 1), w = (1,3)
Find the intersection of the lines r1(t) =(−1, 1) + t(2, 4) and r2(s) = (2, 1) + s(−1, 6) in the plane.
A meteor follows a trajectory r(t) = (2, 1, 4) + t (3, 2, −1) km with t in minutes, near the surface of the earth, which is represented by the xy-plane. Determine at what time the meteor hits the ground.
Express u as a linear combination u = rv + sw. Then sketch u, v,w, and the parallelogram formed by rv and sw.u = (6, −2); v = (1, 1), w = (1, −1)
A laser’s beam shines along the ray given by r1(t) = (1, 2, 4) + t(2, 1, −1) for t ≥ 0. A second laser’s beam shines along the ray given by r2(s) = (6, 3, −1) + s(−5, 2, c) for s ≥ 0, where the value of c allows for the adjustment of the z-coordinate of its direction vector. Find
Calculate the magnitude of the force on cables 1 and 2 in Figure 28. Cable 1 65° 50 kg 25° Cable 2
The line with vector parametrization r(t) = (3, 1, −4) + t(−2, −2, 3) intersects the sphere (x − 1)2 + (y + 3)2 + z2 = 8 in two points. Find them. Determine t such that the point (x(t), y(t), z(t)) satisfies the equation of the sphere, and then find the corresponding points on the line.
Determine the magnitude of the forces F1 and F2 in Figure 29, assuming that there is no net force on the object. F₂ 45° F₁ 30⁰ 20 kg
Show that the line with vector parametrization r(t) = (3, 5, 6) + t(1, −2, −1) does not intersect the sphere of radius 5 centered at the origin.
A plane flying due east at 200 km/h encounters a 40-km/h wind blowing in the northeast direction. The resultant velocity of the plane is the vector sum v = v1 + v2, where v1 is the velocity vector of the plane and v2 is the velocity vector of the wind (Figure 30). The angle between v1 and v2 is
Find the components of the vector v whose tail and head are the midpoints of segments in Figure 20. The midpoint of (a1, a2, a3) and (b1, b2, b3) is AC and BC
Refer to Figure 31, which shows a robotic arm consisting of two segments of lengths L1 and L2.Find the components of the vector in terms of θ1 and θ2. 01 L₁ 02 r L2 P ➤X
Find the components of the vector w whose tail is C and head is the midpoint of in Figure 20. AB
Refer to Figure 31, which shows a robotic arm consisting of two segments of lengths L1 and L2.Let L1 = 5 and L2 = 3. Find r for θ1 = π/3, θ2 = π/4. 01 L₁ 02 r L2 P ➤X
A box that weighs 1000 kg is hanging from a crane at the dock. The crane has a square 20 m by 20 m framework as in Figure 21, with four cables, each of the same length, supporting the box. The box hangs 10 m below the level of the framework. Find the magnitude of the force acting on each cable.
We consider the equations of a line in symmetric form, when a ≠ 0, b ≠ 0, c ≠ 0.Let L be the line through P0 = (x0, y0, z0) with direction vector v = (a, b, c). Show that L is defined by the symmetric equations (10). Use the vector parametrization to show that every point on L satisfies
Refer to Figure 31, which shows a robotic arm consisting of two segments of lengths L1 and L2.Let L1 = 5 and L2 = 3. Show that the set of points reachable by the robotic arm with θ1 = θ2 is an ellipse. 01 L₁ 02 r L2 P ➤X
We consider the equations of a line in symmetric form, when a ≠ 0, b ≠ 0, c ≠ 0.Find the symmetric equations of the line through P0 = (−2, 3, 3) with direction vector v = (2, 4, 3). 0x - x a y - Yo b 02-2 C
Use vectors to prove that the diagonals of a parallelogram bisect each other (Figure 32). Observe that the midpoint of BD is the terminal point of w + 1/2(v − w). AC and BD
We consider the equations of a line in symmetric form, when a ≠ 0, b ≠ 0, c ≠ 0.Find the symmetric equations of the line through P = (1, 1, 2) and Q = (−2, 4, 0). 0x - x a y - Yo b 02-2 C
Use vectors to prove that the segments joining the midpoints of opposite sides of a quadrilateral bisect each other (Figure 33). Show that the midpoints of these segments are the terminal points of 1 4 (2u+ v +z) and (2v + w+u) 1 4 u Z W V
Prove that two nonzero vectors v = (a, b) and w = (c, d) are perpendicular if and only if ac + bd = 0
We consider the equations of a line in symmetric form, when a ≠ 0, b ≠ 0, c ≠ 0.Find a vector parametrization for the line 0x - x a y - Yo b 02-2 C
Find a vector parametrization for the line X y 2 7 || N 100 8
Show that the line in the plane through (x0, y0) of slope m has symmetric equations m y-yo = 0x - x
A median of a triangle is a segment joining a vertex to the midpoint of the opposite side. Referring to Figure 22(A), prove that three medians of triangle ABC intersect at the terminal point P of the vector 1/3 (u + v + w). The point P is the centroid of the triangle. Show, by parametrizing the
A median of a tetrahedron is a segment joining a vertex to the centroid of the opposite face. The tetrahedron in Figure 22(B) has vertices at the origin and at the terminal points of vectors u, v, and w. Show that the medians intersect at the terminal point of 1/4 (u + v + w). u W (B)
We consider the equations of a line in symmetric form, when a ≠ 0, b ≠ 0, c ≠ 0.Find the symmetric equations of the line x = 3 + 2t, y = 4 − 9t, z = 12t 0x - x a y - Yo b 02-2 C
Points R1 and R2 in Figure 25 are defined so that F1R1 and F2R2 are perpendicular to the tangent line.(a) Show, with A and B as in Exercise 70, that(b) Use (a) and the distance formula to show that(c) Use (a) and the equation of the tangent line in Exercise 70 to show thatData From Exercise 70We
Find the area of the inner loop of the limac¸on with polar equation r = 2 cos θ − 1 (Figure 19). 1 -1- ta + 2 X+
Find the area of the shaded region in Figure 19 between the inner and outer loop of the limac¸on r = 2 cos θ − 1. -1 y 2
Find the equation of the given hyperbola.Vertices (±3, 0) and asymptotes y = ± 1/2x
Find the area of the part of the circle r = sin θ + cos θ in the fourth quadrant (see Exercise 30 in Section 11.3).Data From Exercise 30For all a > 0 and b > 1, the inequalities ln n ≤ na, na n are true for n sufficiently large (this can be proved using L’Hôpital’s Rule). Use this,
Find the equation of the given hyperbola.Foci (±3, 0) and eccentricity e = 3
Find the equation of the given hyperbola.Foci (0, ±5) and eccentricity e = 1.5
Find the area between the two curves in Figure 20(A). y (A) r = 2 + cos 20 r = sin 20 -X
Find the area of the region inside the circle r = 2 sin (θ + π/4) and above the line r = sec (θ − π/4).
Find the equation of the given hyperbola.Vertices (−3, 0), (7, 0) and eccentricity e = 3
Find the equation of the given hyperbola.Vertices (0, −6), (0, 4) and foci (0, −9), (0, 7)
Find the area inside both curves in Figure 21. r = 2 + sin 20 P X r= 2 + cos 20
Find the equation of the parabola with the given properties. Vertex (0, 0), focus (2,0) 12
Find the area of the region that lies inside one but not both of the curves in Figure 21. r = 2 + sin 20 X r = 2 + cos 20
Find the equation of the parabola with the given properties.Vertex (0, 0), focus (0, 2)
Find parametric equations for the given curve.y = 9 − 4x
(a) Plot r(θ) = 1 − cos(10θ) for 0 ≤ θ ≤ 2π.(b) Compute the area enclosed inside the ten petals of the graph of r(θ).(c) Explain why, for a positive integer n and 0 ≤ θ ≤ 2π, rn(θ) = 1 − cos(nθ) traces out an n-petal flower inscribed in a circle of radius 2 centered at the
Find the equation of the parabola with the given properties.Vertex (0, 0), directrix y = −5
(a) Plot the spiral r(θ) = θ for 0 ≤ θ ≤ 8π.(b) On your plot, shade in the region that represents the increase in area enclosed by the curve as θ goes from 6π to 8π. Compute the shaded area.(c) Show that the increase in area enclosed by the graph of r(θ) as θ goes from 2nπ to 2(n +
Find the equation of the parabola with the given properties.Vertex (3, 4), directrix y = −2
Calculate the total length of the circle r = 4 sin θ as an integral in polar coordinates.
Find the equation of the parabola with the given properties.Focus (0, 4), directrix y = −4
Sketch the segment r = sec θ for 0 ≤ θ ≤ A. Then compute its length in two ways: as an integral in polar acoordinates and using trigonometry.
Find the equation of the parabola with the given properties.Focus (0, −4), directrix y = 4
Compute the length of the polar curve.The length of r = θ2 for 0 ≤ θ ≤ π
Find the equation of the parabola with the given properties.Focus (2, 0), directrix x = −2
Compute the length of the polar curve.The spiral r = θ for 0 ≤ θ ≤ A
Find the equation of the parabola with the given properties.Focus (−2, 0), vertex (2, 0)
Compute the length of the polar curve.The curve r = sin θ for 0 ≤ θ ≤ π
Find the vertices, foci, center (if an ellipse or a hyperbola), and asymptotes (if a hyperbola).x2 + 4y2 = 16
Compute the length of the polar curve.The equiangular spiral r = eθ for 0 ≤ θ ≤ 2π
Find the vertices, foci, center (if an ellipse or a hyperbola), and asymptotes (if a hyperbola).4x2 + y2 = 16
Compute the length of the polar curve.r = √1 + sin 2θ for 0 ≤ θ ≤ π/4
Find the vertices, foci, center (if an ellipse or a hyperbola), and asymptotes (if a hyperbola). 2 x-3 +5 (*² = ³)²³- ( ² + ³ ) ² - 4 7 = = 1
Sketch the curve r = 1/2 θ (the spiral of Archimedes) for θ between 0 and 2π by plotting the points for θ = 0, π/4, π/2, . . . , 2π.
Compute the length of the polar curve.The cardioid r = 1 − cos θ in Figure 14 -2 G -1
Find the vertices, foci, center (if an ellipse or a hyperbola), and asymptotes (if a hyperbola).3x2 − 27y2 = 12
Compute the length of the polar curve.r = cos2 θ
Find the vertices, foci, center (if an ellipse or a hyperbola), and asymptotes (if a hyperbola).4x2 − 3y2 + 8x + 30y = 215
Compute the length of the polar curve.r = 1 + θ for 0 ≤ θ ≤ π/2
Find the vertices, foci, center (if an ellipse or a hyperbola), and asymptotes (if a hyperbola).y = 4x2
Express the length of the curve as an integral but do not evaluate it.r = eθ + 1, 0 ≤ θ ≤ π/2
Find the vertices, foci, center (if an ellipse or a hyperbola), and asymptotes (if a hyperbola).y = 4(x − 4)2
Express the length of the curve as an integral but do not evaluate it.r = (2 − cos θ)−1, 0≤ θ ≤ 2π
Find the vertices, foci, center (if an ellipse or a hyperbola), and asymptotes (if a hyperbola).8y2 + 6x2 − 36x − 64y + 134 = 0
Express the length of the curve as an integral but do not evaluate it.r = sin3 θ, 0≤ θ ≤ 2π
Find the vertices, foci, center (if an ellipse or a hyperbola), and asymptotes (if a hyperbola).4x2 + 25y2 − 8x − 10y = 20
Express the length of the curve as an integral but do not evaluate it.= sin θ cos θ, 0≤ θ ≤ π
Find the vertices, foci, center (if an ellipse or a hyperbola), and asymptotes (if a hyperbola).16x2 + 25y2 − 64x − 200y + 64 = 0
Use a computer algebra system to calculate the total length to two decimal places.The three-petal rose r = cos 3θ in Figure 18 11 7/1 =cos 38
Use the Discriminant Test to determine the type of the conic section (in each case, the equation is nondegenerate). Use a graphing utility or computer algebra system to plot the curve.4x2 + 5xy + 7y2 = 24
Use a computer algebra system to calculate the total length to two decimal places.The curve r = 2 + sin 2θ in Figure 21 r=2 + sin 20 x r=2+cos20
Use the Discriminant Test to determine the type of the conic section (in each case, the equation is nondegenerate). Use a graphing utility or computer algebra system to plot the curve.x2 − 2xy + y2 + 24x − 8 = 0
Use a computer algebra system to calculate the total length to two decimal places.The curve r = θ sin θ in Figure 22 for 0 ≤ θ ≤ 4π + 5 10+ 5- +X 5

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