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study help
mathematics
calculus 4th
Calculus 4th Edition Jon Rogawski, Colin Adams, Robert Franzosa - Solutions
Convert the points (r, θ) = (1, π/6), (3, 5π/4) from polar to rectangular coordinates.
Write (x + y)2 = xy + 6 as an equation in polar coordinates.
Write r = 2 cos θ cos θ − sin θ as an equation in rectangular coordinates.
Convert the equation 9(x2 + y2) = (x2 + y2 − 2y)2 to polar coordinates, and plot it with a graphing utility.
The equation r = sin(nθ), where n ≥ 2 is even, is a “rose” of 2n petals (Figure 1). Compute the total area of the flower, and show that it does not depend on n. *** n = 4 (8 petals) n = 2 (4 petals) n = 6 (12 petals)
Calculate the length of the curve with polar equation r = θ in Figure 4. π r=0 π 2 y X
Figure 5 shows the graph of r = e0.5θ sin θ for 0 ≤ θ ≤ 2π. Use a computer algebra system to approximate the difference in length between the outer and inner loops. -6 10+ 5+ 3 -X
Show that r = ƒ1(θ) and r = ƒ2(θ) define the same curves in polar coordinates if ƒ1(θ) = −ƒ2(θ +π). Use this to show that the following define the same conic section:
Identify the conic section. Find the vertices and foci. I = + 813
Identify the conic section. Find the vertices and foci.x2 − 2y2 = 4
Identify the conic section. Find the vertices and foci.(2x + 1/2y)2 = 4 − (x − y)2
Identify the conic section. Find the vertices and foci.(y − 3)2 = 2x2 − 1
Find the equation of the conic section indicated.Ellipse with vertices (±8, 0), foci (± √3, 0)
Find the equation of the conic section indicated.Ellipse with foci (±8, 0), eccentricity 1/8
Find the equation of the conic section indicated.Hyperbola with vertices (±8, 0), asymptotes y = ±3/4 x
Find the equation of the conic section indicated.Hyperbola with foci (2, 0) and (10, 0), eccentricity e = 4
Find the equation of the conic section indicated.Parabola with focus (8, 0), directrix x = −8
Find the equation of the conic section indicated.Parabola with vertex (4, −1), directrix x = 15
Find the asymptotes of the hyperbola 3x2 + 6x − y2 − 10y = 1.
Show that the “conic section” with equation x2 − 4x + y2 + 5 = 0 has no points.
Show that the relation dy/dx = (e2 − 1) x/y holds on a standard ellipse or hyperbola of eccentricity e.
The orbit of Jupiter is an ellipse with the sun at a focus. Find the eccentricity of the orbit if the perihelion (closest distance to the sun) equals 740 × 106 km and the aphelion (farthest distance from the sun) equals 816 × 106 km.
Refer to Figure 25 in Section 11.5. Prove that the product of the perpendicular distances F1R1 and F2R2 from the foci to a tangent line of an ellipse is equal to the square b2 of the semiminor axes. C R₁ = (₁.₁) F₁ =(-c. 0) P = (xo-yo) 8₂ R₂ = (0₂.B₂) F₂=(c.0) -X
Calculate.(−4, 6) − (3, −2)
Calculate.(−1/2, 5/3) + (3, 10/3)
Calculate.2.7(−1.4, 0.8) − 3.3(3.1, −2.2)
Calculate the linear combinations.5(2, 2, −3) + 3(1, 7, 2)
Calculate.(2e, 1 − 2π) − (2e − π, 8 − 2π)
Calculate the linear combinations.−2 (8, 11, 3) + 4 (2, 1, 1)
Calculate the linear combinations.6(4j + 2k) − 3(2i + 7k)
Calculate the linear combinations. (4,-2,8) (12,3,3)
Calculate.(ln 6, sin2 3) + (1 − ln 3, cos2 3)
Which of the vectors (A)–(C) in Figure 22 is equivalent to v − w? 2 W (A) (B) (C)
Calculate the linear combinations. 5(i +2j) - 3(2j+ k) + 7(2k - i)
Sketch v + w and v − w for the vectors in Figure 23. W
Calculate the linear combinations. 4 (6,-1, 1)2 (1, 0,-1) + 3(-2, 1, 1)
Sketch 2v, −w, v + w, and 2v − w for the vectors in Figure 24. 5 4- 3 2 1 v = (2, 3) " = w = (4,1) + + 1 2 3 4 5 6 x
Determine whether or not the two vectors are parallel. u = (1, -2,5), v = (-2,4, -10)
Sketch v = (1, 3), w = (2, −2), v + w, v − w.
Determine whether or not the two vectors are parallel.u = (4, 2, −6), v = (2, −1, 3)
Sketch v = (0, 2), w = (−2, 4), 3v + w, 2v − 2w.
Determine whether or not the two vectors are parallel.u = (4, 2, −6), v = (2, 1, 3)
Sketch v = (−2, 1), w =(2, 2), v + 2w, v − 2w.
Sketch the vector v such that v + v1 + v2 = 0 for v1 and v2 in Figure 25(A). V2 -3 3 + 1+ (A) 1 -X
Determine whether or not the two vectors are parallel.u = (−3, 1, 4), v = (6, −2, 8)
Sketch the vector sum v = v1 + v2 + v3 + v4 in Figure 25(B). VA y (B) V1 V3 V2 →X
Find the given vector.ev, where v = (1, 1, 2)
Letwhere P = (−2, 5), Q = (1, −2). Which of the following vectors with the given tails and heads are equivalent to v?(a) (−3, 3), (0, 4) (b) (0, 0), (3, −7)(c) (−1, 2), (2, −5)(d) (4, −5), (1, 4) V = PO,
Find the given vector.ew, where w = (4, −2, −1)
Which of the following vectors are parallel to v = (6, 9) and which point in the same direction? (a) (12, 18) (d) (-6, -9) (b) (3, 2) (e) (-24,-27) (c) (2,3) (f) (-24, -36)
Find the given vector.Unit vector in the direction of u = (1, 0, 7)
Sketch the vectors and determine whether they are equivalent. AB and PO,
Find the given vector.Unit vector in the direction opposite to v = (−4, 4, 2)
Sketch the vectors and determine whether they are equivalent.A = (1, 4), B = (−6, 3), P = (1, 4), Q = (6, 3) AB and PO,
Sketch the following vectors, and find their components and lengths:(a) 4i + 3j − 2k (b) i + j + k(c) 4j + 3k (d) 12i + 8j − k
Describe the surface. x² + y² + (z-2)² = 4, with z > 2
Sketch the vectors and determine whether they are equivalent.A = (−3, 2), B = (0, 0), P = (0, 0), Q = (3, −2) AB and PO,
Describe the surface. x² + y² + z² = 9, with x, y, z ≥ 0
Sketch the vectors and determine whether they are equivalent.A = (5, 8), B = (1, 8), P = (1, 8), Q = (−3, 8) AB and PO,
Describe the surface. x² + y² = 7, with |z| ≤ 7
Are parallel? And if so, do they point in the same direction? AB and PO
Describe the surface. x² + y² = 4, with y, z ≥ 0
Are parallel? And if so, do they point in the same direction?A = (−3, 2), B = (0, 0), P = (0, 0), Q = (3, 2) AB and PO
Are parallel? And if so, do they point in the same direction?A = (2, 2), B = (−6, 3), P = (9, 5), Q = (17, 4) AB and PO
Give an equation for the indicated surface.The sphere of radius 3 centered at (0, 0, −3)
Are parallel? And if so, do they point in the same direction?A = (5, 8), B = (2, 2), P = (2, 2), Q = (−3, 8) AB and PO
Give an equation for the indicated surface.The sphere centered at the origin passing through (1, 2, −3)
Let R = (−2, 7). Calculate the following:The length of OR
Give an equation for the indicated surface.The sphere centered at (6, −3, 11) passing through (0, 1, −4)
Give an equation for the indicated surface.The sphere with diameter PQ where P = (1, 1, -3) and Q = (1,7, 1)
Let R = (−2, 7). Calculate the following:The components of u = PŘ, where P = (1,2)
Let R = (−2, 7). Calculate the following:The point P such that has components (−2, 7) PR
Give an equation for the indicated surface.The cylinder passing through the origin with the vertical line through (1, −1, 0) as its central axis
Let R = (−2, 7). Calculate the following:The point Q such that has components (8, −3) RQ
Give an equation for the indicated surface.The cylinder passing through (0, 2, 1) with the vertical line through (1, 0, 0) as its central axis
Find a vector parametrization for the line with the given description.Passes through P = (1, 2, −8), direction vector v = (2, 1, 3)
Find the given vector.Unit vector ev, where v = (3, 4)
Find a vector parametrization for the line with the given description.Passes through P = (4, 0, 8), direction vector v = (1, 0, 1)
Find the given vector.Unit vector ew, where w = (24, 7)
Find a vector parametrization for the line with the given description.Passes through P = (4, 0, 8), direction vector v = 7i + 4k
Find the given vector.Vector of length 4 in the direction of u = (−1, −1)
Find a vector parametrization for the line with the given description.Passes through O, direction vector v = (3, −1, −4)
Find the given vector.Vector of length 3 in the direction of v = 4i + 3j
Find a vector parametrization for the line with the given description.Passes through (1, 1, 1) and (3, −5, 2)
Find the given vector.Vector of length 2 in the direction opposite to v = i − j
Find the given vector.Unit vector in the direction opposite to v = (−2, 4)
Find a vector parametrization for the line with the given description.Passes through (−2, 0, −2) and (4, 3, 7)
Find the given vector.Unit vector e making an angle of 4π/7 with the x-axis
Find a vector parametrization for the line with the given description.Passes through O and (4, 1, 1)
Find the given vector.Vector v of length 2 making an angle of 30◦ with the x-axis
Find a vector parametrization for the line with the given description.Passes through (1, 1, 1) parallel to the line through (2, 0, −1) and (4, 1, 3)
Find the given vector.A unit vector pointing in the direction from (1, 1) to (0, 3)
Find parametric equations for the lines with the given description.Perpendicular to the xy-plane, passes through the origin
Find the given vector.A unit vector pointing in the direction from (−3, 4) to the origin
Find parametric equations for the lines with the given description.Perpendicular to the yz-plane, passes through (0, 0, 2)
Find all scalars λ such that λ (2, 3) has length 1.
Find parametric equations for the lines with the given description.Parallel to the line through (1, 1, 0) and (0, −1, −2), passes through (0, 0, 4)
What are the coordinates of the point P in the parallelogram in Figure 26(A)? y P (2, 2) (A) (7, 8) (5,4) -X
Find a vector v satisfying 3v + (5, 20) = (11, 17).
Which of the following is a parametrization of the line through P = (4, 9, 8) perpendicular to the xz-plane (Figure 19)? (a) r(t) = (4,9, 8) +1(1,0, 1) (c) r(t)=(4,9, 8) + t (0, 1, 0) (b) r(t)= (4,9, 8) + t (0,0,1) (d) r(t)=(4,9, 8) +1(1, 1,0)
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