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study help
mathematics
calculus early transcendentals
Questions and Answers of
Calculus Early Transcendentals
Graph the curves described by the following functions, indicating the positive orientation.r(t) = (cos t, 0, sin t) for 0 ≤ t ≤ 2π
Find an equation of the line segment joining the first point to the second point.(-1, -8, 4) and (-9, 5, -3)
Find an equation of the line segment joining the first point to the second point.(2, 4, 8) and (7, 5, 3)
Find an equation of the line segment joining the first point to the second point.(1, 0, 1) and (0, -2, 1)
Find an equation of the line segment joining the first point to the second point.(0, 0, 0) and (1, 2, 3)
Find equations of the following lines.The line through (1, 0, -1) that is perpendicular to the lines r1(t) = (3 + 2t, 3t, -4t) and r2(t) = (t, t, -t)
Find equations of the following lines.The line through (1, 2, 3) that is perpendicular to the lines r1(t) = (3 - 2t, 5 + 8t, 7 - 4t) and r2(t) = (-2t, 5 + t, 7 - t)
Find equations of the following lines.The line through (0, 2, 1) that is perpendicular to both u = (4, 3, -5) and the z-axis
Find equations of the following lines.The line through (-2, 5, 3) that is perpendicular to both u = (1, 1, 2) and the x-axis
Find equations of the following lines.The line through (-3, 4, 2) that is perpendicular to both u = (1, 1, -5) and v = (0, 4, 0)
Find equations of the following lines.The line through (0, 0, 0) that is perpendicular to both u = (1, 0, 2) and v = (0, 1, 1)
Find equations of the following lines.The line through (1, -3, 4) that is parallel to the line r(t) = (3 + 4t, 5 - t, 7)
Find equations of the following lines.The line through (0, 0, 0) that is parallel to the line r(t) = (3 - 2t, 5 + 8t, 7 - 4t)
Find equations of the following lines.The line through (0, 4, 8) and (10, -5, -4)
Find equations of the following lines.The line through (-3, 4, 6) and (5, -1, 0)
Find equations of the following lines.The line through (1, 0, 1) and (3, -3, 3)
Find equations of the following lines.The line through (0, 0, 0) and (1, 2, 3)
Find equations of the following lines.The line through (0, 0, 1) parallel to the x-axis
Find equations of the following lines.The line through (0, 0, 1) parallel to the y-axis
Find equations of the following lines.The line through (-3, 2, -1) in the direction of the vector v = (1, -2, 0)
Find equations of the following lines.The line through (0, 0, 1) in the direction of the vector v = (4, 7, 0)
How do you determine whether r(t) = f(t)i + g(t)j + h(t) k is continuous at t = a?
How do you evaluater(t), where r(t) = (f(t), g(t), h(t))? lim r(t),
In what plane does the curve r(t) = ti + t2k lie?
What is an equation of the line through the points P0(x0, y0, z0) and P1(x1, y1, z1)?
Explain how to find a vector in the direction of the line segment from P0(x0, y0, z0) to P1(x1, y1, z1).
Why is r(t) = (f(t), g(t), h(t)) called a vector-valued function?
How many dependent scalar variables does the function r(t) = (f (t), g(t), h(t)) have?
How many independent variables does the function r(t) = (f(t), g(t), h(t)) have?
Suppose u and v are known nonzero vectors in R3.a. Prove that the equation u × z = v has a nonzero solution z if and only if u • v = 0. Take the dot product of both sides with v.b. Explain this
Prove the following identities. Assume that u, v, w, and x are nonzero vectors in R3.(u × v) • (w × x) = (u • w)(v • x) - (u • x)(v • w)
Prove the following identities. Assume that u, v, w, and x are nonzero vectors in R3.u × (v × w) = (u • w)v - (u • v)w Vector triple product
Determine whether the following statements are true using a proof or counterexample. Assume that u, v, and w are nonzero vectors in R3.u • (v × w) = w • (u × v)
Show that the polar equationr2 - 2rr0 cos (θ - θ0) = R2 - r02describes a circle of radius R whose center has polar coordinates (r0, θ0).
Show that the polar equationr2 - 2r(a cos θ + b sin θ) = R2 - a2 - b2describes a circle of radius R centered at (a, b).
Sketch the following sets of points (r, θ).r ≥ 2
Sketch the following sets of points (r, θ).0 < r < 3 and 0 ≤ θ ≤ π
Sketch the following sets of points (r, θ).|θ| ≤ π/3
Sketch the following sets of points (r, θ).1 < r < 2 and π/6 ≤ θ ≤ π/3.
Sketch the following sets of points (r, θ).π/2 ≤ θ ≤ 3π/4
Sketch the following sets of points (r, θ).2 ≤ r ≤ 8
Sketch the following sets of points (r, θ).θ = 2π/3
Sketch the following sets of points (r, θ).r = 3
Convert the following equations to polar coordinates.y = 1/x
Convert the following equations to polar coordinates.(x - 1)2 + y2 = 1
Convert the following equations to polar coordinates.y = x2
Convert the following equations to polar coordinates.y = 3
Determine whether the following statements are true and give an explanation or counterexample.a. The point with Cartesian coordinates (-2, 2) has polar coordinates (2√2, 3π/4), (2√2, 11π/4),
Use a graphing utility to graph the following equations. In each case, give the smallest interval [0, P] that generates the entire curve.r = 1 - 2 sin 5θ
Use a graphing utility to graph the following equations. In each case, give the smallest interval [0, P] that generates the entire curve.r = 1 - 3 cos 2θ
Use a graphing utility to graph the following equations. In each case, give the smallest interval [0, P] that generates the entire curve.r = cos 3θ/7
Use a graphing utility to graph the following equations. In each case, give the smallest interval [0, P] that generates the entire curve.r = cos 3θ/5
Use a graphing utility to graph the following equations. In each case, give the smallest interval [0, P] that generates the entire curve.r = 2 sin 2θ/3
Use a graphing utility to graph the following equations. In each case, give the smallest interval [0, P] that generates the entire curve.r = cos 3θ + cos2 2θ
Use a graphing utility to graph the following equations. In each case, give the smallest interval [0, P] that generates the entire curve.r = 2 - 4 cos 5θ
Use a graphing utility to graph the following equations. In each case, give the smallest interval [0, P] that generates the entire curve.r = sin θ/4
A Cartesian and a polar graph of r = f(θ) are given in the figures. Mark the points on the polar graph that correspond to the points shown on the Cartesian graph.r = cos u + sin 2θ УА y, F 3т х
A Cartesian and a polar graph of r = f(θ) are given in the figures. Mark the points on the polar graph that correspond to the points shown on the Cartesian graph.r = 1/4 - cos 4θ УА K G B Н 2т
A Cartesian and a polar graph of r = f(θ) are given in the figures. Mark the points on the polar graph that correspond to the points shown on the Cartesian graph.r = sin (1 + 3 cos θ) УА D + Н +
A Cartesian and a polar graph of r = f(θ) are given in the figures. Mark the points on the polar graph that correspond to the points shown on the Cartesian graph.r = 1 - 2 sin 3θ yA AAA 3 B D J. L
Graph the following equations. Use a graphing utility to check your work and produce a final graph.r = 2 sin 5θ
Graph the following equations. Use a graphing utility to check your work and produce a final graph.r = sin 3θ
Graph the following equations. Use a graphing utility to check your work and produce a final graph.r2 = 16 sin 2θ
Graph the following equations. Use a graphing utility to check your work and produce a final graph.r2 = 16 cos θ
Graph the following equations. Use a graphing utility to check your work and produce a final graph.r2 = 4 sin θ
Graph the following equations. Use a graphing utility to check your work and produce a final graph.r = sin2 (θ/2)
Graph the following equations. Use a graphing utility to check your work and produce a final graph.r = 2 - 2 sin θ
Graph the following equations. Use a graphing utility to check your work and produce a final graph.r = 1 - sin θ
Tabulate and plot enough points to sketch a graph of the following equations.r = 1 - cos θ
Tabulate and plot enough points to sketch a graph of the following equations.r(sin θ - 2 cos θ) = 0
Tabulate and plot enough points to sketch a graph of the following equations.r = 4 + 4 cos θ
Tabulate and plot enough points to sketch a graph of the following equations.r = 8 cos θ
Convert the following equations to Cartesian coordinates. Describe the resulting curve. 1 2 cos 0 + 3 sin 0
Convert the following equations to Cartesian coordinates. Describe the resulting curve.r = 8 sin θ
Convert the following equations to Cartesian coordinates. Describe the resulting curve.r = sin θ sec2 θ
Convert the following equations to Cartesian coordinates. Describe the resulting curve.r cos θ = sin 2 θ
Convert the following equations to Cartesian coordinates. Describe the resulting curve.sin θ = |cos θ|
Convert the following equations to Cartesian coordinates. Describe the resulting curve.r = 2 sin θ + 2 cos θ
Convert the following equations to Cartesian coordinates. Describe the resulting curve.r = 3 csc θ
Convert the following equations to Cartesian coordinates. Describe the resulting curve.r = 2
Convert the following equations to Cartesian coordinates. Describe the resulting curve.r = cot θ csc θ
Convert the following equations to Cartesian coordinates. Describe the resulting curve.r cos θ = -4
Express the following Cartesian coordinates in polar coordinates in at least two different ways.(4, 4√3)
Express the following Cartesian coordinates in polar coordinates in at least two different ways.(-4, 4√3)
Express the following Cartesian coordinates in polar coordinates in at least two different ways.(-9, 0)
Express the following Cartesian coordinates in polar coordinates in at least two different ways.(1, √3)
Express the following Cartesian coordinates in polar coordinates in at least two different ways.(-1, 0)
Express the following Cartesian coordinates in polar coordinates in at least two different ways.(2, 2)
Express the following polar coordinates in Cartesian coordinates.(4, 5π)
Express the following polar coordinates in Cartesian coordinates.(-4, 3π/4)
Express the following polar coordinates in Cartesian coordinates.(2, 7π/4)
Express the following polar coordinates in Cartesian coordinates.(1, -π/3)
Express the following polar coordinates in Cartesian coordinates.(1, 2π/3)
Express the following polar coordinates in Cartesian coordinates.(3, π/4)
Give two sets of polar coordinates for each of the points A–F in the figure. УА D Ө — Ө — 3 4 12 х El3 ||
Graph the points with the following polar coordinates. Give two alternative representations of the points in polar coordinates.(-4, 3π/2)
Graph the points with the following polar coordinates. Give two alternative representations of the points in polar coordinates.(2, 7π/4)
Graph the points with the following polar coordinates. Give two alternative representations of the points in polar coordinates.(-1, -π/3)
Graph the points with the following polar coordinates. Give two alternative representations of the points in polar coordinates.(3, 2π/3)
Graph the points with the following polar coordinates. Give two alternative representations of the points in polar coordinates.(2, π/4)
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