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mathematics
precalculus
Thomas Calculus Early Transcendentals 13th Edition Joel R Hass, Christopher E Heil, Maurice D Weir - Solutions
Find the lengths of the curves.The cardioid r = 1 + cos θ
Graph the sets of points whose polar coordinates satisfy the equations and inequalitie.0 ≤ θ ≤ π, r = -1
Find a parametrization for the curve.The line segment with endpoints (-1, 3) and (3, -2)
Give equations for ellipses. Put each equation in standard form. Then sketch the ellipse. Include the foci in your sketch.9x2 + 10y2 = 90
Find the eccentricity of the hyperbola. Then find and graph the hyperbola’s foci and directrices.y2 - 3x2 = 3
Find the length of the enclosed loop x = t2, y = (t3/3) - t shown here. The loop starts at t = -√3 and ends at t = √3. = 0 y 0 -1 | 1 2 t> 0 t = 4 t < 0 +√3 X
Find the lengths of the curves.The spiral r = eθ/√2, 0 ≤ θ ≤ π
Find the area enclosed by the y-axis and the curve x = t - t2, y = 1 + e-t.
Give the eccentricities of conic sections with one focus at the origin of the polar coordinate plane, along with the directrix for that focus. Find a polar equation for each conic section.e = 1/3, r sin θ = -6
Graph the sets of points whose polar coordinates satisfy the equations and inequalitie.0 ≤ θ ≤ π, r = 1
Find a parametrization for the curve.The line segment with endpoints (-1, -3) and (4, 1)
Give equations for ellipses. Put each equation in standard form. Then sketch the ellipse. Include the foci in your sketch.3x2 + 2y2 = 6
The area of the region that lies inside the cardioid curve r = cos θ + 1 and outside the circle r = cos θ is notWhy not? What is the area? Give reasons for your answers. 2T [(cos + 1)² cos²0] de = T
Find the eccentricity of the hyperbola. Then find and graph the hyperbola’s foci and directrices.8x2 - 2y2 = 16
Find the lengths of the curves.The spiral r = θ2, 0 ≤ u ≤ √5
Find the area under one arch of the cycloid x = a(t - sin t), y = a(1 - cos t).
Give the eccentricities of conic sections with one focus at the origin of the polar coordinate plane, along with the directrix for that focus. Find a polar equation for each conic section.e = 1/2, r sin θ = 2
Graph the sets of points whose polar coordinates satisfy the equations and inequalitie.θ = π/2, r ≤ 0
Find parametric equations and a parameter interval for the motion of a particle that starts at (a, 0) and traces the ellipse (x2/a2) + (y2/b2) = 1a. Once clockwise. b. Once counterclockwise.c. Twice clockwise. d. Twice counterclockwise.
Give equations for ellipses. Put each equation in standard form. Then sketch the ellipse. Include the foci in your sketch.2x2 + y2 = 4
Find the lengths of the curves. x = 3 cos 0, y = 3 sin 0, 0505 З п 2
Find the eccentricity of the hyperbola. Then find and graph the hyperbola’s foci and directrices.y2 - x2 = 4
a. Find the area of the shaded region in the accompanying figure.b. It looks as if the graph of r = tan θ, -π/2 17 0| tan 0 x 1
Find the slopes of the curves at the given points. Sketch the curves along with their tangents at these points.r = cos 2θ; θ = 0, ±π/2, π
Assuming that the equations define x and y implicitly as differentiable functions x = ƒ(t), y = g(t), find the slope of the curve x = ƒ(t), y = g(t) at the given value of t.t = ln (x - t), y = tet, t = 0
Give the eccentricities of conic sections with one focus at the origin of the polar coordinate plane, along with the directrix for that focus. Find a polar equation for each conic section.e = 1, r cos θ = -4
Graph the sets of points whose polar coordinates satisfy the equations and inequalitie.θ = π/2, r ≥ 0
Find parametric equations and a parameter interval for the motion of a particle that starts at (a, 0) and traces the circle x2 + y2 = a2a. Once clockwise.b. Once counterclockwise.c. Twice clockwise.d. Twice counterclockwise.
Give equations for ellipses. Put each equation in standard form. Then sketch the ellipse. Include the foci in your sketch.2x2 + y2 = 2
Find the eccentricity of the hyperbola. Then find and graph the hyperbola’s foci and directrices.y2 - x2 = 8
Find the slopes of the curves at the given points. Sketch the curves along with their tangents at these points.r = sin 2θ; θ = ±π/4, ±3π/4
Assuming that the equations define x and y implicitly as differentiable functions x = ƒ(t), y = g(t), find the slope of the curve x = ƒ(t), y = g(t) at the given value of t.x = t3 + t, y + 2t3 = 2x + t2, t = 1
What are the standard equations for lines and conic sections in polar coordinates? Give examples.
Give the eccentricities of conic sections with one focus at the origin of the polar coordinate plane, along with the directrix for that focus. Find a polar equation for each conic section.e = 2, r cos θ = 2
Give equations for ellipses. Put each equation in standard form. Then sketch the ellipse. Include the foci in your sketch.7x2 + 16y2 = 112
Graph the sets of points whose polar coordinates satisfy the equations and inequalitie.θ = 11π/4, r ≥ -1
Give parametric equations and parameter intervals for the motion of a particle in the xy-plane. Identify the particle’s path by finding a Cartesian equation for it. Graph the Cartesian equation. Indicate the portion of the graph traced by the particle and the direction of motion.x = 2 sinh t, y =
Find the eccentricity of the hyperbola. Then find and graph the hyperbola’s foci and directrices.9x2 - 16y2 = 144
Find the slopes of the curves at the given points. Sketch the curves along with their tangents at these points.r = -1 + sin θ; θ = 0, π
Find the lengths of the curves.x = t3 - 6t2, y = t3 + 6t2, 0 ≤ t ≤ 1
Find the areas of the region.Inside the circle r = 4 sin θ and below the horizontal line r = 3 csc θ
Assuming that the equations define x and y implicitly as differentiable functions x = ƒ(t), y = g(t), find the slope of the curve x = ƒ(t), y = g(t) at the given value of t.x sin t + 2x = t, t sin t - 2t = y, t = π
Explain the equation PF = e · PD.
Find the length of the curve r = 2 sin3 (θ/3), 0 ≤ θ ≤ 3π, in the polar coordinate plane.
Graph the sets of points whose polar coordinates satisfy the equations and inequalitie.θ = π/3, -1 ≤ r ≤ 3
Give equations for ellipses. Put each equation in standard form. Then sketch the ellipse. Include the foci in your sketch.16x2 + 25y2 = 400
Find the eccentricity of the hyperbola. Then find and graph the hyperbola’s foci and directrices.x2 - y2 = 1
Give parametric equations and parameter intervals for the motion of a particle in the xy-plane. Identify the particle’s path by finding a Cartesian equation for it. Graph the Cartesian equation. Indicate the portion of the graph traced by the particle and the direction of motion.x = -cosh t, y =
Find the lengths of the curves.x = 5 cos t - cos 5t, y = 5 sin t - sin 5t, 0 ≤ t ≤ π/2
Find the areas of the region.Inside the circle r = 4 cos θ and to the right of the vertical line r = sec θ
What points in the xy-plane satisfy the equations and inequalities? Draw a figure for each exercise.(9x2 + 4y2 - 36)(4x2 + 9y2 - 16) > 0
Graph the sets of points whose polar coordinates satisfy the equations and inequalitie.θ = 2π/3, r ≤ -2
a. Find an equation in polar coordinates for the curve x = e2t cos t, y = e2t sin t; -∞ < t < ∞.b. Find the length of the curve from t = 0 to t = 2π.
Give foci and corresponding directrices of ellipses centered at the origin of the xy-plane. In each case, use the dimensions in Figure 11.47 to find the eccentricity of the ellipse. Then find the ellipse’s standard-form equation in Cartesian coordinates.In Figure 11.47 D₁ | Directrix 1 a |x=
Find the slopes of the curves at the given points. Sketch the curves along with their tangents at these points.r = -1 + cos θ; θ = ±π/2
Assuming that the equations define x and y implicitly as differentiable functions x = ƒ(t), y = g(t), find the slope of the curve x = ƒ(t), y = g(t) at the given value of t.x + 2x3/2 = t2 + t, y√t + 1 + 2t√y = 4, t = 0
Assuming that the equations define x and y implicitly as differentiable functions x = ƒ(t), y = g(t), find the slope of the curve x = ƒ(t), y = g(t) at the given value of t. x = √5 - √t, y(t - 1) = √t, t = 4
Give foci and corresponding directrices of ellipses centered at the origin of the xy-plane. In each case, use the dimensions in Figure 11.47 to find the eccentricity of the ellipse. Then find the ellipse’s standard-form equation in Cartesian coordinates.In Figure 11.47 D₁ | Directrix 1 a |x=
What is the eccentricity of a conic section? How can you classify conic sections by eccentricity? How does eccentricity change the shape of ellipses and hyperbolas?
Give equations of parabolas. Find each parabola’s focus and directrix. Then sketch the parabola. Include the focus and directrix in your sketch.x = 2y2
Give parametric equations and parameter intervals for the motion of a particle in the xy-plane. Identify the particle’s path by finding a Cartesian equation for it. Graph the Cartesian equation. Indicate the portion of the graph traced by the particle and the direction of motion.x = -sec t, y =
Graph the lemniscates. What symmetries do these curves have?r2 = -cos 2θ
Find the lengths of the curves.x = (y3/12) + (1/y), 1 ≤ y ≤ 2
Find the areas of the region.Inside the circle r = 6 above the line r = 3 csc θ
Give foci and corresponding directrices of ellipses centered at the origin of the xy-plane. In each case, use the dimensions in Figure 11.47 to find the eccentricity of the ellipse. Then find the ellipse’s standard-form equation in Cartesian coordinates.In Figure 11.47 D₁ | Directrix 1 a |x=
What is a hyperbola? What are the Cartesian equations for hyperbolas centered at the origin with foci on one of the coordinate axes? How can you find the foci, vertices, and directrices of such an ellipse from its equation?
Give parametric equations and parameter intervals for the motion of a particle in the xy-plane. Identify the particle’s path by finding a Cartesian equation for it. Graph the Cartesian equation. Indicate the portion of the graph traced by the particle and the direction of motion. x = √t +
What points in the xy-plane satisfy the equations and inequalities? Draw a figure for each exercise.(9x2 + 4y2 - 36)(4x2 + 9y2 - 16) ≤ 0
Graph the sets of points whose polar coordinates satisfy the equations and inequalitie.0 ≤ θ ≤ π/6, r ≥ 0
Give equations of parabolas. Find each parabola’s focus and directrix. Then sketch the parabola. Include the focus and directrix in your sketch.x = -3y2
Give parametric equations and parameter intervals for the motion of a particle in the xy-plane. Identify the particle’s path by finding a Cartesian equation for it. Graph the Cartesian equation. Indicate the portion of the graph traced by the particle and the direction of motion.x = sec2 t - 1, y
Graph the lemniscates. What symmetries do these curves have?r2 = -sin 2θ
Find the lengths of the curves.y = (5/12)x6/5 - (5/8)x4/5, 1 ≤ x ≤ 32
Find the areas of the region.Inside the circle r = -2 cos θ and outside the circle r = 1
Assuming that the equations define x and y implicitly as differentiable functions x = ƒ(t), y = g(t), find the slope of the curve x = ƒ(t), y = g(t) at the given value of t.x3 + 2t2 = 9, 2y3 - 3t2 = 4, t = 2
What is an ellipse? What are the Cartesian equations for ellipses centered at the origin with foci on one of the coordinate axes? How can you find the foci, vertices, and directrices of such an ellipse from its equation?
What points in the xy-plane satisfy the equations and inequalities? Draw a figure for each exercise.(x2/9) - ( y2/16) ≤ 1
Graph the sets of points whose polar coordinates satisfy the equations and inequalitie.1 ≤ r ≤ 2
Give foci and corresponding directrices of ellipses centered at the origin of the xy-plane. In each case, use the dimensions in Figure 11.47 to find the eccentricity of the ellipse. Then find the ellipse’s standard-form equation in Cartesian coordinates.In Figure 11.47 D₁ | Directrix 1 a |x=
Give equations of parabolas. Find each parabola’s focus and directrix. Then sketch the parabola. Include the focus and directrix in your sketch.y = -8x2
Graph the lemniscates. What symmetries do these curves have?r2 = 4 sin 2θ
Find the lengths of the curves.x = y2/3, 1 ≤ y ≤ 8
Find the areas of the region.Inside the circle r = 3a cos θ and outside the cardioid r = a(1 + cos θ), a > 0
What is a parabola? What are the Cartesian equations for parabolas whose vertices lie at the origin and whose foci lie on the coordinate axes? How can you find the focus and directrix of such a parabola from its equation?
Find an equation for the line tangent to the curve at the point defined by the given value of t. Also, find the value of d2y/dx2 at this point.x = t + et, y = 1 - et, t = 0
What points in the xy-plane satisfy the equations and inequalities? Draw a figure for each exercise.(x2/9) + ( y2/16) ≤ 1
Graph the sets of points whose polar coordinates satisfy the equations and inequalitie.r ≥ 1
Give parametric equations and parameter intervals for the motion of a particle in the xy-plane. Identify the particle’s path by finding a Cartesian equation for it. Graph the Cartesian equation. Indicate the portion of the graph traced by the particle and the direction of motion. x = 1, y =
Find an equation for the line tangent to the curve at the point defined by the given value of t. Also, find the value of d2y/dx2 at this point. X = 1 t + 1' y = t = 1² - t = 2
Give equations of parabolas. Find each parabola’s focus and directrix. Then sketch the parabola. Include the focus and directrix in your sketch.y = 4x2
Which of the series defined by the formulas converge, and which diverge? Give reasons for your answers. α x n=1n
Graph the lemniscates. What symmetries do these curves have?r2 = 4 cos 2θ
Find the lengths of the curves.y = x1/2 - (1/3)x3/2, 1 ≤ x ≤ 4
Which of the series defined by the formulas converge, and which diverge? Give reasons for your answers. α x n=1n
Find the areas of the region.Inside the lemniscate r2 = 6 cos 2θ and outside the circle r = √3
Which of the series defined by the formulas converge, and which diverge? Give reasons for your answers. α x n=1n
Which of the series defined by the formulas converge, and which diverge? Give reasons for your answers. x n=10n
Under what conditions can you find the length of a curve r = ƒ(θ), α ≤ θ ≤ β, in the polar coordinate plane? Give an example of a typical calculation.
Which of the series defined by the formulas converge, and which diverge? Give reasons for your answers. x n=10n
Which of the series defined by the formulas converge, and which diverge? Give reasons for your answers. α x n=1n
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