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mathematics
precalculus

Questions and Answers of Precalculus

Find two different parametric equations for each rectangular equation. y = 4x - 1
Graph the curve whose parametric equations are given and show its orientation. Find the rectangular equation of each curve.x = t2, y = ln t; t > 0
Graph the curve whose parametric equations are given and show its orientation. Find the rectangular equation of each curve. x = sin2t, y = cos2t; 0 ≤ t ≤ 2π
Graph the curve whose parametric equations are given and show its orientation. Find the rectangular equation of each curve.x = csct, y = cot t; π/4 ≤ t ≤ π/2
Graph the curve whose parametric equations are given and show its orientation. Find the rectangular equation of each curve.x = sec t, y = tan t; 0 ≤ t ≤ π/4
Graph the curve whose parametric equations are given and show its orientation. Find the rectangular equation of each curve.x = 2cost, y = sint; 0 ≤ t ≤ π/2
Graph the curve whose parametric equations are given and show its orientation. Find the rectangular equation of each curve.x = 2cost, y = 3sint; -π ≤ t ≤ 0
Graph the curve whose parametric equations are given and show its orientation. Find the rectangular equation of each curve.x = 2cost, y = 3sint; 0 ≤ t ≤ π
Graph the curve whose parametric equations are given and show its orientation. Find the rectangular equation of each curve.x = 2cost, y = 3sint; 0 ≤ t ≤ 2 π
Graph the curve whose parametric equations are given and show its orientation. Find the rectangular equation of each curve.x = t3/2 + 1, y = √t, t ≥ 0
Graph the curve whose parametric equations are given and show its orientation. Find the rectangular equation of each curve.x = √t, y = t3/2, t ≥ 0
Graph the curve whose parametric equations are given and show its orientation. Find the rectangular equation of each curve.x = et, y = e-t; t ≥ 0
Graph the curve whose parametric equations are given and show its orientation. Find the rectangular equation of each curve.x = 2et, y = 1 + et; t ≥ 0
Graph the curve whose parametric equations are given and show its orientation. Find the rectangular equation of each curve.x = 2t – 4, y = 4t2, -∞ < t < ∞
Graph the curve whose parametric equations are given and show its orientation. Find the rectangular equation of each curve.x = 3t2, y = t + 4, -∞ < t < ∞
Graph the curve whose parametric equations are given and show its orientation. Find the rectangular equation of each curve.x = √t + 4, y = √t – 4; t ≥ 0,
Graph the curve whose parametric equations are given and show its orientation. Find the rectangular equation of each curve.x = t2 + 4, y = t2 – 4 , -∞ < t < ∞
Graph the curve whose parametric equations are given and show its orientation. Find the rectangular equation of each curve.x = √2t, y = 4t; t ≥ 0
Graph the curve whose parametric equations are given and show its orientation. Find the rectangular equation of each curve.x = t + 2, y = √t; t ≥ 0
Graph the curve whose parametric equations are given and show its orientation. Find the rectangular equation of each curve.x = t - 3, y = 2t + 4; ; 0 ≤ t ≤ 2
Graph the curve whose parametric equations are given and show its orientation. Find the rectangular equation of each curve.x = 3t + 2, y = t + 1; 0 ≤ t ≤ 4
True or False.Curves defined using parametric equations have an orientation.
True or False. Parametric equations defining a curve are unique.
If a circle rolls along a horizontal line without slippage, a fixed point P on the circle will trace out a curve called a(n) _______.
The parametric equation x = 2sint, y = 3cost define a(n) ______.
Let x = f(t) and y = g (t), where f and g are two function whose common domain is some interval I. The collection of points defined by (x, y) = (f(t), g(t) is called a(n) _______. The variable
The function f(x) = 3sin(4x) has amplitude ______ and period ______.
The planet Mercury travels around the Sun in an elliptical orbit given approximately by where r is measured in miles and the Sun is at the pole. Find the distance from Mercury to the Sun at
Derive equation (d) in Table 5: ep 1 - e sin 0
Derive equation (c) in Table 5: ep 1 + e sin 0
Derive equation (b) in Table 5: ep 1 +еcos 6Ө
Find a polar equation for each conic. For each, a focus is at the pole. e = 5; directrix is perpendicular to the polar axis 5 units to the right of the pole.
Find a polar equation for each conic. For each, a focus is at the pole. e = 6; directrix is perpendicular to the polar axis 2 units to the right of the pole.
Find a polar equation for each conic. For each, a focus is at the pole. e = 2/3; directrix is parallel to the polar axis 3 units above the pole.
Find a polar equation for each conic. For each, a focus is at the pole. e = 4/5; directrix is perpendicular to the polar axis 3 units to the left of the pole.
Find a polar equation for each conic. For each, a focus is at the pole. e = 1; directrix is parallel to the polar axis 2 units below the pole.
Find a polar equation for each conic. For each, a focus is at the pole. e = 1; directrix is parallel to the polar axis 1 unit above the pole.
Convert each polar equation to a rectangular equation. 3 csc 0 csc 0 - 1
Convert each polar equation to a rectangular equation. 6 sec 0 2 sec 0 – 1
Convert each polar equation to a rectangular equation. r(2 - cosθ) = 2
Convert each polar equation to a rectangular equation. r(3 - 2 sinθ) = 6
Convert each polar equation to a rectangular equation. 2 + 4 cos 0
Convert each polar equation to a rectangular equation. 2 - sin 0
Convert each polar equation to a rectangular equation. 12 4 + 8 sin 0
Convert each polar equation to a rectangular equation. 3 – 6 cos 0
Convert each polar equation to a rectangular equation. 10 5 + 4 cos 0
Convert each polar equation to a rectangular equation. 4 + 3 sin 0
Convert each polar equation to a rectangular equation. 1 - sin 0
Convert each polar equation to a rectangular equation. 1 + cos 0
Analyze each equation and graph it. 3 csc 0 csc 0 - 1
Analyze each equation and graph it. 6 sec 0 2 sec 0 – 1
Analyze each equation and graph it. r(2 - cosθ) = 2
Analyze each equation and graph it. r(3 - 2 sinθ) = 6
Analyze each equation and graph it. 2 + 4 cos 0
Analyze each equation and graph it. 2 – sin 0
Analyze each equation and graph it. 12 4 + 8 sin 0
Analyze each equation and graph it. 9. 3 - 6 cos 0
Analyze each equation and graph it. 10 5 + 4 cos 0
Analyze each equation and graph it. 8 4 + 3 sin 0
Analyze each equation and graph it. 3 1- sin 0
Analyze each equation and graph it. 1 + cos 0
Identify the conic that each polar equation represents. Also, give the position of the directrix. 8 + 2 sin 0
Identify the conic that each polar equation represents. Also, give the position of the directrix. 4 - 2 cos 0
Identify the conic that each polar equation represents. Also, give the position of the directrix. 1 + 2 cos 0
Identify the conic that each polar equation represents. Also, give the position of the directrix. 4 2 - 3 sin 0
Identify the conic that each polar equation represents. Also, give the position of the directrix. 3 1 - sin 0
Identify the conic that each polar equation represents. Also, give the position of the directrix. 1 + cos 0
True or False.The eccentricity e of any conic is c/a where a is the distance of a vertex from the center and c is the distance of a focus from the center.
True or False.If (r, θ) are polar coordinates, the equation defines a hyperbola. 2 + 3 sin 0 2.
The eccentricity e of a parabola is_________, of an ellipse it is_________, and of a hyperbola it is__________.
A___________ is the set of points P in the plane such that the ratio of the distance from a fixed point called the_________to P to the distance from a fixed line called the________ to P equals a
Transform the equation r = 6cosθ polar coordinates to rectangular coordinates.
If (x, y) are the rectangular coordinates of a point P and (r, θ) are its polar coordinates, then x = _____ and y = _____.
Show that the graph of the equation x1/2 + y1/2 = a1/2 is part of the graph of a parabola.
Use the rotation formulas (5) to show that distance is invariant under a rotation of axes. That is, show that the distance from P1 = (x1, y1) to P2 = (x2, y2) in the xy-plane equals the distance
Prove that, except for degenerate cases, the equation Ax2 + Bxy + Cy2 + Dx + Ey + F = 0(a) Define a parabola if B2 – 4AC = 0.(b) Defines an ellipse (or a circle) if B2 – 4AC <
Apply the rotation formulas (5) to Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 to obtaine the equation A’x’2 + B’x’y’ + C’y’2 + D’x’ + E’y’ + F’ = 0.Refer to Problem 54.
Apply the rotation formulas (5) to Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 to obtaine the equation A’x’2 + B’x’y’ + C’y’2 + D’x’ + E’y’ + F’ = 0.Show that A + c = A' +
Apply the rotation formulas (5) to Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 to obtaine the equation A’x’2 + B’x’y’ + C’y’2 + D’x’ + E’y’ + F’ = 0.Express A', B', C', D',
Identify the graph of each equation without applying a rotation of axes. 3x2 + 2xy + y2 + 4x – 2y + 10 = 0
Identify the graph of each equation without applying a rotation of axes. Identify the graph of each equation without applying a rotation of axes. .3x2 – 2xy + y2 + 4x + 2y – 1 = 0
Identify the graph of each equation without applying a rotation of axes. 4x2 + 12xy + 9y2 – x – y = 0
Identify the graph of each equation without applying a rotation of axes. 10x2 – 12xy + 4y2 – x – y - 10 = 0
Identify the graph of each equation without applying a rotation of axes. 10x2 + 12xy + 4y2 – x – y + 10 =0
Identify the graph of each equation without applying a rotation of axes. 9x2 + 12xy + 4y2 – x – y – 10 = 0
Identify the graph of each equation without applying a rotation of axes. 2x2 – 3xy + 2y2 – 4x – 2 = 0
Identify the graph of each equation without applying a rotation of axes. x2 – 7xy + 3xy2 – y – 10 = 0
Identify the graph of each equation without applying a rotation of axes. 2x2 – 3xy + 4y2 + 2x + 3y – 5 = 0
Identify the graph of each equation without applying a rotation of axes. x2 + 3xy – 2y2 + 3x + 2y + 5 = 0
Rotate the axes so that the new equation contains no xy-term. Analyze and graph the new equation.16x2 + 24xy + 9y2 – 60x + 80y = 0
Rotate the axes so that the new equation contains no xy-term. Analyze and graph the new equation.16x2 + 24xy + 9y2 – 130x + 90y = 0
Rotate the axes so that the new equation contains no xy-term. Analyze and graph the new equation.34x2 – 24xy + 41y2 – 25 = 0
Rotate the axes so that the new equation contains no xy-term. Analyze and graph the new equation.25x2 – 36xy + 40y2 – 12√13x - 8√13y = 0
Rotate the axes so that the new equation contains no xy-term. Analyze and graph the new equation.x2 + 4xy + 4y2 + 5√5x – 5 = 0
Rotate the axes so that the new equation contains no xy-term. Analyze and graph the new equation.4x2 – 4xy + y2 - 8√5x – 16√5y = 0
Rotate the axes so that the new equation contains no xy-term. Analyze and graph the new equation.11x2 + 10√3xy + y2 – 4 = 0
Rotate the axes so that the new equation contains no xy-term. Analyze and graph the new equation.13x2 – 6√3xy + 7y2 – 16 = 0
Rotate the axes so that the new equation contains no xy-term. Analyze and graph the new equation.3x2 – 10xy + 3y2 – 32 = 0
Rotate the axes so that the new equation contains no xy-term. Analyze and graph the new equation.5x2 + 6xy + 5y2 – 8 = 0
Rotate the axes so that the new equation contains no xy-term. Analyze and graph the new equation.x2 – 4xy + y2 – 3 = 0

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