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mathematics
precalculus

Questions and Answers of Precalculus

Write a polar equation of a conic with the focus at the origin and the given data.Ellipse, eccentricity 1/2, directrix x = 4
(a) Calculate the surface area of the ellipsoid that is generated by rotating an ellipse about its major axis.(b) What is the surface area if the ellipse is rotated about its minor axis?
(a) If an ellipse is rotated about its major axis, find the volume of the resulting solid.(b) If it is rotated about its minor axis, find the resulting volume.
Show that if an ellipse and a hyperbola have the same foci, then their tangent lines at each point of intersection are perpendicular.
Find an equation for the ellipse with foci (1, 1) and (-1, -1) and major axis of length 4.
Show that the function defined by the upper branch of the hyperbola is concave upward. y/a² – x²/b² = 1 v²/a?
Find an equation for the conic that satisfies the givenconditions.Hyperbola, foci (2, 0), (2, 8), asymptotes y = 3 + x and y = 5 – }x ||
Find an equation for the conic that satisfies the given conditions.(±3, 0), asymptotes y = ±2x
Find an equation for the conic that satisfies the given conditions.Hyperbola, vertices (-1, 2), (7, 2), foci (-2, 2), (8, 2)
Find an equation for the conic that satisfies the given conditions.Hyperbola, vertices (-3, -4), (-3, 6), foci (-3, -7), (-3, 9)
Find an equation for the conic that satisfies the given conditions.Hyperbola, vertices (0, ±2), foci (0, ±5)
Find an equation for the conic that satisfies the given conditions.Ellipse, foci (±4, 0), passing through (-4, 1.8)
Find an equation for the conic that satisfies the given conditions.Ellipse, center (-1, 4), vertex (-1, 0), focus (-1, 6)
Find an equation for the conic that satisfies the given conditions.Ellipse, foci (0, -1), (8, -1), vertex (9, -1)
Find an equation for the conic that satisfies the given conditions.Ellipse, foci (0, 2), (0, 6), vertices (0, 0), (0, 8)
Find an equation for the conic that satisfies the given conditions.Ellipse, foci (0, ±√2 ), vertices (0, ±2)
Find an equation for the conic that satisfies the given conditions.Ellipse, foci (±2, 0), vertices (±5, 0)
Find an equation for the conic that satisfies the given conditions.Parabola, vertical axis, passing through (0, 4), (1, 3), and (-2, -6)
Find an equation for the conic that satisfies the given conditions.Parabola, vertex (3, -1), horizontal axis, passing through (-15, 2)
Find an equation for the conic that satisfies the given conditions.Parabola, focus (2, -1), vertex (2, 3)
Find an equation for the conic that satisfies the given conditions.Parabola, focus (-4, 0), directrix x = 2
Find an equation for the conic that satisfies the given conditions.Parabola, focus (0, 0), directrix y = 6
Find an equation for the conic that satisfies the given conditions.Parabola, vertex (0, 0), focus (1, 0)
Identify the type of conic section whose equation is given and find the vertices and foci.x2 - 2x + 2y2 - 8y + 7 = 0
Identify the type of conic section whose equation is given and find the vertices and foci.3x2 - 6x - 2y = 1
Identify the type of conic section whose equation is given and find the vertices and foci.y2 - 2 = x2 - 2x
Identify the type of conic section whose equation is given and find the vertices and foci.x2 = 4y - 2y2
Identify the type of conic section whose equation is given and find the vertices and foci.4x2 = y + 4
Identify the type of conic section whose equation is given and find the vertices and foci.4x2 = y2 + 4
Find the vertices, foci, and asymptotes of the hyperbola and sketch its graph.9y2 - 4x2 - 36y - 8x = 4
Find the vertices, foci, and asymptotes of the hyperbola and sketch its graph.x2 - y2 + 2y = 2
Find the vertices, foci, and asymptotes of the hyperbola and sketch its graph.y2 - 16x2 = 16
Find the vertices, foci, and asymptotes of the hyperbola and sketch its graph.x2 - y2 = 100
Find the vertices, foci, and asymptotes of the hyperbola and sketch its graph. .2 ,2 1 64 36
Find the vertices, foci, and asymptotes of the hyperbola and sketch its graph. 2 х y? 25 9.
Find the vertices and foci of the ellipse and sketch its graph. .2 x² + 3y? + 2x – 12y + 10 = 0 %3D
Find the vertices and foci of the ellipse and sketch its graph.9x2 - 18x + 4y2 = 27
Find the vertices and foci of the ellipse and sketch its graph.100x2 + 36y2 = 225
Find the vertices and foci of the ellipse and sketch its graph.x2 + 9y2 = 9
Find the vertices and foci of the ellipse and sketch its graph. x? 36 00
Find the vertices and foci of the ellipse and sketch its graph. y2 2
Find the vertex, focus, and directrix of the parabola and sketch its graph. 2х2 — 16х — Зу + 38 3 0
Find the vertex, focus, and directrix of the parabola and sketch its graph.(y - 2)2 = 2x + 1
Find the vertex, focus, and directrix of the parabola and sketch its graph. у? + бу + 2х +1 %3D0
Find the vertex, focus, and directrix of the parabola and sketch its graph.3x2 + 8y = 0
Find the vertex, focus, and directrix of the parabola and sketch its graph.2x = -y2
Find the vertex, focus, and directrix of the parabola and sketch its graph.2y2 = 5x
Find the vertex, focus, and directrix of the parabola and sketch its graph.x2 = 6y
Use a calculator to find the length of the curve correct to four decimal places. If necessary, graph the curve to determine the parameter interval.r = tan θ, π/6 < θ < π/3
Use a calculator to find the length of the curve correct to four decimal places. If necessary, graph the curve to determine the parameter interval.One loop of the curve r = cos 2θ
Find the exact length of the curve. Use a graph to determine the parameter interval. r = cos2(θ/2)
Find the exact length of the polar curve.r = 2(1 + cos θ)
Find the exact length of the polar curve.r = 5θ, 0 ≤ θ ≤ 2π
When recording live performances, sound engineers often use a microphone with a cardioid pickup pattern because it suppresses noise from the audience. Suppose the microphone is placed 4 m from the
Find all points of intersection of the given curves.r = 1 + cos θ, r = 1 - sin θ
Find all points of intersection of the given curves.r2 = sin 2θ, r2 = cos 2θ
Find all points of intersection of the given curves.r = sin θ, r = sin 2θ
Find all points of intersection of the given curves.r = cos 3θ, r = sin 3θ
Find all points of intersection of the given curves.r = sin θ, r = 1 - sin θ
Find the area between a large loop and the enclosed smallloop of the curve r = 1 + 2 cos 3θ.
Find the area of the region that lies inside both curves.r = a sin θ, r = b cos θ, a > 0, b > 0
Find the area of the region that lies inside both curves. r2 = 2 sin 2θ, r = 1
Find the area of the region that lies inside both curves.r = 3 + 2 cos θ, r = 3 + 2 sin θ
Sketch the curve with the given polar equation by first sketching the graph of r as a function of θ in Cartesian coordinates.r = 2 sin 6θ
Identify the curve by finding a Cartesian equation for the curve.θ = π/3
Sketch the region in the plane consisting of points whose polar coordinates satisfy the given conditions. 1
The Cartesian coordinates of a point are given.(i) Find polar coordinates (r, θ) of the point, where r > 0 and 0 < θ, 2π.(ii) Find polar coordinates (r, θ) of the point, where r <
(a) Suppose that the dog in Problem 9 runs twice as fast as the rabbit. Find a differential equation for the path of the dog. Then solve it to find the point where the dog catches the rabbit.(b)
A peach pie is taken out of the oven at 5:00 pm. At that time it is piping hot, 100 8C. At 5:10 pm its temperature is 80 8C; at 5:20 pm it is 65 8C. What is the temperature of the room?
A subtangent is a portion of the x-axis that lies directly beneath the segment of a tangent line from the point of contact to the x-axis. Find the curves that pass through the point (c, 1) and whose
A student forgot the Product Rule for differentiation and made the mistake of thinking that (fg)' = f'g'. However, he was lucky and got the correct answer. The function f that he used was f (x) =
Find the orthogonal trajectories of the family of curvesy = ekx
Solve the initial-value problem.
Solve the differential equation. |x²y' – y = 2x²e-1/
(a) A direction field for the differential equation y' = y(y - 2)(y - 4) is shown. Sketch the graphs of the solutions that satisfy the given initial conditions.(i) y(0) = 20.3 (ii) y(0) = 1(iii)
If we ignore air resistance, we can conclude that heavier objects fall no faster than lighter objects. But if we take air resistance into account, our conclusion changes. Use the expression for the
Solve the initial-value problem. | (x² + 1) + 3x(y – 1) = 0, y(0) = 2 dx
Solve the initial-value problem. |ху — у + x? sin х, у(т) — 0 У(т) — 0
Solve the initial-value problem. |ху' + у — хIn x, У(1) — 0
Solve the initial-value problem. du — 1? + Зи, 1> 0, и(2) — 4 t dt
Solve the initial-value problem. dy + 3r?y = cos t, y(7)= 0 dt
Solve the initial-value problem. х*у' + 2ху — n x, у(1) — 2
Solve the differential equation. dr t In t +r= te' dt
Solve the differential equation. dy + 3ty = V1 + t², t>0 dt
Solve the differential equation.y' + 2xy = 1
Solve the differential equation. ху' — 2у — х?, х>0 х> 0
Suppose we alter the differential equation in Exercise 23 as follows:(a) Solve this differential equation with the help of a table of integrals or a CAS.(b) Graph the solution for several values of
An integral equation is an equation that contains an unknown function ysxd and an integral that involves y(x). Solve the given integral equation. dt y(x) = 2 + ty(t)'
A model for tumor growth is given by the Gompertz equation dV/dt = a(ln b - ln V)V where a and b are positive constants and V is the volume of the tumor measured in mm3.(a) Find a family of
A tank contains 1000 L of pure water. Brine that contains 0.05 kg of salt per liter of water enters the tank at a rate of 5 L/min. Brine that contains 0.04 kg of salt per liter of water enters the
The air in a room with volume 180 m3 contains 0.15% carbon dioxide initially. Fresher air with only 0.05% carbon dioxide flows into the room at a rate of 2 m3/min and the mixed air flows out at the
In contrast to the situation of Exercise 40, experiments show that the reaction H2 + Br2 → 2HBr satisfies the rate lawand so for this reaction the differential equation becomeswhere x = [HBr] and a
Find a function f such that f (3) = 2 and (t2 + 1) f'(t) + [f(t)]2 + 1 = 0 t ≠ 1
An integral equation is an equation that contains an unknown function y(x) and an integral that involves y(x). Solve the given integral equation. У) 3 2 + [—у(0)] d
Find the orthogonal trajectories of the family of curves. Use a graphing device to draw several members of each family on a common screen.y = 1/x + k
Find the orthogonal trajectories of the family of curves. Use a graphing device to draw several members of each family on a common screen.y = k/x
Find the orthogonal trajectories of the family of curves. Use a graphing device to draw several members of each family on a common screen.y2 = kx3
Find the orthogonal trajectories of the family of curves. Use a graphing device to draw several members of each family on a common screen.x2 + 2y2 = k2
(a) Use a computer algebra system to draw a direction field for the differential equation. Get a printout and use it to sketch some solution curves without solving the differential equation.(b) Solve
Solve the differential equation xy' = y + xey/x by making the change of variable v = y/x.

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