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mathematics
precalculus
Precalculus 9th edition Michael Sullivan - Solutions
A particle is moving with the given data. Find the position of the particle. a(t) = sin t + 3 cos t, s(0) = , v(0) = 2
A particle is moving with the given data. Find the position of the particle. v(t) = 2t – 1/(1 + t³), s(0) = 1
Find f.
Find f.
Find f. u? + Ju f'(u) f(1) = 3
Find f. f'(1) = 2t – 3 sin t, f(0) = 5
Find the most general antiderivative of the function. f(x) = x + cosh x
Find the most general antiderivative of the function. f() = 2 sin t – 3e'
Find the most general antiderivative of the function. 1 x? + 1 9(х) х
Find the most general antiderivative of the function. – 6x² + 3 f(x) = 4
For what values of the constants a and b is (1, 3) a point of inflection of the curve y = ax3 + bx2?
By applying the Mean Value Theorem to the function f (x) = x1/5 on the interval [32, 33], show that 2 < 33 < 2.0125
Show that the equation 3x + 2 cos x + 5 = 0 has exactly one real root.
Use the graphs of f, f', and f'' to estimate the x-coordinates of the maximum and minimum points and inflection points of f. f(x) = e 0.1x In(x? 1)
Use the graphs of f, f', and f'' to estimate the x-coordinates of the maximum and minimum points and inflection points of f. cos'x Vr? + x + 1' f(x)
Produce graphs of f that reveal all the important aspects of the curve. Use graphs of f' and f'' to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points. In Exercise 35 use calculus to find these quantities exactly.
Produce graphs of f that reveal all the important aspects of the curve. Use graphs of f' and f'' to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points. In Exercise 35 use calculus to find these quantities exactly.
The index of refraction of light in passing from a vacuum into water is 1.33. If the angle of incidence is 40°, determine the angle of refraction.Some Indexes of RefractionMedium ……………………………………… Index of Refraction†Water
The index of refraction of light in passing from a vacuum into dense flint glass is 1.66. If the angle of incidence is 50°, determine the angle of refraction.Light sound, and other waves travel at different speeds, depending on the media (air, water, wood, and so on) through which they pass.
Ptolemy, who lived in the city of Alexandria in Egypt during the second century AD, gave the measured values in the following table for the angle of incidence θ1 and the angle of refraction θ2 for a light beam passing from air into water. Do these values agree with Snell’s Law? If so, what
The speed of yellow sodium light (wavelength, 589 nanometers) in a certain liquid is measured to be 1.92 × 108 meters per second. What is the index of refraction of this liquid, with respect to air, for sodium light? The speed of light in air is approximately 2.998 × 108 meters per second.Light
A beam of light with a wavelength of 589 nanometers traveling in air makes an angle of incidence of 40° on a slab of transparent material, and the refracted beam makes an angle of refraction of 26°. Find the index of refraction of the material.Light sound, and other waves travel at different
A light ray with a wavelength of 589 nanometers (produced by a sodium lamp) traveling through air makes an angle of incidence of 30° on a smooth, flat slab of crown glass. Find the angle of refraction.Light sound, and other waves travel at different speeds, depending on the media (air, water,
A light beam passes through a thick slab of material whose index of refraction is n2.Show that the emerging beam is parallel to the incident beam.Light sound, and other waves travel at different speeds, depending on the media (air, water, wood, and so on) through which they pass. Suppose that light
Brewster's Law If the angle of incidence and the angle of refraction are complementary angles, the angle of incidence is referred to as the Brewster angle θB. The Brewster angle is related to the index of refractions of the two media, n1 and n2, by the equation n1 sin θB = n2 cos θB, where n1 is
Provide a justification as to why no further points of intersection (and therefore solutions) exist in Figure 25 for x < -π or x > 4π.Figure 25 Y, = 5 sin x + x %3D 14 Y2 = 3 4т %3D -8
(a) If you can throw a baseball with an initial speed of 40 meters per second, at what angle of elevation θ should you direct the throw so that the ball travels a distance of 110 meters before striking the ground?(b) Determine the maximum distance that you can throw the ball.(c) Graph R = R(θ),
The horizontal distance that a projectile will travel in the air (ignoring air resistance) is given by the equationwhere υ0 is the initial velocity of the projectile, θ is the angle of elevation, and g is acceleration due to gravity (9.8 meters per second squared).(a) If you can throw a baseball
Two hallways, one of width 3 feet, the other of width 4 feet, meet at a right angle. See the illustration. It can be shown that the length L of the ladder as a function of θ is L(θ) = 4 csc θ + 3sec θ.(a) In calculus, you will be asked to find the length of the longest ladder that can turn the
In the study of heat transfer, the equation x + tan x = 0 occurs. Graph Y1 = -x and Y2 = tan x for x ≥ 0. Conclude that there are an infinite number of points of intersection of these two graphs. Now find the first two positive solutions of x + tan x = 0 rounded to two decimal places.
A golfer hits a golf ball with an initial velocity of 100 miles per hour. The range R of the ball as a function of the angle 0 to the horizontal is given by R(θ) = 672 sin(2θ), where R is measured in feet.(a) At what angle θ should the ball be hit if the golfer wants the ball to travel 450 feet
An airplane is asked to stay within a holding pattern near Chicago’s O’Hare International Airport. The function d(x) = 70 sin(0.65x) + 150 represents the distance d, in miles, of the airplane from the airport at time x, in minutes.(a) When the plane enters the holding pattern, x = 0 how far is
In 1893, George Ferris engineered the Ferris Wheel. It was 250 feet in diameter. If the wheel makes 1 revolution every 40 seconds, then the functionrepresents the height h, in feet, of a seat on the wheel as a function of time t, where t is measured in seconds. The ride begins when t = 0.(a) During
Blood pressure is a way of measuring the amount of force exerted on the walls of blood vessels. It is measured using two numbers: systolic (as the heart beats) blood pressure and diastolic (as the heart rests) blood pressure. Blood pressures vary substantially from person to person, but a typical
(a) Graph f(x) = 2 sin x and g(x) = -2 sin x + 2 on the same Cartesian plane for the interval [0, 2π].(b) Solve f(x) = g(x) on the interval [0, 2π] and label the points of intersection on the graph drawn in part (b).(c) Solve f(x) > g(x) on the interval [0, 2π].(d) Shade the region bounded by
(a) Graph f(x) = -4 cos x and g(x) = 2 cos x + 3 on the same Cartesian plane for the interval [0, 2π].(b) Solve f(x) = g(x) on the interval [0, 2π] and label the points of intersection on the graph drawn in part (b).(c) Solve f(x) > g(x) on the interval [0, 2π].(d) Shade the region bounded by
(a) Graph f(x) = 2 cos x/2 + 3 and g(x) = 4 on the same Cartesian plane for the interval [0, 4π].(b) Solve f(x) = g(x) on the interval [0, 4π] and label the points of intersection on the graph drawn in part (b).(c) Solve f(x) < g(x) on the interval [0, 4π].(d) Shade the region bounded by f(x)
(a) Graph f(x) = 3 sin(2x) + 2 and g(x) = 7/2 on the same Cartesian plane for the interval [0, π].(b) Solve f(x) = g(x) on the interval [0, π] and label the points of intersection on the graph drawn in part (b).(c) Solve f(x) > g(x) on the interval [0, π].(d) Shade the region bounded by f(x)
f(x) = cot x(a) Solve f(x) = -√3.(b) For what values of x is f(x) > - √3 on the interval (0, π)?
f(x) = 4 tan x(a) Solve f(x) = -4.(b) For what values of x is f(x) < -4 on the interval (-π/2, π/2)?
f(x) = 2 cos x(a) Find the zeros of f on the interval [-2π, 4π].(b) Graph f(x) = 2 cos x on the interval [-2π, 4π].(c) Solve f(x) = -√3 on the interval [-2π, 4π]. What points are on the graph of f? Label these points on the graph drawn in part (b).(d) Use the graph drawn in part (b) along
f(x) = 3 sin x(a) Find the zeros of f on the interval [-2π, 4π].(b) Graph f (x) = 3 sin x on the interval [-2π, 4π](c) Solve f(x) = -3/2 on the interval [-2π, 4π]. What points are on the graph of f? Label these points on the graph drawn in part (b).(d) Use the graph drawn in part (b) along
What are the zeros of f(x) = 2 cos (3x) + 1 on the interval [0, π]?
What are the zeros of f(x) = 4 sin2 x - 3 on the interval [0, 2π]?
Use a graphing utility to solve equation. Express the solution(s) rounded to two decimal places.4 cos(3x) - ex = 1, x > 0
Use a graphing utility to solve equation. Express the solution(s) rounded to two decimal places.6 sin x - ex = 2, x > 0
Use a graphing utility to solve equation. Express the solution(s) rounded to two decimal places.x2 = x + 3 cos(2x)
Use a graphing utility to solve equation. Express the solution(s) rounded to two decimal places.x2 - 2 sin(2x) = 3x
Use a graphing utility to solve equation. Express the solution(s) rounded to two decimal places.x2 + 3 sin x = 0
Use a graphing utility to solve equation. Express the solution(s) rounded to two decimal places.x2 - 2 cos x = 0
Use a graphing utility to solve equation. Express the solution(s) rounded to two decimal places.19x + 8 cos x = 2
Use a graphing utility to solve equation. Express the solution(s) rounded to two decimal places.22x - 17 sin x = 3
Use a graphing utility to solve equation. Express the solution(s) rounded to two decimal places.x - 4 sin x = 0
Use a graphing utility to solve equation. Express the solution(s) rounded to two decimal places.x + 5 cos x = 0
Solve equation on the interval 0 ≤ θ ≤ 2π.sec θ = tan θ + cot θ
Solve equation on the interval 0 ≤ θ ≤ 2π.sec2 θ + tan θ = 0
Solve equation on the interval 0 ≤ θ ≤ 2π.csc2 θ = cot θ + 1
Solve equation on the interval 0 ≤ θ ≤ 2π.tan2 θ = 3/2sec θ
Solve equation on the interval 0 ≤ θ ≤ 2π.4(1 + sin θ) = cos2 θ
Solve equation on the interval 0 ≤ θ ≤ 2π.3(1 - cos θ) = sin2 0
Solve equation on the interval 0 ≤ θ ≤ 2π.2 cos2 θ - 7cos θ - 4 = 0
Solve equation on the interval 0 ≤ θ ≤ 2π.2 sin2 θ - 5sin θ + 3 = 0
Solve equation on the interval 0 ≤ θ ≤ 2π.sin2 θ = 2cos θ + 2
Solve equation on the interval 0 ≤ θ ≤ 2π.1 + sin θ = 2cos2 θ
Solve equation on the interval 0 ≤ θ ≤ 2π.tan θ = cot θ
Solve equation on the interval 0 ≤ θ ≤ 2π.tan θ = 2sin θ
Solve equation on the interval 0 ≤ θ ≤ 2π.cos θ - sin(-θ) = 0
Solve equation on the interval 0 ≤ θ ≤ 2π.cos θ = -sin(-θ)
Solve equation on the interval 0 ≤ θ ≤ 2π.2sin2 θ = 3(1 - cos(-θ))
Solve equation on the interval 0 ≤ θ ≤ 2π.sin2 θ = 6(cos(-θ) + 1)
Solve equation on the interval 0 ≤ θ ≤ 2π.cos2 θ - sin2 θ + sin θ = 0
Solve equation on the interval 0 ≤ θ ≤ 2π.sin2 θ - cos2 θ = 1 + cos θ
Solve equation on the interval 0 ≤ θ ≤ 2π.(cot θ + 1)(csc θ – 1/2) = 0
Solve equation on the interval 0 ≤ θ ≤ 2π.(tan θ - 1)(sec θ - 1) = 0
Solve equation on the interval 0 ≤ θ ≤ 2π.2cos2 θ + cos θ - 1 = 0
Solve equation on the interval 0 ≤ θ ≤ 2π.2sin2 θ - sin θ - 1 = 0
Solve equation on the interval 0 ≤ θ ≤ 2π.sin2 θ - 1 = 0
Solve equation on the interval 0 ≤ θ ≤ 2π.2cos2 θ + cos θ = 0
Use a calculator to solve equation on the interval 0 ≤ θ ≤ 2π. Round answers to two decimal places.4 cos θ + 3 = 0
Use a calculator to solve equation on the interval 0 ≤ θ ≤ 2π. Round answers to two decimal places.3 sin θ -2 = 0
Use a calculator to solve equation on the interval 0 ≤ θ ≤ 2π. Round answers to two decimal places.4 cot θ = -5
Use a calculator to solve equation on the interval 0 ≤ θ ≤ 2π. Round answers to two decimal places.5 tan θ + 9 = 0
Use a calculator to solve equation on the interval 0 ≤ θ ≤ 2π. Round answers to two decimal places.csc θ = -3
Use a calculator to solve equation on the interval 0 ≤ θ ≤ 2π. Round answers to two decimal places.sec θ = -4
Use a calculator to solve equation on the interval 0 ≤ θ ≤ 2π. Round answers to two decimal places.sin θ = -0.2
Use a calculator to solve equation on the interval 0 ≤ θ ≤ 2π. Round answers to two decimal places.cos θ = -0.9
Use a calculator to solve equation on the interval 0 ≤ θ ≤ 2π. Round answers to two decimal places.cot θ = 2
Use a calculator to solve equation on the interval 0 ≤ θ ≤ 2π. Round answers to two decimal places.tan θ = 5
Use a calculator to solve equation on the interval 0 ≤ θ ≤ 2π. Round answers to two decimal places.cos θ = 0.6
Use a calculator to solve equation on the interval 0 ≤ θ ≤ 2π. Round answers to two decimal places.sin θ = 0.4
Solve equation. Give a general formula for all the solutions. List six solutions.tan θ/2 = -1
Solve equation. Give a general formula for all the solutions. List six solutions.sin θ/2 = -√3/2
Solve equation. Give a general formula for all the solutions. List six solutions.sin (2θ) = -1
Solve equation. Give a general formula for all the solutions. List six solutions.cos (2θ) = -1/2
Solve equation. Give a general formula for all the solutions. List six solutions.sin θ = √2/2
Solve equation. Give a general formula for all the solutions. List six solutions.cos θ = 0
Solve equation. Give a general formula for all the solutions. List six solutions.cos θ = -√3/2
Solve equation. Give a general formula for all the solutions. List six solutions.tan θ = -√3/3
Solve equation. Give a general formula for all the solutions. List six solutions.tan θ = 1
Solve equation. Give a general formula for all the solutions. List six solutions.sin θ = 1/2
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