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mathematics
precalculus
Questions and Answers of
Precalculus
When we estimate distances from velocity data, it is sometimes necessary to use times t0, t1, t2, t3, . . . that are not equally spaced. We can still estimate distances using the time periods Dti =
Evaluate the upper and lower sums for f (x) = 1 + x2, -1 < x < 1, with n = 3 and 4. Illustrate with diagrams like Figure 14. УА 3. y = дх (4, 0)/ (5, 0) (-4, 0) х 0. (-5, 0) -- - --
(a) By reading values from the given graph of f, use five rectangles to find a lower estimate and an upper estimate for the area under the given graph of f from x = 0 to x = 10. In each case sketch
If a metal ball with mass m is projected in water and the force of resistance is proportional to the square of the velocity, then the distance the ball travels in time t iswhere c is a positive
If an electrostatic field E acts on a liquid or a gaseous polar dielectric, the net dipole moment P per unit volume isShow that e + e-* eE – e-E P(E) =- limo P(E) = 0. E-o+
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
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
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
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
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
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,
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
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
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
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
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
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,
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 +
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.
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
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
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
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
(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
(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
(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
(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
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
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
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
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