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mathematics
calculus early transcendentals
Calculus Early Transcendentals 2nd edition William L. Briggs, Lyle Cochran, Bernard Gillett - Solutions
With snow on the ground and falling at a constant rate, a snowplow began plowing down a long straight road at noon. The plow traveled twice as far in the first hour as it did in the second hour. At what time did the snow start falling? Assume the plowing rate is inversely proportional to the depth
Suppose that r(t) = r0 e-kt, with k > 0, is the rate at which a nation extracts oil, where r0 = 107 barrels/yr is the current rate of extraction. Suppose also that the estimate of the total oil reserve is 2 × 109 barrels. a. Find Q(t), the total amount of oil extracted by the nation after
A 2000-liter cistern is empty when water begins flowing into it (at t = 0) at a rate (in L/min) given by Q'(t) = 3√t, where t is measured in minutes.a. How much water flows into the cistern in 1 hour?b. Find and graph the function that gives the amount of water in the tank at any time t ≥ 0.c.
A strong west wind blows across a circular running track. Abe and Bess start running at the south end of the track, and at the same time, Abe starts running clockwise and Bess starts running counterclockwise. Abe runs with a speed (in units of mi/hr) given by u(φ) = 3 - 2 cos φ and Bess runs with
At noon (t = 0), Alicia starts running along a long straight road at 4 mi/hr. Her velocity decreases according to the function v(t) = 4/(t + 1), for t ≥ 0. At noon, Boris also starts running along the same road with a 2-mi head start on Alicia; his velocity is given by u(t) = 2/(t + 1), for t ≥
Theo and Sasha start at the same place on a straight road, riding bikes with the following velocities (measured in mi/hr). Assume t is measured in hours. Theo: vT(t) = 10, for t ≥ 0 Sasha: vS(t) = 15t, for 0 ≤ t ≤ 1 and vS(t) = 15, for t > 1 a. Graph the velocity functions
Kelly started at noon (t = 0) riding a bike from Niwot to Berthoud, a distance of 20 km, with velocity v(t) = 15/(t + 1)2 (decreasing because of fatigue). Sandy started at noon (t = 0) riding a bike in the opposite direction from Berthoud to Niwot with velocity u(t) = 20/(t + 1)2 (also decreasing
Consider the following velocity functions. In each case, complete the sentence: The same distance could have been traveled over the given time period at a constant velocity of ________.v(t) = t(25 - t2)1/2, for 0 ≤ t ≤ 5
Consider the following velocity functions. In each case, complete the sentence: The same distance could have been traveled over the given time period at a constant velocity of ________.v(t) = 2 sin t, for 0 ≤ t ≤ π
Consider the following velocity functions. In each case, complete the sentence: The same distance could have been traveled over the given time period at a constant velocity of ________.v(t) = 1 - t2/16, for 0 ≤ t ≤ 4
Consider the following velocity functions. In each case, complete the sentence: The same distance could have been traveled over the given time period at a constant velocity of ________.v(t) = 2t + 6, for 0 ≤ t ≤ 8
The figures show velocity functions for motion along a straight line. Assume the motion begins with an initial position of s(0) = 0. Determine the following:a. The displacement between t = 0 and t = 5b. The distance traveled between t = 0 and t = 5c. The position at t = 5 d. A piecewise function
The figures show velocity functions for motion along a straight line. Assume the motion begins with an initial position of s(0) = 0. Determine the following:a. The displacement between t = 0 and t = 5b. The distance traveled between t = 0 and t = 5c. The position at t = 5 d. A piecewise function
Determine whether the following statements are true and give an explanation or counterexample. a. The distance traveled by an object moving along a line is the same as the displacement of the object.b. When the velocity is positive on an interval, the displacement and the distance traveled on
Consider the following marginal cost functions.a. Find the additional cost incurred in dollars when production is increased from 100 units to 150 units.b. Find the additional cost incurred in dollars when production is increased from 500 units to 550 units.C'(x) = 3000 - x - 0.001x2
Consider the following marginal cost functions.a. Find the additional cost incurred in dollars when production is increased from 100 units to 150 units.b. Find the additional cost incurred in dollars when production is increased from 500 units to 550 units.C'(x) = 300 + 10x - 0.01x2
Consider the following marginal cost functions.a. Find the additional cost incurred in dollars when production is increased from 100 units to 150 units.b. Find the additional cost incurred in dollars when production is increased from 500 units to 550 units.C'(x) = 200 - 0.05x
Consider the following marginal cost functions.a. Find the additional cost incurred in dollars when production is increased from 100 units to 150 units.b. Find the additional cost incurred in dollars when production is increased from 500 units to 550 units.C'(x) = 2000 - 0.5x
The daily discharge of the Spokane River as it flows through Spokane, Washington, in April and June is modeled by the functionsr1(t) = 0.25t2 + 37.46t + 722.47 (April) andr2(t) = 0.90t2 - 69.06t + 2053.12 (June),where the discharge is measured in millions of cubic feet per day and t = 1 corresponds
A culture of bacteria in a Petri dish has an initial population of 1500 cells and grows at a rate (in cells/day) of N'(t) = 100e-0.25t. Assume t is measured in days.a. What is the population after 20 days? After 40 days?b. Find the population N(t) at any time t ≥ 0.
The population of a community of foxes is observed to fluctuate on a 10-year cycle due to variations in the availability of prey. When population measurements began (t = 0), the population was 35 foxes. The growth rate in units of foxes/year was observed to bea. What is the population 15 years
When records were first kept (t = 0), the population of a rural town was 250 people. During the following years, the population grew at a rate of P'(t) = 30(1 + √t), where t is measured in years.a. What is the population after 20 years?b. Find the population P(t) at any time t ≥ 0.
Starting with an initial value of P(0) = 55, the population of a prairie dog community grows at a rate of P'(t) = 20 - t/5 (prairie dogs/month), for 0 ≤ t ≤ 200, where t is measured in months.a. What is the population 6 months later?b. Find the population P(t), for 0 ≤ t ≤ 200.
An oil refinery produces oil at a variable rate given bywhere t is measured in days and Q is measured in barrels.a. How many barrels are produced in the first 35 days?b. How many barrels are produced in the first 50 days?c. Without using integration, determine the number of barrels produced over
The owners of an oil reserve begin extracting oil at time t = 0. Based on estimates of the reserves, suppose the projected extraction rate is given by Q'(t) = 3t2 (40 - t)2, where 0 ≤ t ≤ 40, Q is measured in millions of barrels, and t is measured in years. a. When does the peak extraction
At t = 0, a train approaching a station begins decelerating from a speed of 80 mi/hr according to the acceleration function a(t) = -1280(1 + 8t)-3, where t ≥ 0 is measured in hours. How far does the train travel between t = 0 and t = 0.2? Between t = 0.2 and t = 0.4? The units of acceleration are
A car slows down with an acceleration of a(t) = -15 ft/s2. Assume that v(0) = 60 ft/s, s(0) = 0, and t is measured in seconds.a. Determine and graph the position function, for t ≥ 0.b. How far does the car travel in the time it takes to come to rest?
A drag racer accelerates at a(t) = 88 ft/s2. Assume that v(0) = 0, s(0) = 0, and t is measured in seconds. a. Determine and graph the position function, for t ≥ 0.b. How far does the racer travel in the first 4 seconds?c. At this rate, how long will it take the racer to travel 1/4 mi?d. How
Find the position and velocity of an object moving along a straight line with the given acceleration, initial velocity, and initial position.
Find the position and velocity of an object moving along a straight line with the given acceleration, initial velocity, and initial position.a(t) = cos 2t, v(0) = 5, s(0) = 7
Find the position and velocity of an object moving along a straight line with the given acceleration, initial velocity, and initial position. a(t) (t + 2)2 (0) = 20, s(0) = 10
Find the position and velocity of an object moving along a straight line with the given acceleration, initial velocity, and initial position.a(t) = -0.01t, v(0) = 10, s(0) = 0
Find the limit of the following sequences or determine that the limit does not exist.
Find the limit of the following sequences or determine that the limit does not exist.
Find the limit of the following sequences or determine that the limit does not exist.
Find the limit of the following sequences or determine that the limit does not exist. n sin п п
Find the limit of the following sequences or determine that the limit does not exist.{n(1 - cos(1/n))}
Find the limit of the following sequences or determine that the limit does not exist.{ln sin (1/n) + ln n}
Find the limit of the following sequences or determine that the limit does not exist.{ln {n3 + 1) - ln (3n3 + 10n)}
Find the limit of the following sequences or determine that the limit does not exist.{bn}, where (n/(n + 1) if n < 5000 if n > 5000 п —п пе
Find the limit of the following sequences or determine that the limit does not exist. {(1 п п
Find the limit of the following sequences or determine that the limit does not exist.
Find the limit of the following sequences or determine that the limit does not exist. In (1/n) п
Find the limit of the following sequences or determine that the limit does not exist. п en + Зп
Find the limit of the following sequences or determine that the limit does not exist. {(1 +:)'} Зп {(« 4
Find the limit of the following sequences or determine that the limit does not exist. {V(1+=} 2n,
Find the limit of the following sequences or determine that the limit does not exist. {G, п п п+5
Find the limit of the following sequences or determine that the limit does not exist. {(1 - )}
Find the limit of the following sequences or determine that the limit does not exist.{n2/n}
Find the limit of the following sequences or determine that the limit does not exist. tan
Find the limit of the following sequences or determine that the limit does not exist.{√n2 + 1 - n}
Find the limit of the following sequences or determine that the limit does not exist.{tan-1 n}
Find the limit of the following sequences or determine that the limit does not exist. 9k² + 1) I + ¿¥6\
Find the limit of the following sequences or determine that the limit does not exist. 3"+1 + 3 3"
Find the limit of the following sequences or determine that the limit does not exist.
Find the limit of the following sequences or determine that the limit does not exist. Зп3 2n3 + 1)
Find the limit of the following sequences or determine that the limit does not exist. n12 п Зп12 + 4
Find the limit of the following sequences or determine that the limit does not exist. n3 .4 п + 1)
Explain how two sequences that differ only in their first ten terms can have the same limit.
Compare the growth rates of {n100} and {en/100} as n→∞.
Explain how the methods used to find the limit of a function as x→∞ are used to find the limit of a sequence.
For what values of r does the sequence {rn} converge? Diverge?
Consider the functions f(x) = a sin 2x and g(x) = (sin x)/a, where a > 0 is a real number.a. Graph the two functions on the interval [0, π/2], for a = 1/2, 1, and 2.b. Show that the curves have an intersection point x* (other than x = 0) on [0, π/2] that satisfies cos x* = 1/(2a2), provided a
Consider the functions f(x) = xn and g(x) = x1/n, where n ≥ 2 is a positive integer. a. Graph f and g for n = 2, 3, and 4, for x ≥ 0. b. Give a geometric interpretation of the area function
Assume f and g are even, integrable functions on [-a, a], where a > 1. Suppose f(x) > g(x) > 0 on [-a, a] and the area bounded by the graphs of f and g on [-a, a] is 10. What is the value of S(f(x?) – 8(x²)) dx? Va ·
Consider the cubic polynomial f(x) = x(x - a)(x - b), where 0 ≤ a ≤ b.a. For a fixed value of b, find the functionFor what value of a (which depends on b) is F(a) = 0?b. For a fixed value of b, find the function A(a) that gives the area of the region bounded by the graph of f and the x-axis
Find the area of the region bounded by the curve and the line x = 1 in the first quadrant. Express y in terms of x. 1 4y2 х 2y |
Determine the area of the shaded region bounded by the curve x2 = y4(1 - y3) (see figure). УА y, |x? = y^(1 – y³) 1 х -1
Graph the curves y = a2x3 and y = √x for various values of a > 0. Note how the area A(a) between the curves varies with a. Find and graph the area function A(a). For what value of a is A(a) = 16?
Graph the curves y = (x + 1)(x - 2) and y = ax + 1 for various values of a. For what value of a is the area of the region between the two curves a minimum?
Consider the parabola y = x2. Let P, Q, and R be points on the parabola with R between P and Q on the curve. Let ℓP, ℓQ, and ℓR be the lines tangent to the parabola at P, Q, and R, respectively (see figure). Let P' be the intersection point of ℓQ and ℓR, let Q' be the intersection point
A Lorenz curve is given by y = L(x), where 0 ≤ x ≤ 1 represents the lowest fraction of the population of a society in terms of wealth and 0 ≤ y ≤ 1 represents the fraction of the total wealth that is owned by that fraction of the society. For example, the Lorenz curve in the figure shows
Suppose a dartboard occupies the square {(x, y): 0 ≤ |x| ≤ 1, 0 ≤ |y| ≤ 1}. A dart is thrown randomly at the board many times (meaning it is equally likely to land at any point in the square). What fraction of the dart throws land closer to the edge of the board than the center?
For each region R, find the horizontal line y = k that divides R into two subregions of equal area.R is the region bounded by y = √x and y = x.
For each region R, find the horizontal line y = k that divides R into two subregions of equal area.R is the region bounded by y = 4 - x2 and the x-axis.
For each region R, find the horizontal line y = k that divides R into two subregions of equal area.R is the region bounded by y = 1 - |x - 1| and the x-axis
For each region R, find the horizontal line y = k that divides R into two subregions of equal area.R is the region bounded by y = 1 - x, the x-axis, and the y-axis.
Find the area of the following regions, expressing your results in terms of the positive integer n ≥ 2.Let An be the area of the region bounded by f(x) = x1/n and g(x) = xn on the interval [0, 1], where n is a positive integer. Evaluate and interpret the result. lim An >00
Find the area of the following regions, expressing your results in terms of the positive integer n ≥ 2.The region bounded by f(x) = x1/n and g(x) = xn, for x ≥ 0
Find the area of the following regions, expressing your results in terms of the positive integer n ≥ 2.The region bounded by f(x) = x and g(x) = x1/n, for x ≥ 0
Find the area of the following regions, expressing your results in terms of the positive integer n ≥ 2.The region bounded by f(x) = x and g(x) = xn, for x ≥ 0
Find the area of the regions shown in the following figures. (y – 2)2 х - 3 у %3D8 — х х ||
Find the area of the regions shown in the following figures. УА y = x? x = 2 sin? y
Find the area of the regions shown in the following figures. УА у %3D 8х у%39— х? у3х х
Find the area of the regions shown in the following figures. Ул y = 4V2x (y = 2x2 у %3D —4х + 6 х
Let f(x) = xp and g(x) = x1/q, where p > 1 and q > 1 are positive integers. Let R1 be the region in the first quadrant between y = f(x) and y = x and let R2 be the region in the first quadrant between y = g(x) and y = x. a. Find the area of R1 and R2 when p = q, and determine which
Use the most efficient strategy for computing the area of the following regions.The region in the first quadrant bounded by y = x-1, y = 4x, and y = x/4
Use the most efficient strategy for computing the area of the following regions.The region in the first quadrant bounded by х - and y y = 2. ||
Use the most efficient strategy for computing the area of the following regions.The region bounded by y = x2 - 4, 4y - 5x - 5 = 0, and y = 0, for y ≥ 0
Use the most efficient strategy for computing the area of the following regions.The region bounded by y = √x, y = 2x - 15, and y = 0
Use the most efficient strategy for computing the area of the following regions.The region bounded by y = x3, y = -x3, and 3y - 7x - 10 = 0
Use the most efficient strategy for computing the area of the following regions.The region bounded by x = y(y - 1) and y = x/3
Use the most efficient strategy for computing the area of the following regions.The region bounded by x = y(y - 1) and x = -y(y - 1)
Sketch the region and find its area.The region bounded by y = x2 - 2x + 1 and y = 5x - 9
Sketch the region and find its area.The region bounded by y = 2 and Vi - х*
Sketch the region and find its area.The region bounded by y = (x - 1)2 and y = 7x - 19
Sketch the region and find its area.The region bounded by y = sin x and y = x(x - π), for 0 ≤ x ≤ π
Determine whether the following statements are true and give an explanation or counterexample. a. The area of the region bounded by y = x and x = y2 can be found only by integrating with respect to x.b. The area of the region between y = sin x and y = cos x on the interval [0, π/2] is
Use any method (including geometry) to find the area of the following regions. In each case, sketch the bounding curves and the region in question.The region bounded by x = y2 - 4 and y = x/3
Use any method (including geometry) to find the area of the following regions. In each case, sketch the bounding curves and the region in question.The region between the line y = x and the curve y = 2x√1 - x2 in the first quadrant.
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