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mathematics
calculus 4th

Calculus 4th Edition Jon Rogawski, Colin Adams, Robert Franzosa - Solutions

Calculate the approximation for the given function and interval. L6, f(x)= √6x + 2, [1,3]
Write the integral in terms of u and du. Then evaluate. Sx√4. x√4x - 1dx, u = 4x-1
Evaluate the integral using FTC I. S 15/2 dt
Find formulas for the functions represented by the integrals. J 1 t dt
Calculate the indefinite integral. S (4x² - 2x²) dx
A small boutique produces wool sweaters at a marginal cost of 40 − 5[x/5] for 0 ≤ x≤ 20, where [x] is the greatest integer function. Find the cost of producing 20 sweaters. Then compute the average cost of the first 10 sweaters and the last 10 sweaters.
Calculate the approximation for the given function and interval. R6, f(x)=2x-x², [0, 2]
Describe the partition P and the set of sample points C for the Riemann sum shown in Figure 17. Compute the value of the Riemann sum. 34.25 20 15 8- 0.5 1 2 2.5 3 3.2 4.5 5 x
Evaluate the integral using FTC I. S dt 1²
Find formulas for the functions represented by the integrals. x/4 Jx/2 sec² u du
The rate (in liters per minute) at which water drains from a tank is recorded at half-minute intervals. Compute the average of the left- and right-endpoint approximations to estimate the total amount of water drained during the first 3 min. t (min) r (L/min) 0 0.5 1 1.5 2 50 48 46 2.5 3 44 42 40 38
Calculate the indefinite integral. S.x² x2/4 dx
Calculate the approximation for the given function and interval. R₁, f(x) = x² + x, [-1,1]
Calculate the Riemann sum R(ƒ, P,C) for the given function, partition, and choice of sample points. Also, sketch the graph of ƒ and the rectangles corresponding to R(ƒ, P,C). f(x) = x, P = {1, 1.2, 1.5, 2), C = {1.1, 1.4, 1.9}
Evaluate the integral using FTC I. S xªdx -4
Write the integral in terms of u and du. Then evaluate. S sir sin(407) de, u = 40-7
Find formulas for the functions represented by the integrals. 3/2 J3√x t³ dt
The velocity of a car is recorded at half-second intervals (in feet per second). Use the average of the left- and right-endpoint approximations to estimate the total distance traveled during the first 4 s. 0 0.5 v(t) 0 12 t 11 1.5 20 29 2 2.5 3 3.5 4 38 44 32 35 30
Calculate the indefinite integral. S sin(0 - 8) de
Calculate the approximation for the given function and interval. M4, f(x) = 1 x² +1' [1, 5]
Calculate the Riemann sum R(ƒ, P,C) for the given function, partition, and choice of sample points. Also, sketch the graph of ƒ and the rectangles corresponding to R(ƒ, P,C). f(x) = 2x + 3, P = {-4,-1, 1, 4, 8), C = {-3, 0, 2,5}
Write the integral in terms of u and du. Then evaluate. S si sin 0 cos³ 0 d0, u = Cos 0
Find formulas for the functions represented by the integrals. -2x sec² tdt
To model the effects of a carbon tax on CO2 emissions, policymakers study the marginal cost of abatement B(x), defined as the cost of increasing CO2 reduction from x to x + 1 tons (in units of 10,000 tons— Figure 4). Which quantity is represented by the area under the curve over [0, 3] in Figure
Calculate the indefinite integral. S cos(5 - 70) de
Calculate the approximation for the given function and interval. M4, f(x)=√√x, [3,5]
Calculate the Riemann sum R(ƒ, P,C) for the given function, partition, and choice of sample points. Also, sketch the graph of ƒ and the rectangles corresponding to R(ƒ, P,C). f(x) = x² + x, P = {(2,3,4.5, 5), C = {2, 3.5,5}
Write the integral in terms of u and du. Then evaluate. fse secx tan xdx, xtanxdx, u=tanx
Evaluate the integral using FTC I. 1 +3 1-2 x dx
The snowfall rate R (in inches per hour) was tracked during a major 24-hour lake effect snowstorm in Buffalo, New York. The graph in Figure 5 shows R as a function of t (hours) during the storm. What quantity does ∫240 R(t) dt represent? Approximate the integral. 4 3 2 1 R A 0 5 10 X X X X
Verify ∫x0|t| dt = 1/2 x|x|.Consider x ≥ 0 and x ≤ 0 separately.
Calculate the indefinite integral. -4 (4t-³ - 12t-¹)dt
Calculate the approximation for the given function and interval. L4, f(x) = cos²x, []
Calculate the Riemann sum R(ƒ, P,C) for the given function, partition, and choice of sample points. Also, sketch the graph of ƒ and the rectangles corresponding to R(ƒ, P,C). f(x) = sinx, P = {0,,,), C = {0.4, 0.7, 1.2}
Write the integral in terms of u and du. Then evaluate. x sec²(x²) dx, u = x²
Evaluate the integral using FTC I. S² (x²-x²) dx
Calculate the indefinite integral. S(91-²/3 +41713) dt
Verify ∫x0|t|3 dt = 1/4 x|x|3. Consider x ≥ 0 and x ≤ 0 separately.
Figure 6 shows the migration rate M(t) of Ireland in the period 1988–1998. This is the rate at which people (in thousands per year) moved into or out of the country.(a) Is the following integral positive or negative? What does this quantity represent?(b) Did migration in the period 1988–1998
Calculate the approximation for the given function and interval. L6, f(x) = x² + 3x], [-2,1]
Calculate the Riemann sum R(ƒ, P,C) for the given function, partition, and choice of sample points. Also, sketch the graph of ƒ and the rectangles corresponding to R(ƒ, P,C). f(x) = x² + x, P = {0, 1, 2.5, 3.2, 5}, C = {0.5, 2, 3, 4.5}
Write the integral in terms of u and du. Then evaluate. fsec sec’(cos x)sin xdx, u = cosx
Evaluate the integral using FTC I. 1P 7/1-¹ S
Evaluate the integral using FTC I. -27 t+1 √t dt
Calculate the indefinite integral. ၆sec sec² x dx
Let N(d) be the number of asteroids of diameter ≤ d kilometers. Data suggest that the diameters are distributed according to a piecewise power law:(a) Compute the number of asteroids with a diameter between 0.1 and 100 km.(b) Using the approximation N(d + 1) − N(d) ≈ N (d), estimate the
In Example 4, approximate the net APC energy use from midnight to noon. EXAMPLE 4 Let A be the area under the graph of f(x) = 2x²-x+ 3 over [2, 4] (Figure 12). Compute A as the limit lim RN. N→∞
Write the sum in summation notation.47 + 57 + 67 + 77 + 87
Evaluate the integral in the form a sin(u(x)) + C for an appropriate choice of u(x) and constant a. x³ cos(x) dx
Evaluate the integral using FTC I. J8/27 1014/3811/3 1² - dt
Calculate the indefinite integral. Sta tan 30 sec 30 de
Write the sum in summation notation. (2²+ 2) + (3² + 3) + (4² + 4) + (5²+5)
In Example 4, approximate the net APC energy use from noon to midnight. EXAMPLE 4 Let A be the area under the graph of f(x) = 2x²-x+ 3 over [2, 4] (Figure 12). Compute A as the limit lim RN. N→∞
The heat capacity C(T) of a substance is the amount of energy (in joules) required to raise the temperature of 1 g by 1°C at temperature T.(a) Explain why the energy required to raise the temperature from T1 to T2 is the area under the graph of C(T) over [T1, T2].(b) How much energy is required
Evaluate the integral in the form a sin(u(x)) + C for an appropriate choice of u(x) and constant a. [x² cos(x³ + 1) dx
Evaluate the integral in the form a sin(u(x)) + C for an appropriate choice of u(x) and constant a. Sx x¹/2 cos(x³/2) dx
Evaluate the integral using FTC I. S J-π/4 sin Ꮎ dᎾ
Calculate the indefinite integral. (y + 2)¹ dy
Write the sum in summation notation. (2² + 2) + (2³ + 2) + (24 + 2) + (25+2)
Figure 7 shows the rate R(t) of natural gas consumption (billions of cubic feet per day) in the mid-Atlantic states (New York, New Jersey, Pennsylvania). Express the total quantity of natural gas consumed in 2009 as an integral (with respect to time t in days). Then estimate this quantity, given
Sketch the signed area represented by the integral. Indicate the regions of positive and negative area. Soc Jo (4x - x²) dx
Evaluate the integral in the form a sin(u(x)) + C for an appropriate choice of u(x) and constant a. Sco cos x cos(sin x) dx
Evaluate the integral using FTC I. -13л 10 sin x dx
Calculate the indefinite integral. 3x³-9 x² dx
Cardiac output is the rate R of volume of blood pumped by the heart per unit time (in liters per minute).Doctors measure R by injecting A mg of dye into a vein leading into the heart at t = 0 and recording the concentration c(t) of dye (in milligrams per liter) pumped out at short regular time
Write the sum in summation notation. 1+ 1 + 2 + + + 16 32
Sketch the signed area represented by the integral. Indicate the regions of positive and negative area. π/4 J-π/4 tan x dx
Evaluate the integral using FTC I. WT/3 0 cost dt
Let A(x) = ∫x0 ƒ(t) dt for ƒ(x) in Figure 8.(a) Calculate A(2), A(3), A'(2), and A'(3).(b) Find formulas for A(x) on [0, 2] and [2, 4], and sketch the graph of A. 43 2 1 y 1 y = f(x) ... 2 3 نا 4 - x
Calculate the indefinite integral. (cos 0 - 0) de
Write the sum in summation notation. 1 2.3 + 2 3.4 + n (n + 1)(n + 2)
A study suggests that the extinction rate r(t) of marine animal families during the Phanerozoic Eon can be modeled by the function r(t) = 3130/(t + 262) for 0 ≤ t ≤ 544, where t is time elapsed (millions of years) since the beginning of the eon 544 million years ago. Thus, t = 544 refers to the
Sketch the signed area represented by the integral. Indicate the regions of positive and negative area. K 2r sin x dx
Evaluate the integral using FTC I. T/6 10 sec 0 tan 0 de
Calculate the indefinite integral. sec² (12-250) de
Express the area under the graph as a limit using the approximation indicated (in summation notation), but do not evaluate. LN, f(x) = cos x over [3,4]
Combine to write as a single integral: + [ f(x) dx + f f(x) dx -2 8 * f f(x) dx +
A water balloon is dropped from a high building. It falls for 5 seconds before hitting the ground. Determine the velocity it is traveling when it is about to hit the ground, assuming an acceleration due to gravity of −9.8 m/s2 and no wind resistance.
Express the area under the graph as a limit using the approximation indicated (in summation notation), but do not evaluate. MN, f(x) = tan x over [1,1]
Express the area under the graph as a limit using the approximation indicated (in summation notation), but do not evaluate. MN, f(x) = x2 over [3,5]
Let A(x) =∫x0 ƒ(x) dx, where f is the function shown in Figure 4. Identify the location of the local minima, the local a maxima, and points of inflection of A on the interval [0, E], as well as the intervals where A is increasing, decreasing, concave up, or concave down. Where does the
A hammer is dropped and it falls for 2 seconds before hitting the ground. Determine how far it falls, assuming an acceleration due to gravity of −9.8 m/s2 and no wind resistance.
Evaluate by interpreting it as the area of part of a familiar geometric figure. 1 lim N-0 N N j=1 1- N 2
Find the local minima, the local maxima, and the inflection points of A(x) = S tdt (t² + 1)²
A mass oscillates at the end of a spring. Let s(t) be the displacement of the mass from the equilibrium position at time t. Assuming that the mass is located at the origin at t = 0 and has velocity v(t) = sin t m/s, state the differential equation with initial condition satisfied by s(t), and find
Let ƒ(x) = x2 and let RN, LN, and MN be the approximations for the interval [0, 1]. Show that RN = - 1 1 + 3 2N + 1 6N² 1 Interpret the quantity + 2N 1 6N² as the area of a region.
A particle starts at the origin at time t = 0 and moves with velocity v(t) as shown in Figure 5.(a) How many times does the particle return to the origin in the first 12 seconds?(b) What is the particle’s maximum distance from the origin?(c) What is the particle’s maximum distance to the left
Let ƒ(x) = x2 and let RN, LN, and MN be the approximations for the interval [0, 1]. Show thatThen, given that the area under the graph of y = x2 over [0, 1] is 1/3, rank the three approximations RN, LN, and MN in order of increasing accuracy (use Exercise 74).Data From Exercise 74Let ƒ(x) =
Beginning at t = 0 with initial velocity 4 m/s, a particle moves in a straight line with acceleration a(t) = 3t1/2 m/s2. Find the distance traveled after 25 s.
For the function f illustrated in Figure 6, do the following: 8 4 0 -4 -8 y = f(x). 5 . 10
Use the Comparison Theorem to show that THEOREM 5 Comparison Theorem If f and g are integrable and g(x) ≤ f(x) for x in [a,b], then [² 8(x) dx = ["^/(x). dx
At time t = 0 a car traveling 25 m/s begins to accelerate at a constant rate of −4 m/s2. After how many seconds does the car come to a stop and how far will the car have traveled between t = 0 and the time it stopped?
Let The sine integral function Si is an area function defined by Si(x) = ∫x 0 ƒ(t) dt.(a) Explain why Si has critical points at nπ for all nonzero integers n.(b) Use Riemann sums to approximate Si(x) for x = π, 2π, . . . , 8π and sketch a graph of Si(x) for 0 ≤ x ≤ 8π. f(x)
For each of RN, LN, and MN, find the smallest integer N for which the error is less than 0.001.
Use the Change of Variables Formula to evaluate the definite integral. S x√x² +9dx
At time t = 1 second, a particle is traveling at 72 m/s and begins to accelerate at the rate a(t) = −t−1/2 until it stops. How far does the particle travel from t = 1 until the time it stopped?
Use the Graphical Insight on page 290 to obtain bounds on the area. Let A be the area under ƒ(x) = √x over [0, 1]. Prove that 0.51 ≤ A ≤ 0.77 by computing R4 and L4. Explain your reasoning.
A 900-kg rocket is released from a space station. As it burns fuel, the rocket’s mass decreases and its velocity increases. Let v(m) be the velocity (in meters per second) as a function of mass m. Find the velocity when m = 729 kg if dv/dm = −50m−1/2. Assume that v(900) = 0 m/s.
Use the Change of Variables Formula to evaluate the definite integral. S 4x + 12 (x² + 6x + 1)² dx
On a typical day, a city consumes water at the rate of r(t) = 100 + 72t − 3t2 (in thousands of gallons per hour), where t is the number of hours past midnight. What is the daily water consumption? How much water is consumed between 6 pm and midnight?

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