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mathematics
applied calculus
Applied Calculus 6th Edition Deborah Hughes Hallett, Patti Frazer Lock, Andrew M. Gleason, Daniel E. Flath, Sheldon P. Gordon, David O. Lomen, David Lovelock, William G. McCallum, Brad G. Osgood, Andrew Pasquale - Solutions
Let y = y(t) be the downward speed (in feet per second) of a skydiver after t seconds of free fall. This function satisfies the differential equation y′ = .2(160 - y), y(0) = 0. What is the skydiver’s acceleration when her downward speed is 60 feet per second?
A lake is stocked with 100 fish. Let f (t) be the number of fish after t months, and suppose that y = f (t) satisfies the differential equation y′ = .0004y(1000 - y). Figure 7 shows the graph of the solution to this differential equation. The graph is asymptotic to the line y = 1000, the maximum
Describe Euler’s method for approximating the solution of a differential equation.
L. F. Richardson proposed the following model to describe the spread of war fever. If y = f (t) is the percentage of the population advocating war at time t, the rate of change of f (t) at any time is proportional to the product of the percentage of the population advocating war and the percentage
Rework Exercise 13 for a metal with a constant of proportionality k = .2. Which rod cools faster, the rod with a constant of proportionality k = .1 or the rod with a constant of proportionality k = .2? What can you say about the effect of varying the constant of proportionality in a cooling
One or more initial conditions are given for each differential equation in the following exercises. Use the qualitative theory of autonomous differential equations to sketch the graphs of the corresponding solutions. Include a yz-graph if one is not already provided. Always indicate the constant
The differential equation y′ = .5(y - 1)(4 - y) has five types of solutions labeled A–E. For each of the following initial values, graph the solution of the differential equation and identify the type of solution. Use a small value of h, let t range from 0 to 4, and let y range from -1 to 5.(a)
Solve the given equation using an integrating factor. Take t > 0.y′ - 2y = e2t
Sketch the solutions of the differential equations in Exercises. In each case, also indicate the constant solutions.y′ = 2 cos y, y(0) = 0
Use the graph in Fig. 20 to sketch the solutions to the Gompertz growth equationsatisfying y(0) = 10 and y(0) = 150. dy dt || 1 10 y 100 In-
The differential equation y′ = et - 2y, y(0) = 1, has solution y = 1/3(2e-2t + et ). In the following table, fill in the second row with the values obtained from the use of a numerical method and the third row with the actual values calculated from the solution. What is the greatest difference
In economic theory, the following model is used to describe a possible capital investment policy. Let f(t) represent the total invested capital of a company at time t. Additional capital is invested whenever f(t) is below a certain equilibrium value E, and capital is withdrawn whenever f(t) exceeds
The differential equation y′ = 2ty + et2, y(0) = 5, has solution y = (t + 5)et2. In the following table, fill in the second row with the values obtained from the use of a numerical method and the third row with the actual values calculated from the solution. What is the greatest difference
A body was found in a room when the room’s temperature was 70 F. Let f (t) denote the temperature of the body t hours from the time of death. According to Newton’s law of cooling, f satisfies a differential equation of the form y′ = k(T - y).(a) Find T.(b) After several measurements of the
Solve the given equation using an integrating factor. Take t > 0.(1 + t)y′ + y = -1
Sketch the solutions of the differential equations in Exercises. In each case, also indicate the constant solutions.y′ = 5 + 4y - y2, y(0) = 1
Consider a certain commodity that is produced by many companies and purchased by many other firms. Over a relatively short period, there tends to be an equilibrium price p0 per unit of the commodity that balances the supply and the demand. Suppose that, for some reason, the price is different from
Derive the formula for the population in Example 3, if the population in 1995 was 2 million.Example 3In 2005, people in a country suffering from economic problems started to emigrate to other countries. Let P(t) denote the population of the country in millions t years after 2005. Sociologists
The graph of z = - 1/2 y ln(y/30) has the same general shape as the graph in Fig. 20 with relative maximum point at y ≈ 11.0364 and y-intercept at y = 30; y(0) = 1, y(0) = 20, and y(0) = 40. Sketch the solutions to the Gompertz growth equation dy [ dt || 1 y y ln 30 2
Solve the given equation using an integrating factor. Take t > 0.y′ = e-t(y + 1)
The fish population in a pond with carrying capacity 1000 is modeled by the logistic equationHere, N(t) denotes the number of fish at time t in years. When the number of fish reached 275, the owner of the pond decided to remove 75 fish per year.(a) Modify the differential equation to model the
Sketch the solutions of the differential equations in Exercises. In each case, also indicate the constant solutions.y′ = y2 + y, y(0) = - 1/3
A certain piece of news is being broadcast to a potential audience of 200,000 people. Let f (t) be the number of people who have heard the news after t hours. Suppose that y = f (t) satisfiesDescribe this initial-value problem in words. y' = .07(200,000 - y), y (0) = = 10.
In an experiment, a certain type of bacteria was being added to a culture at the rate of e0.03t + 2 thousand bacteria per hour. Suppose that the bacteria grow at a rate proportional to the size of the culture at time t, with constant of proportionality k = .45. Let P(t) denote the number of
Solve the given equation using an integrating factor. Take t > 0.6y′ + ty = t
Sketch the solutions of the differential equations in Exercises. In each case, also indicate the constant solutions.y′ = y2 - 2y + 1, y(0) = -1
Let f (t) be the size of a paramecium population after t days. Suppose that y = f (t) satisfies the differential equationDescribe this initial-value problem in words. y' = .003y (500 - y), y (0) = 20.
A continuous annuity is a steady stream of money that is paid to some person. Such an annuity may be established, for example, by making an initial deposit in a savings account and then making steady withdrawals to pay the continuous annuity. Suppose that an initial deposit of $5400 is made into a
Find a formula for P(t) in Exercise 17 if, initially, 10,000 bacteria were present in the culture.Exercise 17In an experiment, a certain type of bacteria was being added to a culture at the rate of e0.03t + 2 thousand bacteria per hour. Suppose that the bacteria grow at a rate proportional to the
Solve the given equation using an integrating factor. Take t > 0.ety′ + y = 1
Sketch the solutions of the differential equations in Exercises. In each case, also indicate the constant solutions.y′ = ln y, y(0) = 2
Solve the following differential equations with the given initial conditions.y′ = 2te-2y - e-2y, y(0) = 3
A company wishes to set aside funds for future expansion and so arranges to make continuous deposits into a savings account at the rate of $10,000 per year. The savings account earns 5% interest compounded continuously.(a) Set up the differential equation that is satisfied by the amount f (t) of
According to the National Kidney Foundation, in 1997 more than 260,000 Americans suffered from chronic kidney failure and needed an artificial kidney (dialysis) to stay alive. When the kidneys fail, toxic waste products such as creatinine and urea build up in the blood. One way to remove these
Let f (t) denote the amount of capital invested by a certain business firm at time t. The rate of change of invested capital, f′(t), is sometimes called the rate of net investment. The management of the firm decides that the optimum level of investment should be C dollars and that, at any time,
Solve the given equation using an integrating factor. Take t > 0.y′ + y = 2 - et
Sketch the solutions of the differential equations in Exercises. In each case, also indicate the constant solutions.y′ = 1 + cos y, y(0) = - 3/4
Solve the following differential equations with the given initial conditions.y′ = y2 - e3t y2, y(0) = 1
The slope field in Fig. 4(a) suggests that the solution curve of the differential equation y′ = t - y through the point (0, -1) is a straight line.(a) Assuming that this is true, find the equation of the line.(b) Verify that the function that you found in part (a) is a solution by plugging its
A single dose of iodine is injected intravenously into a patient. The iodine mixes thoroughly in the blood before any is lost as a result of metabolic processes (ignore the time required for this mixing process). Iodine will leave the blood and enter the thyroid gland at a rate proportional to the
Solve the following differential equations with the given initial conditions.3y2y′ = -sin t, y(π/2) = 1
Sketch the solutions of the differential equations in Exercises. In each case, also indicate the constant solutions.y′ = .4y2(1 - y), y(0) = -1, y(0) = .1, y(0) = 2
Solve the initial-value problem.ty′ + y = ln t, y(e) = 0, t > 0
You make an initial deposit of $500 in a savings account and plan on making future deposits at a gradually increasing annual rate given by 90t + 810 dollars per year, t years after the initial deposit. Assume that the deposits are made continuously and that interest is compounded continuously at
Sketch the solutions of the differential equations in Exercises. In each case, also indicate the constant solutions. y' 3 y+3² (0)=2
A certain drug is administered intravenously to a patient at the continuous rate of 5 milligrams per hour. The patient’s body removes the drug from the bloodstream at a rate proportional to the amount of the drug in the blood. Write a differential equation that is satisfied by the amount f (t) of
Solve the following differential equations with the given initial conditions.y′ = t2 e-3y, y(0) = 2
Solve the initial-value problem.y′ + 2y = 1, y(0) = 1
Sketch the solutions of the differential equations in Exercises. In each case, also indicate the constant solutions. y' = 1 2 y ²+1' y(0) = -1
When the breath is held, carbon dioxide (CO2) diffuses from the blood into the lungs at a steadily decreasing rate. Let P0 and Pb denote the pressure of CO2 in the lungs, respectively, in the blood at the moment when the breath is held. Suppose that Pb is constant during breath holding, and let
Solve the given equation using an integrating factor. Take t > 0. 1 Vt + 1 =y' + y = 1
The air in a crowded room full of people contains .25% carbon dioxide (CO2). An air conditioner is turned on that blows fresh air into the room at the rate of 500 cubic feet per minute. The fresh air mixes with the stale air, and the mixture leaves the room at the rate of 500 cubic feet per minute.
Solve the following differential equations with the given initial conditions.y2y′ = t cos t, y(0) = 2
A cool object is placed in a room that is maintained at a constant temperature of 20°C. The rate at which the temperature of the object rises is proportional to the difference between the room temperature and the temperature of the object. Let y = f (t) be the temperature of the object at time t;
Radium 226 is a radioactive substance with a decay constant .00043. Suppose that radium 226 is being continuously added to an initially empty container at a constant rate of 3 milligrams per year. Let P(t) denote the number of grams of radium 226 remaining in the container after t years.(a) Find an
A company arranges to make continuous deposits into a savings account at the rate of P dollars per year. The savings account earns 5% interest compounded continuously. Find the approximate value of P that will make the savings account balance amount to $50,000 in 4 years.
Using Figure 5.89, list from least to greatest,(a) f'(1).(b) The average value of f(x) on 0 ≤ x ≤ a.(c) The average value of the rate of change of f(x), for 0 ≤ x ≤ a.(d) ∫a0f(x) dx. 1 f(x) Figure 5.89 a +x 2
At 11:57 pm on March 12, 1928, the two-year-old St Francis dam on the outskirts of Los Angeles failed catastrophically and the dam emptied. The resulting flood was one of the worst US civil engineering disasters of the 20th century, claiming over 400 lives. The volume of water discharging from the
Using Figure 5.32, find the value of ∫61 f(x) dx. 3 2 1 f(x) 1 2 3 4 5 6 Figure 5.32 X
Two cars travel in the same direction along a straight road. Figure 5.20 shows the velocity, v, of each car at time t. Car B starts 2 hours after car A and car B reaches a maximum velocity of 50 km/hr.(a) For approximately how long does each car travel?(b) Estimate car A’s maximum velocity.(c)
Problems refer to a May 2, 2010, article:“The crisis began around 10 am yesterday when a 10-foot wide pipe in Weston sprang a leak, which worsened throughout the afternoon and eventually cut off GreaterBoston from the Quabbin Reservoir, where most of its water supply is stored. . .Before water
Two cars start at the same time and travel in the same direction along a straight road. Figure 5.21 gives the velocity, v, of each car as a function of time, t. Which car:(a) Attains the larger maximum velocity?(b) Stops first?(c) Travels farther? v (km/hr) Car A Car B Figure 5.21 t (hr)
Figure 5.58 gives your velocity during a trip starting from home. Positive velocities take you away from home and negative velocities take you toward home. Where are you at the end of the 5 hours? When are you farthest from home? How far away are you at that time? v
In Problems, use an integral to find the specified area.Between y = sin x + 2 and y = 0.5 for 6 ≤ x ≤ 10.
Without calculation, what can you say about the relationship between the values of the two integrals: Jo 2 e-R dx and 2 Jo e dt?
If we know ∫52 f(x) dx = 4, what is the value of 3 ³ (1₂ 2 5 f(x) dx + 1? dx) +
“The crisis began around 10 am yesterday when a 10-foot wide pipe in Weston sprang a leak, which worsened throughout the afternoon and eventually cut off GreaterBoston from the Quabbin Reservoir, where most of its water supply is stored. . .Before water was shut off to the ruptured pipe [at 6:40
In Problems, use an integral to find the specified area.Between y = cos x + 7 and y = ln(x − 3), 5 ≤ x ≤ 7.
Use a calculator or computer to evaluate the integral. J₁ 4 1 V1 + x² dx
Figure 5.60 shows the rate of growth of two trees. If the two trees are the same height at time t = 0, which tree is taller after 5 years? After 10 years? rate (feet per year) 12 6 5 Figure 5.60 B 10 1 (years)
Use an integral to find the specified area. Above the curve y = x4 − 8 and below the x-axis.
The velocity of a car, in ft/sec, is v(t) = 10t for t in seconds, 0 ≤ t ≤ 6.(a) Use Δt = 2 to give upper and lower estimates for the distance traveled. What is their average?(b) Find the distance traveled using the area under the graph of v(t). Compare it to your answer for part (a).
List the expressions (I)–(III) in order from smallest to largest, where r(t) is the hourly rate that an animal burns calories and R(t) is the total number of calories burned since time t = 0. Assume r(t) > 0 and r'(t) < 0 for 0 ≤ t ≤ 12. I. -8 5 r(t) dt II. 8 11 r(t) dt III. R(12)-R(9)
List the expressions (I)–(III) in order from smallest to largest, where r(t) is the hourly rate that an animal burns calories and R(t) is the total number of calories burned since time t = 0. Assume r(t) > 0 and r'(t) < 0 for 0 ≤ t ≤ 12.I. R(10) II. R(12) III. R(10)+r(10)⋅2
Concern the future of the US Social Security Trust Fund, out of which pensions are paid. Figure 5.61 shows the rates (billions of dollars per year) at which income.I(t), from taxes and interest is projected to flow into the fund and at which expenditures, E(t), flow out of the fund.
For Problems, compute the definite integral and interpret the result in terms of areas. 4 1*(x (x-3 ln x) dx.
Compute the definite integral and interpret the result in terms of areas. 4 1. x²-3 2 X dx.
Use a calculator or computer to evaluate the integral. -1 et dt
Use an integral to find the specified area. Above the curve y = −ex+e2(x−1) and below the x-axis, for x ≥ 0.
Problems concern the future of the US Social Security Trust Fund, out of which pensions are paid. Figure 5.61 shows the rates (billions of dollars per year) at which income.I(t), from taxes and interest is projected to flow into the fund and at which expenditures, E(t), flow out of the
Your velocity is given by v(t) = t2 + 1 in m/sec, with t in seconds. Estimate the distance, s, traveled between t = 0 and t = 5. Explain how you arrived at your estimate.
Use a calculator or computer to evaluate the integral. 1 x²+1 2 J-1 x² - 4 dx
Use a calculator or computer to evaluate the integral. -1.7 1.1 e' ln t dt
An old rowboat has sprung a leak. Water is flowing into the boat at a rate, r(t), given in the table.(a) Compute upper and lower estimates for the volume of water that has flowed into the boat during the 15 minutes.(b) Draw a graph to illustrate the lower estimate. 7 minutes r(1) liters/min 0 5 10
Find the integral by finding the area of the region between the curve and the horizontal axis. -10 (x - 5) dx
A car moving with velocity v has a stopping distance proportional to v2.(a) If a car going 20mi/hr has a stopping distance of 50 feet, what is its stopping distance going 40 mi/hr? What about 60 mi/hr?(b) After applying the brakes, a car going 30 ft/sec stops in 5 seconds and has v = 30 − 6t.
Use a calculator or computer to evaluate the integral. T. In(y² + 1) dy: 0
The rates of consumption of stores of protein and fat in the human body during 8 weeks of starvation are shown in Figure 5.63. Does the body burn more fat or more protein during this period? rate of consumption of stored foods (kg/week) 2 0 - 2 3 + 4 5 6 7 Figure 5.63 Protein Fat +weeks of
The value of a mutual fund increases at a rate of R = 500e0.04t dollars per year, with t in years since 2010.(a) Using t = 0, 2, 4, 6, 8, 10, make a table of values for R.(b) Use the table to estimate the total change in the value of the mutual fund between 2010 and 2020.
Figure 5.64 shows the number of sales per month made by two salespeople. Which person has the most total sales after 6 months? After the first year? At approximately what times (if any) have they sold roughly equal total amounts? Approximately how many total sales has each person made at the end of
Find the integral by finding the area of the region between the curve and the horizontal axis. 8 Jo (6 - 2x) dx
Find the integral by finding the area of the region between the curve and the horizontal axis. 6 1 L ( 3x + 3) d² dx -8 2
The rate of change of the world’s population, in millions of people per year, is given in the following table.(a) Use this data to estimate the total change in the world’s population between 1950 and 2000.(b) The world population was 2555 million people in 1950 and 6085 million
Height velocity graphs are used by endocrinologists to follow the progress of children with growth deficiencies. Figure 5.65 shows the height velocity curves of an average boy and an average girl between ages 3 and 18.(a) Which curve is for girls and which is for boys? Explain how you can tell.(b)
Find the integral by finding the area of the region between the curve and the horizontal axis. 1 -10 - 4x - 16 3 dx
Use a calculator or computer to evaluate the integral. 3 4 Ve² + zdz
A car speeds up at a constant rate from 10 to 70 mph over a period of half an hour. Its fuel efficiency (in miles per gallon) increases with speed; values are in the table. Make lower and upper estimates of the quantity of fuel used during the half hour. Speed (mph) 10 Fuel efficiency
Concern hybrid cars such as the Toyota Prius that are powered by a gas-engine, electric-motor combination, but can also function in Electric-Vehicle (EV) only mode. Figure 5.22 shows the velocity, v, of a 2010 Prius Plug-in Hybrid Prototype operating in normal hybrid mode and EV-only mode,
A healthy human heart pumps about 5 liters of blood per minute. Problems refer to Figure 5.66, which shows the response of the heart to bleeding. The pumping rate drops and then returns to normal if the person recovers fully, or drops to zero if the person dies.(a) If the body is bled 2 liters, how
Use a calculator or computer to evaluate the integral. 2 2 1 dx Jo 1+x²4x + / T y² dy
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