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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
Draw a contour diagram for C(d, m) = 40d + 0.15m. Include contours for C = 50, 100, 150, 200.
Figure 8.5 shows the annual energy production E(d, w) (in kilowatt-hours) of a wind turbine as a function of its diameter, d, (in feet) and three different wind speeds, w, (in mph).(a) Estimate E(15, 12) and explain what information it gives us.(b) For a wind turbine of diameter 10 feet, how much
For a cost function, f(x, y), the minimum cost for a production of 50 is given by f(33, 87) = 1200, with λ = 15. Estimate the cost if the production quota is:(a) Raised to 51 (b) Lowered to 49
Find all the critical points and determine whether each is a local maximum, local minimum, a saddle point, or none of these.f(x, y) = x3 + y3 − 3x2 − 3y + 10
A person’s body mass index (BMI) is a function of their weight W (in kg) and height H (in m) given by B(W,H) = W∕H2.What is the BMI of a 1.72 m tall man weighing 72 kg?
People commuting to a city can choose to go either by bus or by train. The number of people who choose either method depends in part upon the price of each. Let f(P1, P2) be the number of people who take the bus when P1 is the price of a bus ride and P2 is the price of a train ride. What can you
In Problems(a) Find fx(1, 1) and fy(1, 1).(b) Use part (a) to match f(x, y) with one of the contour diagrams (I)–(III), each shown centered at (1, 1) with the same scale in the x and y directions.f(x, y) = x2 + y4 (11) (1) (11)
Maple syrup production is highest when the nights are cold and the days are warm. Make a possible contour diagram for maple syrup production as a function of the high (daytime) temperature and the low (nighttime) temperature. Label the contours with 10, 20, 30, and 40 (in liters of maple syrup).
The wind chill tells you how cold it feels as a function of the air temperature and wind speed. Figure 8.35 is a contour diagram of wind chill (◦F).(a) If the wind speed is 15mph, what temperature feels like −20◦F?(b) Estimate the wind chill if the temperature is 0◦F and the wind speed is
A company has the production function P(x, y), which gives the number of units that can be produced for given values of x and y; the cost function C(x, y) gives the cost of production for given values of x and y.(a) If the company wishes to maximize production at a cost of $50,000, what is the
The pressure of a fixed amount of compressed nitrogen gas in a cylinder is given, in atmospheres, bywhere T is the temperature of the gas, in Kelvin, and V is the volume of the cylinder, in liters. Figures 8.6 and 8.7 give cross sections of the function f.(a) Which figure shows cross-sections of f
Find all the critical points and determine whether each is a local maximum, local minimum, a saddle point, or none of these.f(x, y) = x2 − 2xy + 3y2 − 8y
Table 8.6 shown below gives the percent of rats surviving, P, as a function of time, t, in months and concentration of formaldehyde, c, in ppm, so P = f(t, c). Use partial derivatives to estimate the percent surviving after 26 months when the concentration is 15. Table 8.6 Percent, P, of rat
Give the rates of growth of two populations, x and y, measured in thousands.(a) Describe in words what happens to the population of each species in the absence of the other.(b) Describe in words how the species interact with one another. Give reasons why the populations might behave as described by
You have set aside 20 hours to work on two class projects. You want to maximize your grade (measured in points), which depends on how you divide your time between the two projects.(a) What is the objective function for this optimization problem and what are its units?(b) What is the constraint?(c)
Find solutions to the differential equations in Problems subject to the given initial condition. dP dt = 0.02P, P(0) = 20
A person’s body mass index (BMI) is a function of their weight W (in kg) and height H (in m) given by B(W,H) = W∕H2.A 1.58 m tall woman has a BMI of 23.2. What is her weight?
Figure 8.48 shows contours of f(x, y) with values of f on the contours omitted. If fx(P) > 0, find the sign:(a) fy(P)(b) fy(Q)(c) fx(Q) X- f(x, y) P Figure 8.48 tin X
A person’s body mass index (BMI) is a function of their weight W (in kg) and height H (in m) given by B(W,H) = W∕H2.With a BMI less than 18.5, a person is considered underweight. What is the possible range of weights for an underweight person 1.58 m tall?
The quantity, q, of a product manufactured depends on the number of workers, W , and the amount of capital invested, K, and is given by the Cobb-Douglas function q = 6W3∕4K1∕4.In addition, labor costs are $10 per worker and capital costs are $20 per unit and the budget is $3000.(a) What are the
Give the rates of growth of two populations, x and y, measured in thousands.(a) Describe in words what happens to the population of each species in the absence of the other.(b) Describe in words how the species interact with one another. Give reasons why the populations might behave as described by
Sketch three solution curves for each of the slope fields in Figure 9.17. 17 1/ //// //// 11/ 71 XX 11 y 11 1 1/ 11/ 771 1 1 7/1 !!! X X X X X XXXXAN ANIZA Figure 9.17 y www. X
Find solutions to the differential equations in Problems subject to the given initial condition. dw dr 3w, w = 30 when r = 0
Find particular solutions in Problems = 0.5(y-200), y = 50 when t = 0 dt
Let I be the number of infected people and S be the number of susceptible people in an outbreak of a disease. Explain why it is reasonable to model the interaction between these two groups by the differential equationsWhy have the signs been chosen this way? Why is the constant a the same in both
Sketch the slope field for dy∕dx = y2 at the points marked in Figure 9.18. -1 2 1 Figure 9.18 1 X Figure 9.19
Decide whether or not each of the following is a solution to the differential equation xy' − 2y = 0.(a) y = x2(b) y = x3
Find particular solutions in Problems dP dt = P+4, P 100 when t = 0
Give the rates of growth of two populations, x and y, measured in thousands.(a) Describe in words what happens to the population of each species in the absence of the other.(b) Describe in words how the species interact with one another. Give reasons why the populations might behave as described by
Find solutions to the differential equations in Problems subject to the given initial condition. dy dx = -0.14y, y = 5.6 when x = 0
Check that y = t4 is a solution to the differential equation t dy/dt = 4y.
Find the general solution to the differential equation dy = dt || 21.
Explain how you can tell from the graph of the trajectory shown in Figure 9.46 that most people at the British boarding school eventually got sick. I (infecteds) 400 200 / 1/ / (762, 1) 192 S (susceptibles) 400 Figure 9.46: Trajectory for So = 762, 10 = 1 0
Find particular solutions in Problems dH dt 3(H-75), H = 0 when t = 0
The following system of differential equations represents the interaction between two populations, x and y.(a) Describe how the species interact. How would each species do in the absence of the other? Are they helpful or harmful to each other?(b) If x = 2 and y = 1, does x increase or decrease?
Find particular solutions in Problems dm = 0.1m +200, m(0) = 1000 dt
Find solutions to the differential equations in Problems subject to the given initial condition. dQ dt || alin Q = 50 when t = 0
Figure 9.20 is a slope field for dy∕dx = y − 10.(a) Draw the solution curve for each of the following initial conditions:(i) y = 8 when x = 0(ii) y = 12 when x = 0(iii) y = 10when x = 0(b) Since dy∕dx = y − 10, when y = 10, we have dy∕dx = 10 − 10 = 0. Explain why this matches your
A population of insects grows at a rate proportional to the size of the population. Write a differential equation for the size of the population, P, as a function of time, t. Is the constant of proportionality positive or negative.
Use the fact that the derivative gives the slope of a curve to decide which of the graphs (A)–(F) in Figure 9.6 could represent a solution to the differential equation. (A) (C) (E) y y X x (B) (D) y y X X (F) V.X. Figure 9.6
Find solutions to the differential equations in Problems subject to the given initial condition. dp dq = -0.1p, p= 100 when q = 5
Figure 9.21 is the slope field for the equation y' = x + y.(a) Sketch the solutions that pass through the points(i) (0, 0) (ii) (−3, 1) (iii) (−1, 0)(b) From your sketch, guess the equation of the solution passing through (−1, 0).(c) Check your solution to part (b) by substituting
Find particular solutions in Problems dB dt = 4B 100, B= 20 when t = 0 -
(a) In a school of 150 students, one of the students has the flu initially. What is I0? What is S0?(b) Use these values of I0 and S0 and the equationto determine whether the number of infected people initially increases or decreases. What does this tell you about the spread of the disease?
Find solutions to the differential equations in Problems subject to the given initial condition. dy dx + y — — = = 0, y(0) = 10 3
Use the fact that the derivative gives the slope of a curve to decide which of the graphs (A)–(F) in Figure 9.6 could represent a solution to the differential equation. (A) (C) (E) y y X x (B) (D) y y X X (F) V.X. Figure 9.6
Money in a bank account earns interest at a continuous annual rate of 5% times the current balance. Write a differential equation for the balance, B, in the account as a function of time, t, in years.
Find particular solutions in Problems dQ dt = 0.30-120, Q = 50 when t = 0
Use the fact that the derivative gives the slope of a curve to decide which of the graphs (A)–(F) in Figure 9.6 could represent a solution to the differential equation. (A) (C) (E) y y X x (B) (D) y y X X (F) V.X. Figure 9.6
(a) On the slope field for dI∕dS in Figure 9.45, draw the trajectory through the point where I = 1 and S = 400.(b) How many susceptible people are there when the number of infected people is at its maximum? I (infecteds) 400 200 1// S (susceptibles) 400 800 Figure 9.45: Slope field for dI/dS = -1
Create a system of differential equations to model the situations in Problems. You may assume that all constants of proportionality are 1.Two businesses are in competition with each other. Both businesses would do well without the other one, but each hurts the other’s business. The values of the
Find particular solutions in Problems dB dt +2B = 50, B(1) = 100
Two companies, A and B, are in competition with each other. Let x represent the net worth (in millions of dollars) of Company A, and y represent the net worth (in millions of dollars) of Company B. Four trajectories are given in Figure 9.43. For each trajectory: Describe the initial conditions.
Use the fact that the derivative gives the slope of a curve to decide which of the graphs (A)–(F) in Figure 9.6 could represent a solution to the differential equation. (A) (C) (E) y y X x (B) (D) y y X X (F) V.X. Figure 9.6
Match the slope fields in Figure 9.22 with their differential equations:(a) y' = 1 + y2 (b) y' = x (c) y' = sin x(d) y' = y(e) y' = x − y(f) y' = 4 − y e (III) (V) A A A X X (11) E (IV) (VI) **** www XXXX a Figure 9.22: Each slope field is graphed for -5 ≤ x ≤ 5, -5 ≤ y ≤ 5 X X
(a) For dy∕dx = x2 − y2, find the slope at the following points:(1, 0), (0, 1), (1, 1), (2, 1), (1, 2), (2, 2)(b) Sketch the slope field at these points.
A bank account that initially contains $25,000 earns interest at a continuous rate of 4% per year. Withdrawals are taken out at a constant rate of $2000 per year. Write a differential equation for the balance, B, in the account as a function of the number of years, t.
Use Figure 9.47 to estimate the maximum number of infecteds. What does this represent? When does it occur? Figure 9.47: Progress of the flu over time number of people 762 300 Susceptibles Infecteds 5 10 15 20 t (days)
A deposit of $5000 is made to a bank account paying 1.5% annual interest, compounded continuously.(a) Write a differential equation for the balance in the account, B, as a function of time, t, in years.(b) Solve the differential equation.(c) How much money is in the account in 10 years?
Let w be the number of worms (in millions) and r the number of robins (in thousands) living on an island. Suppose w and r satisfy the following differential equations, which correspond to the slope field in Figure 9.44.Explain why these differential equations are a reasonable model for interaction
Find particular solutions in Problems dB dt +0.1B 100 B(2) = 3
Use the fact that the derivative gives the slope of a curve to decide which of the graphs (A)–(F) in Figure 9.6 could represent a solution to the differential equation. (A) (C) (E) y y X x (B) (D) y y X X (F) V.X. Figure 9.6
Match the slope fields in Figure 9.23 with their differential equations. Explain your reasoning.(a) y' = −y(b) y' = y (c) y' = x(d) y' = 1∕y(e) y' = y2 (1) (V) a a (IV) X e Figure 9.23: Each slope field is graphed for -5 ≤ x ≤ 5, -5 ≤ y ≤ 5 X X
A pollutant spilled on the ground decays at a rate of 8% a day. In addition, cleanup crews remove the pollutant at a rate of 30 gallons a day. Write a differential equation for the amount of pollutant, P, in gallons, left after t days.
Check that y = A+Cekt is a solution to the differential equation dy = k(y - A). dt
Which one of the following differential equations best fits the slope field shown in Figure 9.24? Explain.I. dP∕dt = P − 1 II. dP ∕dt = P(P − 1)III. dP ∕dt = 3P(1 − P)IV. dP ∕dt = 1∕3P(1−P) 2 11 XXXXXII// ANNATT ////|\ XXXX Figure 9.24 1 ***** " AAAA AAVAA ر لے
The value of an investment grows continuously at an annual rate of r (when the interest rate is 5%, r = 0.05, and so on). Suppose $2000 was invested in 2010.(a) Write a differential equation satisfied by M, the value of the investment at time t, measured in years since 2010.(b) Solve the
Use the fact that the derivative gives the slope of a curve to decide which of the graphs (A)–(F) in Figure 9.6 could represent a solution to the differential equation. (A) (C) (E) y y X x (B) (D) y y X X (F) V.X. Figure 9.6
Let w be the number of worms (in millions) and r the number of robins (in thousands) living on an island. Suppose w and r satisfy the following differential equations, which correspond to the slope field in Figure 9.44.Solve these differential equations in the two special cases when there are no
Morphine is administered to a patient intravenously at a rate of 2.5 mg per hour. About 34.7% of the morphine is metabolized and leaves the body each hour. Write a differential equation for the amount of morphine, M, in milligrams, in the body as a function of time, t, in hours.
A bank account that earns 10% interest compounded continuously has an initial balance of zero. Money is deposited into the account at a continuous rate of $1000 per year.(a) Write a differential equation that describes the rate of change of the balance B = f(t).(b) Solve the differential equation.
Use the fact that the derivative gives the slope of a curve to decide which of the graphs (A)–(F) in Figure 9.6 could represent a solution to the differential equation. (A) (C) (E) y y X x (B) (D) y y X X (F) V.X. Figure 9.6
Let w be the number of worms (in millions) and r the number of robins (in thousands) living on an island. Suppose w and r satisfy the following differential equations, which correspond to the slope field in Figure 9.44.Describe and explain the symmetry you observe in the slope field. What
Use the fact that the derivative gives the slope of a curve to decide which of the graphs (A)–(F) in Figure 9.6 could represent a solution to the differential equation. (A) (C) (E) y y X x (B) (D) y y X X (F) V.X. Figure 9.6
Alcohol is metabolized and excreted from the body at a rate of about one ounce of alcohol every hour. If some alcohol is consumed, write a differential equation for the amount of alcohol, A (in ounces), remaining in the body as a function of t, the number of hours since the alcohol was consumed.
The amount of ozone, Q, in the atmosphere is decreasing at a rate proportional to the amount of ozone present. If time t is measured in years, the constant of proportionality is −0.0025. Write a differential equation for Q as a function of t, and give the general solution for the differential
During World War I, a particularly lethal form of flu killed about 40 million people around the world. The epidemic started in an army camp of 45,000 soldiers outside of Boston, where the first soldier fell sick on September 7, 1918. With time, t, in days since September 7, values of the constants
Using the model in the text and the data in Table 9.6 shown below, find how long it would take for 90% of the pollution to be removed from Lake Michigan and from Lake Ontario, assuming no new pollutants are added. Explain how you can tell which lake will take longer to be purified just by looking
A bank account earns 2% annual interest, compounded continuously. Money is deposited in a continuous cash flow at a rate of $1200 per year into the account.(a) Write a differential equation that describes the rate at which the balance B = f(t) is changing.(b) Solve the differential equation given
(a) Consider the slope field for dy∕dx = xy. What is the slope of the line segment at the point (2, 1)? At (0, 2)? At (−1, 1)? At (2,−2)?(b) Sketch part of the slope field by drawing line segments with the slopes calculated in part (a).
Toxins in pesticides can get into the food chain and accumulate in the body. A person consumes 10 micrograms a day of a toxin, ingested throughout the day. The toxin leaves the body at a continuous rate of 3% every day. Write a differential equation for the amount of toxin, A, in micrograms, in the
Let w be the number of worms (in millions) and r the number of robins (in thousands) living on an island. Suppose w and r satisfy the following differential equations, which correspond to the slope field in Figure 9.44.Assume w = 2 and r = 2 when t = 0. Do the numbers of robins and worms increase
Use the model in the text and the data in Table 9.6 on page 409 to determine which of the Great Lakes would require the longest time and which would require the shortest time for 80% of the pollution to be removed, assuming no new pollutants are being added. Find the ratio of these two times. Table
Let w be the number of worms (in millions) and r the number of robins (in thousands) living on an island. Suppose w and r satisfy the following differential equations, which correspond to the slope field in Figure 9.44.For the case discussed in Problem12, estimate the maximum and the minimum values
Use the fact that the derivative gives the slope of a curve to decide which of the graphs (A)–(F) in Figure 9.6 could represent a solution to the differential equation. (A) (C) (E) y y X x (B) (D) y y X X (F) V.X. Figure 9.6
Money in an account earns interest at a continuous rate of 8%per year, and payments are made continuously out of the account at the rate of $5000 a year. The account initially contains $50,000. Write a differential equation for the amount of money in the account, B, in t years. Solve the
A cup of coffee contains about 100 mg of caffeine. Caffeineis metabolized and leaves the body at a continuous rate of about 17% every hour.(a) Write a differential equation for the amount, A, of caffeine in the body as a function of the number of hours, t, since the coffee was consumed.(b) Use the
Fill in the missing values in Table 9.2 given that dy∕dt = 4 − y. Assume the rate of growth, given by dy∕dt, is approximately constant over each unit time interval. Table 9.2 1 y 0 1 2 3 4 8
A company earns 2% per month on its assets, paid continuously, and its expenses are paid out continuously at a rate of $80,000 per month.(a) Write a differential equation for the value, V, of the company as a function of time, t, in months.(b) What is the equilibrium solution for the differential
A quantity W satisfies the differential equation(a) Is W increasing or decreasing at W = 10? W = 2?(b) For what values of W is the rate of change of W equal to zero? MP dt = = 5W - 20.
Let w be the number of worms (in millions) and r the number of robins (in thousands) living on an island. Suppose w and r satisfy the following differential equations, which correspond to the slope field in Figure 9.44.On the same axes, graph w and r (the worm and the robin populations) against
A person deposits money into an account at a continuous rate of $6000 a year, and the account earns interest at a continuous rate of 7% per year.(a) Write a differential equation for the balance in the account, B, in dollars, as a function of years, t.(b) Use the differential equation to calculate
The rate at which a drug leaves the bloodstream and passes into the urine is proportional to the quantity of the drug in the blood at that time. If an initial dose of Q0 is injected directly into the blood, 20% is left in the blood after 3 hours.(a) Write and solve a differential equation for the
Fill in the missing values in Table 9.3 given that dy∕dt = 0.5t. Assume the rate of growth, given by dy∕dt, is approximately constant over each unit time interval. Table 9.3 t y 0 1 2 3 8 4
A bank account earns 7% annual interest compounded continuously. You deposit $10,000 in the account, and withdraw money continuously from the account at a rate of $1000 per year.(a) Write a differential equation for the balance, B, in the account after t years.(b) What is the equilibrium solution
In some chemical reactions, the rate at which the amount of a substance changes with time is proportional to the amount present. For example, this is the case as δ-glucono-lactone changes into gluconic acid.(a) Write a differential equation satisfied by y, the quantity of δ-glucono-lactone
A quantity y satisfies the differential equationUnder what conditions is y increasing? Decreasing? dt = -0.5y.
One theory on the speed an employee learns a new task claims that the more the employee already knows, the more slowly he or she learns. Suppose that the rate at which a person learns is equal to the percentage of the task not yet learned. If y is the percentage learned by time t, the percentage
Let w be the number of worms (in millions) and r the number of robins (in thousands) living on an island. Suppose w and r satisfy the following differential equations, which correspond to the slope field in Figure 9.44.People on the island like robins so much that they decide to import 200 robins
Let w be the number of worms (in millions) and r the number of robins (in thousands) living on an island. Suppose w and r satisfy the following differential equations, which correspond to the slope field in Figure 9.44.Assume that w = 3 and r = 1 when t = 0. Do the numbers of robins and worms
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