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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
Fill in the missing values in Table 9.4 given that dy∕dt = 0.5y. Assume the rate of growth, given by dy∕dt, is approximately constant over each unit time interval. Table 9.4 1 y 0 1 2 3 4 8
Oil is pumped continuously from a well at a rate proportional to the amount of oil left in the well. Initially there were 1 million barrels of oil in the well; six years later 500,000 barrels remain.(a) At what rate was the amount of oil in the well decreasing when there were 600,000 barrels
A patient is given the drug theophylline intravenously at a rate of 43.2 mg/hour to relieve acute asthma. The rate at which the drug leaves the patient’s body is proportional to the quantity there, with proportionality constant 0.082 if time, t, is in hours. The patient’s body contains none of
For a certain quantity y, assume that dy∕dt = √y. Fill in the value of y in Table 9.5. Assume that the rate of growth, dy∕dt, is approximately constant over each unit time interval. Table 9.5 0 1 2 3 1 y 100 3 4
An early model of the growth of Wikipedia assumed that every day a constant number, B, of articles is added by dedicated Wikipedians and that other articles are created by the general public at a rate proportional to the number of articles already there. Express this model as a differential
Hydrocodone bitartrate is used as a cough suppressant. After the drug is fully absorbed, the quantity of drug in the body decreases at a rate proportional to the amount left in the body. The half-life of hydrocodone bitartrate in the body is 3.8 hours and the dose is 10 mg.(a) Write a differential
Consider a solution curve for each of the slope fields in Problem 7. Write one or two sentences describing qualitatively the long-run behavior of y. For example, as x increases, does y → ∞, or does y remain finite? You may get different limiting behavior for different starting points. In each
A chain smoker smokes five cigarettes every hour. From each cigarette, 0.4 mg of nicotine is absorbed into the person’s bloodstream. Nicotine leaves the body at a rate proportional to the amount present, with constant of proportionality −0.346 if t is in hours.(a) Write a differential equation
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.At t = 0 there are 2.2 million worms and 1 thousand robins.(a) Use 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.(a) Assume that there are 3 million worms and 2 thousand robins. Locate the
(a) Find the equilibrium solution of the equation(b) Find the general solution of this equation.(c) Graph several solutions with different initial values.(d) Is the equilibrium solution stable or unstable? dy dt = 0.5y - 250.
The Gompertz equation, which models growth of animal tumors, is y' = −ay ln(y∕b), where a and b are positive constants. Use Figures 9.25 and 9.26 to write a paragraph describing the similarities and/or differences between solutions to the Gompertz equation with a = 1 and b = 2 and solutions to
Suppose Q = Cekt satisfies the differential equationWhat (if anything) does this tell you about the values of C and k? dQ dt = -0.03Q. =
A country’s infrastructure is its transportation and communication systems, power plants, and other public institutions. The Solow model asserts that the value of national infrastructure K increases due to investment and decreases due to capital depreciation. The rate of increase due to
Figure 9.38 gives the slope field for a differential equation. Estimate all equilibrium solutions and indicate whether each is stable or unstable. 4 3 2 y 11111 1 / / / / / / / / / / / / 1 1 1 1 1 1 1 1 XXXXX 1 2 3 3 Figure 9.38 4 4 5
Show that, for any constant P0, the function P = P0et satisfies the equation dP dt = P.
The amount of land in use for growing crops increases as the world’s population increases. Suppose A(t) represents the total number of hectares of land in use in year t. (A hectare is about 2 1/2 acres.)(a) Explain why it is plausible that A(t) satisfies the equation A'(t) = kA(t).What
If the initial population of fish is 70 million, use the differential equation dP∕dt = 0.2P − 10 to estimate the fish population after 1, 2, 3 years.
As you know, when a course ends, students start to forget the material they have learned. One model (called the Ebbinghaus model) assumes that the rate at which a student forgets material is proportional to the difference between the material currently remembered and some positive constant, a.(a)
(a) What are the equilibrium solutions for the differential equation(b) Use a graphing calculator or computer to sketch a slope field for this differential equation. Use the slope field to determine whether each equilibrium solution is stable or unstable. dy dt = 0.2(y - 3)(y + 2)?
A yam is put in a 200◦C oven and heats up according to the differential equationfor k a positive constant.(a) If the yam is at 20◦C when it is put in the oven, solve the differential equation.(b) Find k using the fact that after 30 minutes the temperature of the yam is 120◦C. dH dt =-k(H -
Suppose x and y are the populations of two different species. Describe in words how each population changes with time.
Suppose x and y are the populations of two different species. Describe in words how each population changes with time. У X
Suppose x and y are the populations of two different species. Describe in words how each population changes with time. y X
Is there a value of n which makes y = xn a solution to the equation 13x(dy∕dx) = y? If so, what value?
Find the values of k for which y = x2 + k is a solution to the differential equation 2y − xy' = 10.
Suppose x and y are the populations of two different species. Describe in words how each population changes with time. y X
At 1:00 pm one winter afternoon, there is a power failure at your house in Wisconsin, and your heat does not work without electricity. When the power goes out, it is 68◦F in your house. At 10:00 pm, it is 57◦F in the house, and you notice that it is 10◦F outside.(a) Assuming that the
Some people write the solution of the initial value problemin the formShow that this formula gives the correct solution for y, assuming y0 ≠ A. dy dt =k(y - A) y = y at t = 0
A detective finds a murder victim at 9 am. The temperature of the body is measured at 90.3◦F. One hour later, the temperature of the body is 89.0◦F. The temperature of the room has been maintained at a constant 68◦F.(a) Assuming the temperature, T , of the body obeys Newton’s Law of
Suppose x and y are the populations of two different species. Describe in words how each population changes with time.A kidney removes toxins from the blood. If a kidney does not function, the toxins can be removed by dialysis. This problem explores a model for Q1(t), the quantity of toxins in the
Figure 1.118 shows the levels of the hormones estrogen and progesterone during the monthly ovarian cycles in females. Is the level of both hormones periodic? What is the period in each case? Approximately when in the monthly cycle is estrogen at a peak? Approximately when in the monthly cycle is
Use the variable u for the inside function to express each of the following as a composite function: (a) y = (5t2 − 2)6 (b) P = 12e−0.6t (c) C = 12 ln(q3 + 1)
Find f(5). 10 8 6 4 2 f(x) 2 4 6 8 10 X
Oil consumption in China grew exponentially from 8.938 million barrels per day in 2010 to 10.480 million barrels per day in 2013. Assuming exponential growth continues at the same rate, what will oil consumption be in 2025?
Solve for t using natural logarithms.2P = P e0.3t
In the 1990s, Brazil had a deforestation rate of about 0.48% per year. (This is the rate at which land covered by forests is shrinking.) Assuming the rate continues, what percent of the land in Brazil covered by forests in 1990 will be forested in 2030?
Most breeding birds in the northeast US migrate else- where during the winter. The number of bird species in an Ohio forest preserve oscillates between a high of 28 in June and a low of 10 in December.(a) Graph the number of bird species in this preserve as a function of t, the number of months
Simplify the quantities in Problem using m(z) = z2.m(z + 1) − m(z)
Let y = f(x) = x2 + 2. (a) Find the value of y when x is zero. (b) What is f(3)? (c) What values of x give y a value of 11? (d) Are there any values of x that give y a value of 1?
You want to invest money for your child’s education in a certificate of deposit (CD). You want it to be worth $12,000 in 10 years. How much should you invest if the CD pays interest at a 9% annual rate compounded (a) Annually? (b) Continuously?
Table 1.13 gives the net sales of The Gap, a clothing retailer. (a) Find the change in net sales between 2013 and 2015. (b) Find the average rate of change in net sales between 2011 and 2015. Give units and interpret your answer. (c) From 2011 to 2015, were there any one-year
Solve for t using natural logarithms.5e3t = 8e2t
An amusement park charges an admission fee of $21 per person as well as an additional $4.50 for each ride.(a) For one visitor, find the park’s total revenue R(n) as a function of the number of rides, n, taken.(b) Find R(2) and R(8) and interpret your answers in terms of amusement park fees.
Which of the following tables could represent linear functions? (a)(b)(c) X y 0 1 27 25 2 23 3 21
The consumer price index (CPI) for a given year is the amount of money in that year that has the same purchasing power as $100 in 1983. At the start of 2015, the CPI was 234. Write a formula for the CPI as a function of 푡 years after 2015, assuming that the CPI increases by 1.8% every year.
Table 1.15 shows attendance at NFL football games.(a) Find the average rate of change in the attendance from 2011 to 2015. Give units. (b) Find the annual increase in the average attendance for each year from 2011 to 2015. (Your answer should be four numbers.)(c) Show that the average rate of
Annual revenue R from McDonald’s restaurants worldwide between 2013 and 2015 can be estimated by R = 28.1 − 1.35t, where R is in billion dollars and t is in years since January 1, 2013. (a) What is the slope of this function? Include units. Interpret the slope in terms of McDonald’s
A company has cost function C(q) = 4000 + 2q dollars and revenue function R(q) = 10q dollars.(a) What are the fixed costs for the company?(b) What is the marginal cost?(c) What price is the company charging for its product?(d) Graph C(q) and R(q) on the same axes and label the break-even point, q0.
Graph y = 100e−0.4x. Describe what you see.
Find a possible formula for the graph. 7 y -7 - 5 10 t
Simplify the quantities in Problem using m(z) = z2.m(z + ℎ) − m(z)
The function S = f(t) gives the average annual sea level, S, in meters, in Aberdeen, Scotland,7 as a function of t, the number of years before 2012. Write a mathematical expression that represents the given statement.In 2000 the average annual sea level in Aberdeen was 7.049 meters.
According to the EPA, sales of electronic devices in the US doubled between 1997 and 2009, when 438 million electronic devices sold. (a) Find an exponential function, S(t), to model sales in millions since 1997. (b) What was the annual percentage growth rate between 1997 and 2009?
Use shifts of power functions to find a possible formula for each of the graphs: (a)(b) У (2, 1) X
Solve for t using natural logarithms.Ae2t = Bet
Table 1.14 shows world bicycle production.(a) Find the change in bicycle production between 1950 and 2000. Give units. (b) Find the average rate of change in bicycle production between 1950 and 2000. Give units and interpret your answer in terms of bicycle production.Table 1.14 World bicycle
A photocopying company has two different price lists. The first price list is $100 plus 3 cents per copy; the second price list is $200 plus 2 cents per copy.(a) For each price list, find the total cost as a function of the number of copies needed.(b) Determine which price list is cheaper for 5000
A firm decides to increase output at a constant relative rate from its current level of 20,000 to 30,000 units during the next five years. Calculate the annual percent rate of increase required to achieve this growth.
At time t in seconds, a particle’s distance s(t), in cm, from a point is given in the table. What is the average velocity of the particle from t = 3 to t = 10? t s(t) 0 0 3 6 72 92 10 144 13 180
A company’s pricing schedule in Table 1.5 is designed to encourage large orders. (A gross is 12 dozen.) Find a formula for: (a) q as a linear function of p. (b) p as a linear function of q.Table 1.5 q (order size, gross) p (price/dozen) 3 4 5 6 15 12 9 6
find all the tables that have the given characteristic.y could be an exponential function of x. (A) (B) (C) (D) X y X y 51 X 0 2.2 2.2 X -8 y 40 2.2 -4 62 80 2.2 -4 -3 4 6 y 18 0 4.5 -2.25 160 2.2 0 8 73 95 3 4 5 6 18 9 4.5 2.25
A $50,000 tractor has a resale value of $10,000 twenty years after it was purchased. Assume that the value of the tractor depreciates linearly from the time of purchase.(a) Find a formula for the value of the tractor as a function of the time since it was purchased.(b) Graph the value of the
Find a possible formula for the graph. ८. + -2π y 3 Y 1 + x 2л
use the graphs in Figure 1.90Estimate f(g(4)). 2 2 -f(x)- 4 6 X 4 2 Figure 1.90 2 -g(x)- 4 6 + x
An object is put outside on a cold day at time t = 0 minutes. Its temperature, H = f(t), in ◦C, is graphed in Figure 1.8. (a) What does the statement f(30) = 10 mean in terms of temperature? Include units for 30 and for 10 in your answer. (b) Explain what the vertical intercept, a, and
The functions in Problems represent exponential growth or decay. What is the initial quantity? What is the growth rate? State if the growth rate is continuous.P = 7.7(0.92)t
The antidepressant fluoxetine (or Prozac) has a half-life of about 3 days. What percentage of a dose remains in the body after one day? After one week?
Allometry is the study of the relative size of different parts of a body as a consequence of growth. In this problem, you will check the accuracy of an allometric equation: the weight of a fish is proportional to the cube of its length. Table 1.39 relates the weight, y, in gm, of plaice (a type of
Figure 1.42 shows a particle’s distance from a point as a function of time, t. What is the particle’s average velocity from t = 1 to t = 3?Figure 1.42 distance (meters) 7 5 3 ليا 1 2 s(t) 4 t (sec)
US imports of crude oil and petroleum have been increasing. There have been many ups and downs, but the general trend is shown by the line in Figure 1.26.(a) Find the slope of the line. Include its units of measurement.(b) Write an equation for the line. Define your variables, including their
A $15,000 robot depreciates linearly to zero in 10 years.(a) Find a formula for its value as a function of time.(b) How much is the robot worth three years after it is purchased?
Find all the tables that have the given characteristic.y could be a linear function of x. (A) (B) (C) (D) X y X y 51 X 0 2.2 2.2 X -8 y 40 2.2 -4 62 80 2.2 -4 -3 4 6 y 18 0 4.5 -2.25 160 2.2 0 8 73 95 3 4 5 6 18 9 4.5 2.25
Find a possible formula for the graph. 4 y П x
use the graphs in Figure 1.90.Estimate g(f(1)). 2 2 -f(x)- 4 6 X 4 2 Figure 1.90 2 -g(x)- 4 6 + x
(a) A potato is put in an oven to bake at time t = 0. Which of the graphs in Figure 1.7 could represent the potato’s temperature as a function of time? (b) What does the vertical intercept represent in terms of the potato’s temperature?Figure 1.7 (1) (11) (IV) t
The functions in Problems represent exponential growth or decay. What is the initial quantity? What is the growth rate? State if the growth rate is continuous.P = 5(1.07)t
Air pressure, P, decreases exponentially with height, ℎ, above sea level. If P0 is the air pressure at sea level and ℎ is in meters, then P = P0e−0.00012ℎ. (a) At the top of Denali, height 6194 meters (about 20,320 feet), what is the air pressure, as a percent of the pressure at sea
Use Figure 1.25 showing how the quantity, Q, of grass (kg/hectare) in different parts of Namibia depended on the average annual rainfall, r, (mm), in two different years.Figure 1.25Which of the two functions in Figure 1.25 has the larger difference quotient ΔQ∕Δr? What does this tell us about
Kleiber’s Law states that the metabolic needs (such as calorie requirements) of a mammal are proportional to its body weight raised to the 0.75 power. Surprisingly, the daily diets of mammals conform to this relation well. Assuming Kleiber’s Law holds: (a) Write a formula for C, daily
Find a possible formula for the function represented by the data. 0 1 4.40 1 g(t) 5.50 2 3.52 3 2.82
Production costs for manufacturing running shoes consist of a fixed overhead of $650,000 plus variable costs of $20 per pair of shoes. Each pair of shoes sells for $70.(a) Find the total cost, C(q), the total revenue, R(q), and the total profit, π(q), as a function of the number of pairs of shoes
Find a possible formula for the graph. 2 y 8л X
Use the graphs in Figure 1.90 Estimate f(g(1)). 2 2 -f(x)- 4 6 X 4 2 Figure 1.90 2 -g(x)- 4 6 + x
The function S = f(t) gives the average annual sea level, S, in meters, in Aberdeen, Scotland,7 as a function of t, the number of years before 2012. Write a mathematical expression that represents the given statement.The average annual sea level in Aberdeen decreased by 8 millimeters from 2011 to
A population, currently 200, is growing at 5% per year. (a) Write a formula for the population, P, as a function of time, t, years in the future. (b) Graph P against t. (c) Estimate the population 10 years from now.(d) Use the graph to estimate the doubling time of the population.
The blood mass of a mammal is proportional to its body mass. A rhinoceros with body mass 3000 kilograms has blood mass of 150 kilograms. Find a formula for the blood mass of a mammal as a function of the body mass and estimate the blood mass of a human with body mass 70 kilograms.
Solve for t using natural logarithms.Pe4t − Qe−t = 0
Figure 1.40 shows the total value of US imports, in trillions of dollars.(a) Was the value of the imports higher in 2001 or in 2015? Approximately how much higher? (b) Estimate the average rate of change of US imports between 2001 and 2015. Give units and interpret your answer in terms of
Use Figure 1.25 showing how the quantity, Q, of grass (kg/hectare) in different parts of Namibia depended on the average annual rainfall, r, (mm), in two different years.Figure 1.25(a) For 1997, find the slope of the line, including units. (b) Interpret the slope in this context. (c) Find
A company producing jigsaw puzzles has fixed costs of $6000 and variable costs of $2 per puzzle. The company sells the puzzles for $5 each.(a) Find formulas for the cost function, the revenue function, and the profit function.(b) Sketch a graph of R(q) and C(q) on the same axes. What is the
Find a possible formula for the function represented by the data. x 0 f(x) 4.30 1 2 6.02 8.43 3 11.80
Find a possible formula for the graph. 5 y + x бл
Simplify the quantities in Problem using m(z) = z2.m(z + ℎ) − m(z − ℎ)
determine whether or not the function is a power function. If it is a power function, write it in the form y = kxp and give the values of k and p. y = 3 8x
If you deposit $10,000 in an account earning interest at an 8% annual rate compounded continuously, how much money is in the account after five years?
The following table shows values of a periodic function f(x). The maximum value attained by the function is 5.(a) What is the amplitude of this function?(b) What is the period of this function?(c) Find a formula for this periodic function. 02 50 x f(x) 5 4 68 10 -5 0 5 0 12 -5
(a) Which two lines in Figure 1.23 have the same slope? Of these two lines, which has the larger y-intercept? (b) Which two lines have the same y-intercept? Of these two lines, which has the larger slope? Figure 1.23 y EL 15 IA X
A demand curve is given by 75p + 50q = 300, where p is the price of the product, in dollars, and q is the quantity demanded at that price. Find the p- and q-intercepts and interpret them in terms of consumer demand.
The gross domestic product, G, of Switzerland was 685.4 billion dollars in 2013. Give a formula for G (in billions of dollars) t years after 2013 if G increases by(a) 2% per year(b) 7 billion dollars per year
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