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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
Write the Leibniz notation for the derivative of the given function and include units.The distance to the ground, D, in feet, of a skydiver is a function of the time t in minutes since the skydiver jumped out of the airplane.
Refer to the graph in Figure 2.11.Does each quantity appear to be positive, negative, or zero?(a) f(−2) (b) f(0) (c) f(4)(d) f(5) (e) f'(−2) (f) f'(2)(g) f'(4) (h) f'(5) 20+ PH -20- f(x) 2 Figure 2.11 6 X
You put a yam in a hot oven, maintained at a constant temperature of 200◦C. As the yam picks up heat from the oven, its temperature rises.(a) Draw a possible graph of the temperature T of the yam against time t (minutes) since it is put into the oven. Explain any interesting features of the
It costs $4800 to produce 1295 items and it costs $4830 to produce 1305 items. What is the approximate marginal cost at a production level of 1300 items?
For the function g(x) graphed in Figure 2.45, are the following nonzero quantities positive or negative?(a) g'(0) (b) g"(0) 4 2 1 -4 -3 -2 -1 0 1 2 3 4 Figure 2.45 X
On Figure 2.73, mark lengths that represent the quantities in parts (a)–(e). (Pick any ℎ, with ℎ > 0.)(a) a + ℎ (b) ℎ (c) f(a)(d) f(a + ℎ)(e) f(a+ℎ)−f(a)(f) Using your answers to parts (a)–(e), show how the quantityf(a + ℎ) − f(a)/ℎ can be represented as the
The graph of f(x) is given in Figure 2.27. Estimate f'(1), f'(2), f'(3), f'(4), and f'(5). 6 5 4 3 2 1 f(x) 1 2 3 4 Figure 2.27 5 6 X
Write the Leibniz notation for the derivative of the given function and include units.The cost, C, of a steak, in dollars, is a function of the weight, W, of the steak, in pounds.
Alone in your dim, unheated room, you light a single candle rather than curse the darkness. Depressed with the situation, you walk directly away from the candle, sighing. The temperature (in degrees Fahrenheit) and illumination (in % of one candle power) decrease as your distance (in feet) from the
Refer to the graph in Figure 2.11.Which of the two quantities appears to be larger?(a) f(1) or f(2) (b) f(5) or f(6)(c) f(3) or f(4) (d) f'(−2) or f'(2)(e) f'(4) or f'(5) (f) f'(−1) or f'(0) 20+ PH -20- f(x) 2 Figure 2.11 6 X
The function C(q) gives the cost in dollars to produce q barrels of olive oil.(a) What are the units of marginal cost?(b) What is the practical meaning of the statement MC = 3 for q = 100?
At one of the labeled points on the graph in Figure 2.46 both dy∕dx and d2y∕dx2 are negative. Which is it? y A B C D E Figure 2.46 - x
On Figure 2.74, mark lengths that represent the quantities in parts (a)–(e). (Pick any ℎ, with ℎ > 0.)(a) a + ℎ(b) ℎ(c) f(a)(d) f(a + ℎ)(e) f(a+ℎ)−f(a)(f) Using your answers to parts (a)–(e), represent the quantity f(a + ℎ) − f(a)/ℎ as the slope of a line on the graph.
Write the Leibniz notation for the derivative of the given function and include units.The number, N, of gallons of gas left in a gas tank is a function of the distance, D, in miles, the car has been driven.
For Problem graph the derivative of the given function. -4 4 -4 4+
Chlorofluorocarbons (CFCs) were used as propellants in spray cans until their buildup in the atmosphere started destroying the ozone, which protects us from ultraviolet rays. Since the 1987 Montreal Protocol (an agreement to curb CFCs), the CFCs in the atmosphere above the US have been reduced from
In Figure 2.61, estimate the marginal revenue when the level of production is 600 units and interpret it. 20,000 10,000 $ GA 0 600 Figure 2.61 R(q) 9
For the expressions in Problems(a) What is the result when we substitute ℎ = 0?(b) Find the values when we substitute ℎ = 0.1, ℎ = 0.01, ℎ = 0.001, and ℎ = 0.0001.(c) Use the results of part (b) to estimate the limit as ℎ goes to zero. esh - 1 h
Write the Leibniz notation for the derivative of the given function and include units.An employee’s pay, P, in dollars, for a week is a function of the number of hours worked, H.
For Problem graph the derivative of the given function. -4 T 4 -4 4 x
Which of the functions A, B, C, D in Figure 2.12 satisfy the given condition?f(4) < 0 (A) (C) y y +N 2 -2. + 4/6 X (B) (D) Figure 2.12 y y +N ++ 4 ए — x 6 4 6 X
In Figure 2.62, is marginal cost greater at q = 5 or at q = 30? At q = 20 or at q = 40? Explain. 500 400 300 200 100 s C(q) 10 20 30 40 50 Figure 2.62 9
Give the signs of the first and second derivatives for the function. Each derivative is either positive everywhere, zero everywhere, or negative everywhere. 1 f(x) X
For the expressions in Problems(a) What is the result when we substitute ℎ = 0?(b) Find the values when we substitute ℎ = 0.1, ℎ = 0.01, ℎ = 0.001, and ℎ = 0.0001.(c) Use the results of part (b) to estimate the limit as ℎ goes to zero. 3h 2e³h - 2 h
The time for a chemical reaction, T (in minutes), is a function of the amount of catalyst present, a (in milliliters), so T = f(a).(a) If f(5) = 18, what are the units of 5?What are the units of 18? What does this statement tell us about the reaction?(b) If f'(5) = −3, what are the units of 5?
Graph the derivative of the given function. -4 4." -4 4 X
Which of the functions A, B, C, D in Figure 2.12 satisfy the given condition?f'(2) > 0 (A) (C) y y +N 2 -2. + 4/6 X (B) (D) Figure 2.12 y y +N ++ 4 ए — x 6 4 6 X
Use the graphs in Figure 1.90.Estimate f(f(2)). 2 2 -f(x)- 4 6 X 4 2 Figure 1.90 2 -g(x)- 4 6 + x
The specific heat, s, of an element is the number of calories of heat required to raise the temperature of one gram of the element by one degree Celsius. Use the following table to decide if s is proportional or inversely proportional to the atomic weight, w, of the element. If so, find the
Find a possible formula for the graph. -1 y 5 x
In Problems 25–28, a quantity P is an exponential function of time t. Use the given information about the function P = P0at to:(a) Find values for the parameters a and P0. (b) State the initial quantity and the percent rate of growth or decay.P0a3 = 75 and P0a2 = 50
A corporate office provides the demand curve in Figure 1.63 to its ice cream shop franchises. At a price of $1.00 per scoop, 240 scoops per day can be sold.(a) Estimate how many scoops could be sold per day at a price of 50c/ per scoop. Explain.(b) Estimate how many scoops per day could be sold at
Table 1.17 gives sales of Pepsico, which operates two major businesses: beverages (including Pepsi) and snack foods.(a) Find the change in sales between 2003 and 2015. (b) Find the average rate of change in sales between 2003 and 2015. Give units and interpret your answer.Table 1.17 Pepsico
Use Figure 1.94 to graph the function.y = f(x) + 1 -3 -2 -1 y y = f(x) 12 Figure 1.94 LX 3
One of the main contaminants of a nuclear accident, such as that at Chernobyl, is strontium-90, which decays exponentially at a rate of approximately 2.4% per year. (a) Write the percent of strontium-90 remaining, P, as a function of years, t, since the nuclear accident.(b) Graph P against
With time, t, in years since the start of 1980, textbook prices have increased at 6.7% per year while inflation has been 3.3% per year. Assume both rates are continuous growth rates. (a) Find a formula for B(t), the price of a textbook in year t if it cost $B0 in 1980. (b) Find a formula
In 2014, the populations of China and India were approximately 1.355 and 1.255 billion people, respectively. However, due to central control the annual population growth rate of China was 0.44% while the population of India was growing by 1.25% each year. If these growth rates remain constant, when
The US Consumer Price Index (CPI) is a measure of the cost of living. The inflation rate is the annual relative rate of change of the CPI. Use the January data in Table 1.24 to estimate the inflation rate for each of the years 2007–2012.Table 1.24 Year 2007 CPI
The functions r = f(t) and V = g(r) give the radius and the volume of a commercial hot air balloon being inflated for testing. The variable t is in minutes, r is in feet, and V is in cubic feet. The inflation begins at t = 0. In each case, give a mathematical expression that represents the given
During 2008 the US economy stopped growing and began to shrink. Table 1.25 gives quarterly data on the US Gross Domestic Product (GDP), which measures the size of the economy.(a) Estimate the relative growth rate (percent per year) at the first four times in the table.(b) Economists often say an
The functions r = f(t) and V = g(r) give the radius and the volume of a commercial hot air balloon being inflated for testing. The variable t is in minutes, r is in feet, and V is in cubic feet. The inflation begins at t = 0. In each case, give a mathematical expression that represents the given
An alternative to petroleum-based diesel fuel, biodiesel, is derived from renewable resources such as food crops, algae, and animal oils. The table shows the recent annual percent growth in US biodiesel exports.(a) Find the largest time interval over which the percentage growth in the US exports of
The functions r = f(t) and V = g(r) give the radius and the volume of a commercial hot air balloon being inflated for testing. The variable t is in minutes, r is in feet, and V is in cubic feet. The inflation begins at t = 0. In each case, give a mathematical expression that represents the given
Hydroelectric power is electric power generated by the force of moving water. Figure 1.48 shows the annual percent growth in hydroelectric power consumption by the US industrial sector between 2006 and 2014.(a) Find the largest time interval over which the percentage growth in the US consumption of
Cyanide is used in solution to isolate gold in a mine. This may result in contaminated groundwater near the mine, requiring the poison be removed, as in the following table, where t is in years since 2012. (a) Find an exponential model for c(t), the concentration, in parts per million, of
Solar panels are arrays of photovoltaic cells that convert solar radiation into electricity. The table shows the annual percent change in the US price per watt of a solar panel. (a) Find the largest time interval over which the percentage growth in the US price per watt of a solar panel was a
School organizations raise money by selling candy door-to-door. When the price is $1 a school organization sells 2765 candies and when the price goes up to $1.25 the quantity of candy sold drops down to 2440.(a) Find the relative change in the price of candy.(b) Find the relative change in the
The functions r = f(t) and V = g(r) give the radius and the volume of a commercial hot air balloon being inflated for testing. The variable t is in minutes, r is in feet, and V is in cubic feet. The inflation begins at t = 0. In each case, give a mathematical expression that represents the given
On January 22, 2017, the cost to mail a letter in the US was raised from 47 cents to 49 cents. Find the relative change in the cost.
You are buying a car that comes with a one-year warranty and are considering whether to purchase an extended warranty for $375. The extended warranty covers the two years immediately after the one-year warranty expires. You estimate that the yearly expenses that would have been covered by the
Morphine, a pain-relieving drug, is administered to a patient intravenously starting at 8 am. The drug saturation curve Q = f(t) in Figure 1.97 gives the quantity, Q, of morphine in the blood t hours after 8 am. (a) Draw the drug saturation curve Q = g(t) if the IV line is started at noon
You are considering whether to buy or lease a machine whose purchase price is $12,000. Taxes on the machine will be $580 due in one year, $464 due in two years, and $290 due in three years. If you buy the machine, you expect to be able to sell it after three years for $5,000. If you lease the
Find the relative change of a population if it changes (a) From 1000 to 2000 (b) From 2000 to 1000 (c) From 1,000,000 to 1,001,000
Use Figure 1.95 to graph the function. y = 2f(x) − 1 -5 -4 -3 -2 -1 2 y 1 Figure 1.95 2 y = f(x) 3 درا 4 5 X
Use Figure 1.95 to graph the function. y = 2f(x) -5 -4 -3 -2 -1 2 y 1 Figure 1.95 2 y = f(x) 3 درا 4 5 X
The population of a city is 50,000 in 2014 and is growing at a continuous rate of 4.5% per year. (a) Give the population of the city as a function of the number of years since 2014. Sketch a graph of the population against time. (b) What will be the city’s population in the year
(a) Use the Rule of 70 to predict the doubling time of an investment which is earning 8% interest per year. (b) Find the doubling time exactly, and compare your answer to part (a).
Find the relative, or percent, change.W changes from 0.3 to 0.05
During a recession a firm’s revenue declines continuously so that the revenue, R (measured in millions of dollars), in t years’ time is given by R = 5e−0.15t.(a) Calculate the current revenue and the revenue in two years’ time. (b) After how many years will the revenue decline to $2.7
Use Figure 1.95 to graph the function. y = f(x) + 2 -5 -4 -3 -2 -1 2 y 1 Figure 1.95 2 y = f(x) 3 درا 4 5 X
A car starts slowly and then speeds up. Eventually the car slows down and stops. Graph the distance that the car has traveled against time.
Experiments suggest that the male maximum heart rate (the most times a male’s heart can safely beat in a minute) decreases by 9 beats per minute during the first 21 years of his life, and by 26 beats per minute during the first 33 years. If you model the maximum heart rate as a function of age,
The gross world product is W = 74.31(1.029)t, where W is in trillions of dollars and t is years since 2013. Find a formula for gross world product using a continuous growth rate.
Concern biodiesel, a fuel derived from renewable resources such as food crops, algae, and animal oils. The table shows the percent growth over the previous year in US biodiesel consumption.(a) According to the US Department of Energy, the US consumed 322 million gallons of biodiesel in 2009.
Ask you to plot graphs based on the following story: “As I drove down the highway this morning, at first traffic was fast and uncongested, then it crept nearly bumper to-bumper until we passed an accident, after which traffic flow went back to normal until I exited.”Driving speed against time
In November 2010, a “tiger summit” was held in St. Petersburg, Russia. In 1900, there were 100,000 wild tigers worldwide; in 2010 the number was 3200. (a) Assuming the tiger population has decreased exponentially, find a formula for f(t), the number of wild tigers t years since
A picture supposedly painted by Vermeer (1632–1675) contains 99.5% of its carbon-14 (half-life 5730 years). From this information decide whether the picture is a fake. Explain your reasoning.
In 2010, there were about 246 million vehicles (cars and trucks) and about 308.7 million people in the US. The number of vehicles grew 15.5% over the previous decade, while the population has been growing at 9.7% per decade. If the growth rates remain constant, when will there be, on average, one
Figure 1.96 shows the concentration of a drug in the body, f(t) = a − bekt, over time, t:(a) What does the constant a represent in this context? (b) What is the sign of the constant a? (c) What is the relation between the constants a and b? (d) What is the sign of the constant k?
You win $38,000 in the state lottery to be paid in two installments—$19,000 now and $19,000 one year from now. A friend offers you $36,000 in return for your two lottery payments. Instead of accepting your friend’s offer, you take out a one-year loan at an interest rate of 8.25% per year,
Use Figure 1.95 to graph the function. y = 2 − f(x) -5 -4 -3 -2 -1 2 y 1 Figure 1.95 2 y = f(x) 3 درا 4 5 X
Which relative change is bigger in magnitude? Justify your answer.An increase in sales from $100,000 to $500,000; an increase in sales from $20,000,000 to $20,500,000.
Big Tree McGee is negotiating his rookie contract with a professional basketball team. They have agreed to a three-year deal which will pay Big Tree a fixed amount at the end of each of the three years, plus a signing bonus at the beginning of his first year. They are still haggling about the
Which relative change is bigger in magnitude? Justify your answer.An increase in class size from 5 to 10; an increase in class size from 30 to 50.
A company is considering whether to buy a new machine, which costs $97,000. The cash flows (adjusted for taxes and depreciation) that would be generated by the new machine are given in the following table: (a) Find the total present value of the cash flows. Treat each year’s cash flow as a
Which relative change is bigger in magnitude? Justify your answer.The change in the US population from 5.2 million to 7.2 million from 1800 to 1810; the change in the US population from 151.3 to 179.3 from 1950 to 1960.
The concentration of the car exhaust fume nitrous oxide, NO2, in the air near a busy road is a function of distance from the road. The concentration decays exponentially at a continuous rate of 2.54% per meter. At what distance from the road is the concentration of NO2 half what it is on the road?
Use Figure 1.95 to graph the function. y = −3f(x) -5 -4 -3 -2 -1 2 y 1 Figure 1.95 2 y = f(x) 3 درا 4 5 X
A person is to be paid $2000 for work done over a year. Three payment options are being considered. Option 1 is to pay the $2000 in full now. Option 2 is to pay $1000 now and $1000 in a year. Option 3 is to pay the full $2000 in a year. Assume an annual interest rate of 5% a year, compounded
Which relative change is bigger in magnitude? Justify your answer.The change in the Dow Jones average from 164.6 to 77.9 in 1931; the change in the Dow Jones average from 13,261.8 to 8776.4 in 2008.
Use Figure 1.95 to graph the function. y = f(x − 1) -5 -4 -3 -2 -1 2 y 1 Figure 1.95 2 y = f(x) 3 درا 4 5 X
For children and adults with diseases such as asthma, the number of respiratory deaths per year increases by 0.33% when pollution particles increase by a microgram per cubic meter of air.(a) Write a formula for the number of respiratory deaths per year as a function of quantity of pollution in the
A business associate who owes you $3000 offers to pay you $2800 now, or else pay you three yearly installments of $1000 each, with the first installment paid now. If you use only financial reasons to make your decision, which option should you choose? Justify your answer, assuming a 3% interest
Find the relative, or percent, change.R changes from 50 to 47
Hydroelectric power is electric power generated by the force of moving water. The table shows the annual percent change in hydroelectric power consumption by the US industrial sector. (a) According to the US Department of Energy, the US industrial sector consumed about 2.65 quadrillion BTUs of
The demand and supply curves for a product are given in terms of price, p, byq = 2500 − 20p and q = 10p − 500.(a) Find the equilibrium price and quantity. Represent your answers on a graph.(b) A specific tax of $6 per unit is imposed on suppliers. Find the new equilibrium price and quantity.
Find the present value of an $8000 payment to be made in 5 years. The interest rate is 4% per year compounded continuously.
An Australian study found that, if other factors are constant (education, experience, etc.), taller people receive higher wages for the same work. The study reported a “height premium” for men of 3% of the hourly wage for a 10 cm increase in height; for women the height premium reported was 2%.
The population of the world can be represented by P = 7.32(1.0109)t, where P is in billions of people and t is years since 2015. Find a formula for the population of the world using a continuous growth rate.
Concern wind power, which has been used for centuries to propel ships and mill grain. Modern wind power is obtained from windmills that convert wind energy into electricity. Figure 1.74 shows the annual percent growth in US wind power consumption between 2009 and 2014.(a) According to the US
The demand and supply curves are given by q = 100 − 2p and q = 3p − 50, respectively; the equilibrium price is $30 and the equilibrium quantity is 40 units. A sales tax of 5% is imposed on the consumer.(a) Find the equation of the new demand and supply curves.(b) Find the new equilibrium price
Find the present value of a $20,000 payment to be made in 10 years. Assume an interest rate of 3.2% per year compounded continuously.
A fishery stocks a pond with 1000 young trout. The number of trout t years later is given by P(t) = 1000e−0.5t.(a) How many trout are left after six months? After 1 year? (b) Find P(3) and interpret it in terms of trout. (c) At what time are there 100 trout left? (d) Graph the
Find the relative, or percent, change.B changes from 12,000 to 15,000
Concern wind power, which has been used for centuries to propel ships and mill grain. Modern wind power is obtained from windmills that convert wind energy into electricity. Figure 1.74 shows the annual percent growth in US wind power consumption between 2009 and 2014.(a) According to Figure 1.74,
Answer the questions in Problem 44, assuming that the 5% sales tax is imposed on the supplier instead of the consumer.Data from in Problem 44The demand and supply curves are given by q = 100 − 2p and q = 3p − 50, respectively; the equilibrium price is $30 and the equilibrium quantity is 40
Interest is compounded annually. Consider the following choices of payments to you: Choice 1: $1500 now and $3000 one year from now Choice 2: $1900 now and $2500 one year from now (a) If the interest rate on savings were 5% per year, which would you prefer? (b) Is there an
The Hershey Company is the largest US producer of chocolate. In 2011, annual net sales were 6.1 billion dollars and were increasing at a continuous rate of 7% per year. (a) Write a formula for annual net sales, S, as a function of time, t, in years since 2011. (b) Estimate annual net
Find the relative, or percent, change. S changes from 400 to 450
The island of Manhattan was sold for $24 in 1626. Suppose the money had been invested in an account which compounded interest continuously. (a) How much money would be in the account in the year 2012 if the yearly interest rate was(i) 5%? (ii) 7%? (b) If the yearly interest rate was
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