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mathematics
calculus with applications

Questions and Answers of Calculus With Applications

In the exponential growth or decay function y = y0 ekt, what does y0 represent? What does k represent?
Determine whether each of the following statements is true or false, and explain why.The function g(x) = xπ is an exponential function.
Determine whether each of the following statements is true or false, and explain why.The function ƒ(x) = x2 - 6x + 4 has a vertex at x = 3.
Determine whether each statement is true or false, and explain why.The effective rate of interest is always greater than the nominal rate when interest is compounded.
Determine whether each of the following statements is true or false, and explain why.The function ƒ(x) = 3x2 - 75x + 2 is a quadratic function.
Determine whether each statement is true or false, and explain why.An equation of the form y = y0ekt can also be written as y = y0at, where a = ek.
Determine whether each of the following statements is true or false, and explain why.A rational function is an example of an exponential function.
Determine whether each statement is true or false, and explain why.A substance that decays exponentially will be reduced to 1/8 of its original amount after 3 half-lives.
Determine whether each of the following statements is true or false, and explain why.A linear function is an example of a polynomial function.
Determine whether each statement is true or false, and explain why.If a substance decays according to the equation y = y0ekt, the constant k must be negative.
A music theorist associates the fundamental frequency of a pitch ƒ with a real number defined by p = 69 + 12 log2(ƒ/440).(a) Standard concert pitch for an A is 440 cycles per second. Find the
When a pharmaceutical drug is injected into the bloodstream, its concentration at time t can be approximate by C(t) = C0e-kt, where C0 is the concentration at t = 0. Suppose the drug is ineffective
The enrollment in Head Start for some recent years is included in the table.(a) Plot the points from the table using 0 for 1970, and so on.(b) Use the quadratic regression feature of a graphing
The strength of a contact lens is given in units known as diopters, as well as in mm of arc. The table below is taken from a chart used by optometrists to convert diopters to mm of arc.(a) Notice
Solve each equation in Exercises. Round decimal answers to four decimal places.3x+1 = 5x
From 2002 to 2016, the age-adjusted incidence rate of invasive lung and bronchial cancer among women can be closely approximated by ƒ(t) = -0.0723t2 + 0.723t + 52.0, where t is the number of years
A study done by an advertising agency reveals that when x thousands of dollars are spent on advertising, it results in a sales increase in thousands of dollars given by the function(a) Find the
For the years 2000 to 2019, the number of applicants to U.S. medical schools can be closely approximated by A(t) = 0.1953t4 - 13.70t3 + 282.9t2 - 836.7t + 35,842, where t is the number of years since
Solve each equation in Exercises. Round decimal answers to four decimal places.5x = 12
The atmospheric pressure (in millibars) at a given altitude (in meters) is listed in the table.(a) Find functions of the form P = a • bx, P = mx + b, and P = 1/(ax + b) that fit the data at x = 0
Suppose a cost-benefit model is given bywhere y is the cost in thousands of dollars of removing x percent of a given pollutant.(a) Find the cost of removing each percent of pollutants: 50%; 70%; 80%;
The loudness of sounds is measured in a unit called a decibel. To do this, a very faint sound, called the threshold sound, is assigned an intensity I0. If a particular sound has intensity I, then the
According to the Shannon-Hartley theorem, the capacity of a communications channel in bits per second is given bywhere B is the frequency bandwidth of the channel in hertz and s/n is its
The graph below shows the basal metabolism rate (in cm3 of oxygen per gram per hour) for marsupial carnivores, which include the Tasmanian devil. This rate is inversely proportional to body mass
The allometric formula is used to describe a wide variety of growth patterns. It says that y = nxm, where x and y are variables, and n and m are constants. For example, the famous biologist J. S.
A common measure for the acidity of a solution is its pH. It is defined by pH = -log [H+], where H+ measures the concentration of hydrogen ions in the solution. The pH of pure water is 7. Solutions
A story on the National Public Radio program All Things Considered discussed a proposal to lower the noise limit in Austin, Texas, from 85 decibels to 75 decibels. A manager for a restaurant was
Researchers found a close correlation between an index of universal health care (UHC) and per capita Gross Domestic Product (GDP) given by the equation U = 8.9603 ln x - 13.911, where U is the index
The U.S. Census Bureau has reported that the United States is becoming more diverse. In Exercise 55 of the previous section, the projected Hispanic population (in millions) was modeled by the
Find the value of the index of diversity for populations with n species and 1/n of each if(a) n = 3; (b) n = 4.(c) Verify that your answers for parts (a) and (b) equal ln 3 and ln 4,
Complete the following table, and use the results to discuss when the rule of 70 gives a better approximation for the doubling time, and when the rule of 72 gives a better approximation. r (In
A virgin forest in northwestern Pennsylvania has 4 species of large trees with the following proportions of each: hemlock, 0.521; beech, 0.324; birch, 0.081; maple, 0.074. Find the index of diversity
Suppose a sample of a small community shows two species with 50 individuals each.(a) Find the index of diversity H.(b) What is the maximum value of the index of diversity for two species?(c) Does
An article in Science stated that the number of insect species of a given mass is proportional to m-0.6, where m is the mass in grams. A graph accompanying the article shows the common logarithm of
You are offered two jobs starting July 1, 2021. Humongous Enterprises offers you $45,000 a year to start, with a raise of 4% every July 1. At Crabapple Inc. you start at $30,000, with an annual
In Exercise of the previous section, the Gross Domestic Product (GDP) for the United States was modeled by the exponential function ƒ(t) = 03(1.07022)t, where ƒ(t) was the GDP (in billions of
Andrea plans to invest $600 into a money market account. Find the interest rate that is needed for the money to grow to $1240 in 14 years if the interest is compounded continuously.
Allison invests $15,000 in an account paying 7% per year compounded annually.(a) How many years are required for the compound amount to at least double?(b) In how many years will the amount at least
Assuming annual compounding, find the time it would take for the general level of prices in the economy to double at the following annual inflation rates.(a) 3% (b) 6% (c) 8%(d) Check your
Draw a sketch of the arch or culvert on coordinate axes, with the horizontal and vertical axes through the vertex of the parabola. Use the given information to label points on the parabola. Then give
The following table gives the median age at the first marriage of women in the United States for some selected years.(a) Plot the data using t = 50 for 1950, and so on.(b) Would a linear or quadratic
According to recent data from the Teachers Insurance and Annuity Association (TIAA), the survival function for life after 65 is approximately given by S(x) = 1 - 0.058x - 0.076x2, where x is measured
In Exercises , find the domain of each function.ƒ(x) = ln(x2 - 9)
What would be the maximum area that could be enclosed by the college’s 380 ft of fencing if it decided to close the entrance by enclosing all four sides of the lot? (See Exercise 73.)Data from
In Exercises , find the domain of each function.ƒ(x) = log(5 - x)
Glenview Community College wants to construct a rectangular parking lot on land bordered on one side by a highway. It has 380 ft of fencing to use along the other three sides. What should be the
According to data from the National Traffic Safety Institute, the stopping distance y in feet of a car traveling x mph can be described by the equation y = 0.056057x2 + 1.06657x.(a) Find the stopping
Approximate each expression in the form ax without using e.e-4x
If an object is thrown upward with an initial velocity of 32 ft /second, then its height after t seconds is given by h(t) = 32t - 16t2.(a) Find the maximum height attained by the object.(b) Find the
The number of males per 100 females, age 70 or over, in the United States for some recent years is shown in the following table.(a) Plot the data, letting t be the years since 1970.(b) Would a linear
Approximate each expression in the form ax without using e.e3x
Write each expression using base e rather than base 10.10x2
Write each expression using base e rather than base 10.10x+1
Solve each equation in Exercises. Round decimal answers to four decimal places.1.51(.05)x = 2(1.01)x
Blood flow to the fetal spleen is of research interest because several diseases are associated with increased resistance in the splenic artery (the artery that goes to the spleen). Researchers have
The mass (in grams) of the human brain during the last trimester of gestation and the first two years after birth can be approximated by the functionwhere c is the circumference of the head in
The functionis used in biology to give the growth rate of a population in the presence of a quantity x of food. This is called Michaelis- Menten kinetics.(a) What is a reasonable domain for this
Solve each equation in Exercises. Round decimal answers to four decimal places.e2y = 15
The following table gives the amount of coal (in thousand short tons) consumed by U. S. utilities to produce electricity.(a) Plot the data by letting t be the years since 2000. Would a linear or
The functionis used in population models to give the size of the next generation (ƒ(x)) in terms of the current generation (x).(a) What is a reasonable domain for this function, considering what x
Solve each equation in Exercises. Round decimal answers to four decimal places.ek-1 = 6
The length (in mm) of the mesiodistal crown of the first molar for human fetuses can be approximated by L(t) = -0.01t2 + 0.788t - 7.048, where t is the number of weeks since conception.(a) What does
Solve each equation in Exercises. Round decimal answers to four decimal places.e2y = 15
The table below gives the total world wind energy capacity (in megawatts) in recent years.(a) Let t be the number of years since 2000, and C the capacity (in MW). Find functions of the form C = mt +
Solve each equation in Exercises. Round decimal answers to four decimal places.ek-1 = 6
Suppose a cost-benefit model is given bywhere y is the cost in thousands of dollars of removing x percent of a certain pollutant.(a) Find the cost of removing each percent of pollutants: 0%; 50%;
The table below gives U.S. coal consumption for selected years.(a) Draw a scatterplot, letting t = 0 represent 1950.(b) Use the quadratic regression feature of a graphing calculator to get a
According to Deuteronomy 32:30, “How could one man chase a thousand, or two put ten thousand to flight, unless their Rock had sold them, unless the Lord had given them up?” Letting x represent
The polynomial function A(x) = 0.003631x3 - 0.03746x2 + 0.1012x + 0.009 gives the approximate blood alcohol concentration in a 170-lb woman x hours after drinking 2 oz of alcohol on an empty
Solve each equation in Exercises. Round decimal answers to four decimal places.2x = 6
Solve each equation in Exercises. Round decimal answers to four decimal places.ln (x + 1) - ln x = 1
Solve each equation in Exercises. Round decimal answers to four decimal places.ln x + ln 3x = -1
Solve each equation in Exercises. Round decimal answers to four decimal places.ln(5x + 4) = 2
Exercises refer to the Laffer curve, originated by the economist Arthur Laffer. An idealized version of this curve is shown here. According to this curve, decreasing a tax rate, say from x2 percent
Let C(x) be the cost to produce x batches of widgets, and let R(x) be the revenue in thousands of dollars. For each exercise, (a) Graph both functions, (b) Find the minimum break-even
Solve each equation in Exercises. Round decimal answers to four decimal places.log2 (x2 - 1) - log2 (x + 1) = 2
The incidence of the disease shingles increases with age, as shown by the following table.(a) Plot the data. Does fitting an exponential curve to the data seem reasonable?(b) Use the data for 7 years
The revenue of a charter bus company depends on the number of unsold seats. If the revenue R(x) is given by R(x) = 8000 + 70x - x2 , where x is the number of unsold seats, find the maximum revenue
The quantity (in grams) of a sample of radon-222 present at time t, measured in days, is given by A(t) = 1.2(10)-0.0788t.(a) How much will be present in 10 days?(b) How long will it take to reduce
Exercises refer to the Laffer curve, originated by the economist Arthur Laffer. An idealized version of this curve is shown here. According to this curve, decreasing a tax rate, say from x2 percent
Solve each equation in Exercises. Round decimal answers to four decimal places.log3 (x2 + 17) - log3 (x + 5) = 1
Let C(x) be the cost to produce x batches of widgets, and let R(x) be the revenue in thousands of dollars. For each exercise, (a) Graph both functions, (b) Find the minimum break-even
Exercises refer to the Laffer curve, originated by the economist Arthur Laffer. An idealized version of this curve is shown here. According to this curve, decreasing a tax rate, say from x2 percent
According to the U.S. Census Bureau, the United States is becoming more diverse. Based on U.S. Census population projections for 2016 to 2060, the projected Hispanic population (in millions)
Let C(x) be the cost to produce x batches of widgets, and let R(x) be the revenue in thousands of dollars. For each exercise, (a) Graph both functions, (b) Find the minimum break-even
Solve each equation in Exercises. Round decimal answers to four decimal places.log3 (x - 2) + log3 (x + 6) = 2
The U.S. Gross Domestic Product (GDP) is the output of goods and service produced by labor and property located in the United States. The following table gives the GDP (in billions of current
In a recent year, the cost per ton, y, to build an oil tanker of x thousand deadweight tons was approximated byfor x > 0.(a) Find C(25), C(50), C(100), C(200), C(300), and C(400).(b) Find any
Solve each equation in Exercises. Round decimal answers to four decimal places.log(x + 5) + log(x + 2) = 1
Since 1960, the growth in world population (in millions) closely fits the exponential function defined by A(t) = 3184e0.0158t, where t is the number of years since 1960.(a) World population was about
Solve each equation in Exercises. Round decimal answers to four decimal places.log9 m - log9 (m - 4) = -2
Let C(x) be the cost to produce x batches of widgets, and let R(x) be the revenue in thousands of dollars. For each exercise, (a) Graph both functions, (b) Find the minimum break-even
Solve each equation in Exercises. Round decimal answers to four decimal places.log4 x - log4 (x + 3) = -1
Solve each equation in Exercises. Round decimal answers to four decimal places.log5 (9x - 4) = 1
Suppose the fixed cost to make chocolate croissants is $90, and the marginal cost to make one croissant is $0.68.(a) Write a formula for the average cost function.(b) As the number of croissants
Lucky Larry was faced with solvinglog(2x + 1) - log(3x - 1) = 0.Larry just dropped the logs and proceeded:(2x + 1) - (3x - 1) = 0-x + 2 = 0x = 2.Although Lucky Larry is wrong in dropping the logs,
Using the graph of f (x) in Figure 10, show the graph of f (ax) where a satisfies the given condition.1 < a Figure 10 y=f(x) X
Janille needs to choose between two investments: One pays 6% compounded annually, and the other pays 5.9, compounded monthly. If she plans to invest $18,000 for 2 years, which investment should
Use natural logarithms to evaluate each logarithm to the nearest thousandth.log1.2 0.95
In Exercises, find any horizontal and vertical asymptotes and any holes that may exist for each rational function. Draw the graph of each function, including any x- and y-intercepts. y = x² + 7x +

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