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mathematics
calculus with applications
Calculus With Applications 10th Edition Margaret L Lial, Raymond N Greenwell, Nathan P Ritchey - Solutions
Find derivatives of the functions defined as follows.s = 2 · 3√t
Use graphical differentiation to verify that d/dx (ex) = ex.
Find each derivative.f'(–2) if f(x) = x4/6 – 3x
Find derivatives of the functions defined as follows.s = 5 · 2√t–2
Find each derivative.f'(3) if f(x) = x3/9 – 7x2
Find derivatives of the functions defined as follows.y = tet + 2/e2t + 1
Find the derivative of each function defined as follows.y = (3x4 + 1)4 (x3 + 4)
Find derivatives of the functions defined as follows.y = t2e2t/t + e3t
Find the derivative of each function defined as follows.y = (x3 + 2)(x2 – 1)4
In Exercises 31–34, find the slope of the tangent line to the graph of the given function at the given value of x. Find the equation of the tangent line.y = x4 – 5x3 + 2; x = 2
In Exercises 31–34, find the slope of the tangent line to the graph of the given function at the given value of x. Find the equation of the tangent line.y = –3x5 – 8x3 + 4x2; x = 1
Find derivatives of the functions defined as follows.f(x) = ex√3x + 2
In Exercises 31–34, find the slope of the tangent line to the graph of the given function at the given value of x.y = –2x1/2 + x3/2; x = 9
Find the derivative of each function defined as follows.q(y) = 4y2(y2 + 1)5/4
Find an equation of the line tangent to the graph of f(x) = x/(x – 2) at (3, 3).
Find derivatives of the functions defined as follows.f(x) = ex2/(x3 + 2)
Find the derivative of each function defined as follows.p(z) = z(6z + 1)4/3
In Exercises 31–34, find the slope of the tangent line to the graph of the given function at the given value of x.y = –x–3 + x–2; x = 2
Consider the functiona. Find the derivative using the quotient rule.b. Find the derivative by first simplifying the function c. Compare your answers from parts a and b and explain any discrepancies. f(x): = 3x³ + 6 X-2/3
Find the derivative of each function defined as follows.y = 1/(3x2 – 4)5
Find the derivative of each function defined as follows.r(t) = (5t – 6)4/3t2 + 4
The sales of a new personal computer (in thousands) are given by S(t) = 100 – 90e–0.3t, where t represents time in years. Find the rate of change of sales at each time.a. After 1 year b. After 5 yearsc. What is happening to the rate of change of sales as time goes on?d. Does the rate of
The cost in dollars to produce x DVDs can be approximated byFind the marginal cost when the following quantities are made.a. 0 b. 20c. What happens to the marginal cost as the number produced becomes larger and larger? C(x) = √900-800-1.1¯*.
Find the derivative of each function defined as follows.p(t) = (2t + 3)3/4t2 – 1
Find the derivative of each function defined as follows.y = 3x2 – x/(2x – 1)5
After the introduction of a new product for tanning without sun exposure, the percent of the public that is aware of the product is approximated by A(t) = 10t22–t, where t is the time in months. Find the rate of change of the percent of the public that is aware of the product after the following
Find the derivative of each function defined as follows.y = x2 + 4x/(3x3 + 2)4
Using data in a car magazine, we constructed the mathematical model y = 100e–0.03045t for the percent of cars of a certain type still on the road after t years. Find the percent of cars on the road after the followingnumbers of years.a. 0 b. 2 c. 4 d. 6Find the rate of change of
The generalized power rule says that if g(x) is a function of x and y = [g(x)]n for any real number n, thenExplain why the generalized power rule is a consequence of the chain rule and the power rule. dy dx = n. [g(x)] -1.g(x).
The value of a particular investment changes over time according to the function S(t) = 5000e0.1(e0.25t), where S(t) is the value after t years. Calculate the rate at which the value of the investment is changing after 8 years. (Choose one of the following.)a. 618 b. 1934 c. 2011 d.
The growth in the number (in millions) of Internet users in the United States between 1990 and 2015 can be approximated by a logistic function with k = 0.0018, where t is the number of years since 1990. In 1990 (when t = 0), the number of users was about 2 million, and the number is expected to
It has been observed that the following formula accurately models the relationship between the size of a breast tumor and the amount of time that it has been growing.where t is in months and V(t) is measured in cubic centimeters.a. Find the tumor volume at 240 months.b. Assuming that the shape of a
The growth of a population of rare South American beetles is given by the logistic function with k = 0.00001 and t in months. Assume that there are 200 beetles initially and that the maximum population size is 10,000.a. Find the growth function G(t) for these beetles. Find the population and rate
The population of a bed of clams in the Great South Bay off Long Island is described by the logistic function with k = 0.0001 and t in years. Assume that there are 400 clams initially and that the maximum population size is 5200.a. Find the growth function G(t) for the clams. Find the population
The concentration of pollutants (in grams per liter) in the east fork of the Big Weasel River is approximated by P(x) = 0.04e–4x, where x is the number of miles downstream from a paper mill that the measurement is taken. Find the following values.a. The concentration of pollutants 0.5 mile
The percentage of people of any particular age group that will die in a given year may be approximated by the formulawhere t is the age of the person in years.a. Find P(25), P(50), and P(75)b. Find P'(25), P'(50) and p'(75)c. Interpret your answers for parts a and b. Are there any limitations of
It has been observed that there has been an increase in the proportion of medical research papers that use the word “novel” in the title or abstract, and that this proportion can be accurately modeled by the functionwhere x is the number of years since 1970.a. Find p(40)b. If this phenomenon
Assume that the total number (in millions) of bacteria present in a culture at a certain time t (in hours) is given by N(t) = 3t(t – 10)2 + 40a. Find N'(t)Find the rate at which the population of bacteria is changing at the following times.b. 8 hours c. 11 hoursd. The answer in part b is
Researchers have compared two models that are used to predict the weight of beef cattle of various ages,andwhere w1(t) and w2(t) represent the weight (in kilograms) of a t-day-old beef cow.a. What is the maximum weight predicted by each function for the average beef cow? Is this difference
The age/weight relationship of female Arctic foxes caught in Svalbard, Norway, can be estimated by the function where t is the age of the fox in days and M(t) is the weight of the fox in grams. Source: Journal of Mammalogy.a. Estimate the weight of a female fox that is 200 days old.b. Use M(t)
Suppose that for the situation in Exercise 53 the cost in dollars of producing q stereo systems is given by C(q) = 0.2q2 + 6q + 50 Find the marginal profit when the demand is 10.Data from Exercises 53.If the price in dollars of a stereo system is given bywhere q represents the demand for the
Researchers have found that the risk of coronary heart disease rises as blood cholesterol increases. This risk may be approximated by the functionwhere R is the risk in terms of coronary heart disease incidence per 1000 per year, and c is the cholesterol in mg/dL. Suppose a person’s cholesterol
Often sales of a new product grow rapidly at first and then level off with time. This is the case with the sales represented by the functionwhere t represents time in years. Find the rate of change of sales for the following numbers of years.a. 1 b. 10 S(t) 100 100t-¹, =
According to work by the psychologist C. L. Hull, the strength of a habit is a function of the number of times the habit is repeated. If N is the number of repetitions and H(N) is the strength of the habit, thenwhere k is a constant. Find H'(N) if K = 0.1 and the number of times the habit is
Paleontologist John Cisne has demonstrated that the survival of ancient manuscripts can be modeled by the logistic equation. For example, the number of copies of the Venerable Bede’s De Temporum Ratione was found to approach a limiting value over the five centuries after its publication in the
The total amount of money in circulation for the years 1950–2009 can be closely approximated bywhere t represents the number of years since 1950 and M(t) is in billions of dollars. Find the derivative of M(t) and use it to find the rate of change of money in circulation in the following years.a.
The growth of the number of students taking at least one online course can be approximated by a logistic function with k = 0.0440, where t is the number of years since 2002. In 2002 (when t = 0), the number of students enrolled in at least one online course was 1.603 million. Assume that the number
The amount (in grams) of a sample of lead 214 present after t years is given byFind the rate of change of the quantity present after each of the following years.a. 4 b. 6c. 10d. What is happening to the rate of change of the amount present as the number of years increases?e. Will the substance
In a series resistance-capacitance DC circuit, the instantaneous charge Q on the capacitor as a function of time (where t = 0 is the moment the circuit is energized by closing a switch) is given by the equationwhere C, V, and R are constants. Further, the instantaneous charging current Ic is the
The heat index is a measure of how hot it really feels under different combinations of temperature and humidity. The heat index, in degrees Fahrenheit, can be approximated bywhere the temperature T and dewpoint D are both expressed in degrees Fahrenheit. Source: American Meteorological Society.a.
In 1958, L. Lucy developed a method for predicting the world record for any given year that a human could run a distance of 1 mile. His formula is given as follows:where t(n) is the world record (in seconds) for the mile run in year 1950 + n. Thus, n = 5 corresponds to the year 1955.a. Find the
Suppose a person is going up in a hot air balloon. The surrounding air temperature in degrees Fahrenheit decreases with height according to the formulawhere h is the height in feet. How fast is the temperature decreasing when the person is at a height of 1000 ft and rising at a height of 800 ft/hr?
The Gateway Arch in St. Louis, Missouri, is approximately 630 ft wide and 630 ft high. At first glance, the arch resembles a parabola, but its shape is actually known as a modified catenary. The height s (in feet) of the arch, measured from the ground to the middle of the arch, is approximated by
Consider an experiment in which equal numbers of male and female insects of a certain species are permitted to intermingle. Assume thatrepresents the number of matings observed among the insects in an hour, where t is the temperature in degrees Celsius.a. Find the number of matings when the
The number (in billions) of pieces of mail handled by the U.S. Post Office each year from 1980 through 2009 can be approximated bywhere t is the number of years since 1980. Find and interpret the rate of change in the volume of mail for the following years.a. 1995 b. 2005 P(t) = -0.00132t4
Gwartney et al. listed the following data relating the price of a monthly cellular bill (in dollars) and the demand (in millions of subscribers).a. Using a graphing calculator, plot the natural logarithm of the price against the natural logarithm of the quantity. Does the relationship appear to be
For many years researchers thought that the basal metabolic rate (BMR) in mammals was a power function of the mass, with disagreement on whether the power was 0.67 or 0.75. More recently, White et al. proposed that the power may vary for mammals with different evolutionary lineages. The following
Determine whether each of the following statements is true or false, and explain why.A rational function is an example of an exponential function.
What is meant by the half-life of a quantity?
Find the present value of each amount. $10,000 if interest is 6% compounded quarterly for 8 years
Tami Dreyfus bought a television set with money borrowed from the bank at 9% interest compounded semiannually. What effective interest rate did she pay?
Robin Kim deposits $7500 of lottery winnings in an account paying 6% interest compounded monthly. What effective rate does the account earn?
How do you find a vertical asymptote? A horizontal asymptote?
Christine O’Brien, who is self-employed, wants to invest $60,000 in a pension plan. One investment offers 8% compounded quarterly. Another offers 7.75% compounded continuously.a. Which investment will earn the most interest in 5 years?b. How much more will the better plan earn?c. What is the
Greg Tobin wishes to invest a $5000 bonus check into a savings account that pays 6.3% interest. Find how many years it will take for the $5000 to grow to at least $11,000 if interest is compoundeda. Quarterly.b. Continuously.
Describe in words what a logarithm is.
The population of the world in the year 1650 was about 500 million, and in the year 2010 was 6756 million.a. Assuming that the population of the world grows exponentially, find the equation for the population p(t)in millions in the year t.b. Use your answer from part a to find the population of the
A culture contains 25,000 bacteria, with the population increasing exponentially. The culture contains 40,000 bacteria after 10 hours.a. Write a function in the form y = y0ekt giving the number of bacteria after t hours.b.Write the function from part a in the form y = y0at.c. How long will it be
When an antibiotic is introduced into a culture of 50,000 bacteria, the number of bacteria decreases exponentially. After 9 hours, there are only 20,000 bacteria.a. Write an exponential equation to express the growth function y in terms of time t in hours.b. In how many hours will half the number
If money loses value at the rate of 8% per year, the value of $1 in t years is given bya. Use a calculator to help complete the following table.b. Graph y = (0.92)t.c. Suppose a house costs $165,000 today. Use the results of part a to estimate the cost of a similar house in 10 years.d. Find the
The half-life of plutonium-241 is approximately 13 years.a. How much of a sample weighing 4 g will remain after 100 years?b. How much time is necessary for a sample weighing 4 g to decay to 0.1 g?
A group of Tasmanian botanists have claimed that a King’s holly shrub, the only one of its species in the world, is also the oldest living plant. Using carbon-14 dating of charcoal found along with fossilized leaf fragments, they arrived at an age of 43,000 years for the plant, whose exact
Graph the following by hand.y = log2(x – 1)
Graph the following by hand.y = –ln(x + 3)
Graph the following by hand.y = 2 – lnx2
Solve each equation.2x+2 = 1/8
Solve each equation.(9/16)x = 3/4
Solve each equation.92y+3 = 27y
Solve each equation.1/2 = (b/4)1/4
The number of males per 100 females, age 65 or over, in the United States for some recent years is shown in the following table.a. Plot the data, letting x be the years since 1900.b. Would a linear or quadratic function best model this data? Explain.c. If your graphing calculator has a quadratic
Write each equation using logarithms.e0.8 = 2.22554
The following table lists the reported number of cases of infants born in the United States with HIV in recent years because their mother was infected.a. Plot the data on a graphing calculator, letting t = 0 correspond to the year 1995.b. Using the regression feature on your calculator, find a
Find the present value of each amount.Frank Steed wants to open a camera shop. How much must he deposit now at 6% interest compounded monthly to have $25,000 at the end of 3 years?
In 1960 in an article in Science magazine, H. Van Forester, P. M. Mora, and W. Amiot predicted that world population would be infinite in the year 2026. Their projection was based on the rational function defined bywhere p(t) gives population in year t. This function has provided a relatively good
A population of 15,000 small deer in a specific region has grown exponentially to 17,000 in 4 years.a. Write an exponential equation to express the population growth y in terms of time t in years.b. At this rate, how long will it take for the population to reach 45,000?
a. Suppose g(x) = 3√x. Use the graph of g(x) to find g'(0).b. Explain why the derivative of a function does not exist at a point where the tangent line is vertical.
Ifwhere is f not differentiable? f(x) x² - 1 x + 2
The graph shows annual sales (in thousands of dollars) of a Nintendo game at a particular store. Sketch a graph of the rate of change of sales as a function of time. Sales 12 10 00 6 4 2 0 3 1. 10 10 8 6 3 2 H + + 123 4 5 6 7 8 9 10 11 12 Time (in years)
LetUse the limit rules to find each limit. lim f(x) = 9 and lim g(x) = 27. x-4 x-4
The graph in the next column shows how the number of arthropod species resistant to insecticides has varied with time. Sketch a graph of the rate of change of the insecticide–resistant species as a function of time. Number of resistant species 600 500 400 300 200 100 1940 1950 1960 Neonicotinoids
LetUse the limit rules to find each limit. lim f(x) = 9 and lim g(x) = 27. x-4 x-4
LetUse the limit rules to find each limit. lim f(x) = 9 and lim g(x) = 27. x-4 x-4
LetUse the limit rules to find each limit. lim f(x) = 9 and lim g(x) = 27. x-4 x-4
LetUse the limit rules to find each limit. lim f(x) = 9 and lim g(x) = 27. x-4 x-4
LetUse the limit rules to find each limit. lim f(x) = 9 and lim g(x) = 27. x-4 x-4
LetUse the limit rules to find each limit. lim f(x) = 9 and lim g(x) = 27. x-4 x-4
LetUse the limit rules to find each limit. lim f(x) = 9 and lim g(x) = 27. x-4 x-4
If $1000 is invested in an account that pays 5% compounded annually, the total amount, A(t), in the account after t years isa. Find the average rate of change per year of the total amount in the account for the first five years of the investment (from t = 0 to t = 5).b. Find the average rate of
If $1000 is invested in an account that pays 5% compounded continuously, the total amount, A(t), in the account after t years isa. Find the average rate of change per year of the total amount in the account for the first five years of the investment (from t = 0 to t = 5).b. Find the average rate of
LetUse the limit rules to find each limit. lim f(x) = 9 and lim g(x) = 27. x-4 x-4
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