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mathematics
calculus with applications

Calculus With Applications 12th Edition Margaret L. Lial - Solutions

An oil well off the Gulf Coast is leaking, with the leak spreading oil over the surface as a circle. At any time t (in minutes) after the beginning of the leak, the radius of the circular oil slick on the surface is r(t) = t2 feet. Let A(r) = πr2 represent the area of a circle of radius r.(a) Find
The cumulative horn volume for certain types of bighorn rams, found in the Rocky Mountains, can be described by the quadratic functionV(t) = -2159 + 1313t - 60.82t2,where V(t) is the horn volume (in cm3) and t is the year of growth, 2 ≤ t ≤ 9.(a) Find the horn volume for a 3-year-old ram.(b)
Let h(x) = u(x)v(x).(a)Using the fact that use the chain rule, the product rule, and the formula for the derivative of ln x to show that(b) Use the result from part (a) and the fact thatto show thatThe idea of taking the logarithm of a function before differentiating is known as logarithmic
When there is a thermal inversion layer over a city (as happens often in Los Angeles), pollutants cannot rise vertically but are trapped below the layer and must disperse horizontally. Assume that a factory smokestack begins emitting a pollutant at 8 a.m. Assume that the pollutant disperses
Insulin affects the glucose, or blood sugar, level of some diabetics according to the function G(x) = -0.2x2 + 450,where G(x) is the
Consider a child waiting at a street corner for a gap in traffic that is large enough so that he can safely cross the street. A mathematical model for traffic shows that if the expected waiting time for the child is to be at most 1 minute, then the maximum traffic flow, in cars per hour, is given
The Richter scale provides a measure of the magnitude of an earthquake. In fact, the largest Richter number M ever recorded for an earthquake was 8.9 from the 1933 earthquake in Japan. The following formula shows a relationship between the amount of energy released and the Richter number.where E is
Acoustical experts have found that clapping one’s hands near the staircases of the Pyramid of Kukulkan at Chichen Itza results in an echo that sounds like a chirp. The initial frequency of the chirp ƒ depends on the tread length of the stairs T according to the formulawhere c is the speed of
Scientists have developed a model to predict the growth of bacteria in bologna sausage at 32°C. The number of bacteria is given bywhere N0 is the number of bacteria present at the beginning of the experiment and N(t) is the number of bacteria present at time t (in hours).(a) Use the properties of
The passage of the Social Security Amendments of 1965 resulted in the creation of the Medicare and Medicaid programs. Since then, the percent of persons 65 years and over with family income below the poverty level has declined. The percent can be approximated by the following function:P(t) = 30.60
Suppose that the population of a certain collection of rare Brazilian ants is given byP(t) = (t + 100) ln(t + 2),where t represents the time in days. Find and interpret the rates of change of the population on the second day and on the eighth day.
If the cost function in dollars for q units of the item in Exercise 71 is C(q) = 100q + 100, find the following.(a) The marginal cost(b) The profit function P(q)(c) The approximate profit from one more unit when 8 units are sold(d) How might a manager use the information from part (c)?Data from
The body mass index (BMI) is a number that can be calculated for any individual as follows: Multiply a person’s weight by 703 and divide by the person’s height squared. That is,where w is in pounds and h is in inches. The National Heart, Lung, and Blood Institute uses the BMI to determine
The field metabolic rate (FMR), or the total energy expenditure per day in excess of growth, can be calculated for pronghorn fawns using Nagy’s formula,F(x) = 0.774 + 0.727 log x,where x is the mass (in grams) of the fawn and F(x) is the energy expenditure (in kJ/day).(a) Determine the total
Suppose the cost in dollars to make x oboe reeds is given byC(x) = 5 log2 x + 10.Find and interpret the marginal average cost when the following numbers of reeds are produced.(a) 10 (b) 20 CORPORA agt af
A study of the relation between the rate of reproduction in Drosophila (fruit flies) bred in bottles and the density of the mated population found that the number of imagoes (sexually mature adults) per mated female per day (y) can be approximated bylog y = 1.54 - 0.008x - 0.658 log x,where x is
In 1906 Kennelly developed a simple formula for predicting an upper limit on the fastest time that humans could ever run distances from 100 yards to 10 miles. His formula is given bywhere s is the distance in meters and t is the time to run that distance in seconds.(a) Find Kennelly’s estimate
There is a mathematical relationship between an infant’s weight and total body surface area (BSA), given byA(w) = 4.688w0.8168-0.0154 log w,where w is the weight (in grams) and A(w) is the BSA in square centimeters.(a) Find the BSA for an infant who weighs 4000 g.(b) Find A′(4000) and
If the total revenue received from the sale of x items is given byR(x) = 30 ln(2x + 1),while the total cost to produce x items is C(x) = x/2, find the following.(a) The marginal revenue(b) The profit function P(x)(c) The marginal profit when x = 60(d) Interpret the results of part (c).
Suppose the demand function for q units of a certain item iswhere p is in dollars.(a) Find the marginal revenue.(b) Approximate the revenue from one more unit when 8 units are sold.(c) How might a manager use the information from part (b)? p = D(q) = 100 + 50 In q q> 1,
The volume and surface area of a “jawbreaker” for any radius is given by the formulasrespectively. Roger Guffey estimates the radius of a jawbreaker while in a person’s mouth to bewhere r(t) is in millimeters and t is in minutes.(a) What is the life expectancy of a jawbreaker?(b) Find dV/dt
Zenzizenzicube is another obsolete word that represents the square of the square of a cube. In symbols, zenzizenzicube is written as ((x3)2)2.(a) Use the chain rule twice to find the derivative.(b) Use the properties of exponents to first simplify the expression, and then find the derivative.
To increase the velocity of the air flowing through the trachea when a human coughs, the body contracts the windpipe, producing a more effective cough. Tuchinsky formulated that the velocity of air that is flowing through the trachea during a cough isV = C(R0 - R)R2,where C is a constant based on
This exercise shows another way to derive the formula for the derivative of the natural logarithm function using the definition of the derivative.(a) Using the definition of the derivative, show that(b) Eliminate h from the result in part (a) using the substitution h = x/m to show that(c) What
To test an individual’s use of calcium, a researcher injects a small amount of radioactive calcium into the person’s bloodstream. The calcium remaining in the bloodstream is measured each day for several days. Suppose the amount of the calcium remaining in the bloodstream (in milligrams per
Zenzizenzizenzic is an obsolete word with the distinction of containing the most z’s of any word found in the Oxford English Dictionary. It was used in mathematics, before powers were written as superscript numbers, to represent the square of the square of the square of a number. In symbols,
The strength of a person’s reaction to a certain drug is given bywhere Q represents the quantity of the drug given to the patient and C is a constant.(a) The derivative R′(Q) is called the sensitivity to the drug. Find R′(Q).(b) Find the sensitivity to the drug if C = 59 and a patient is
In Exercise, use the ideas from Exercise 67 to find the derivative of each unction.h(x) = (x2 + 1)5xData from Exercise 67Let h(x) = u(x)v(x).Using the fact that use the chain rule, the product rule, and the formula for the derivative of ln x to show thatUse the result from part (a) and
The left ventricular length (viewed from the front of the heart) of a human fetus that is at least 18 weeks old can be estimated by∫(x) = -2.318 + 0.2356x - 0.002674x2,where ∫(x) is the ventricular length (in centimeters) and x is the age (in weeks) of the fetus.(a) Determine a meaningful
The brain mass of a human fetus during the last trimester can be accurately estimated from the circumference of the head bywhere m(c) is the mass of the brain (in grams) and c is the circumference (in centimeters) of the head.(a) Estimate the brain mass of a fetus that has a head circumference of
The total number of bacteria (in millions) present in a culture is given by N(t) = 2t(5t + 9)1/2 + 12, where t represents time (in hours) after the beginning of an experiment. Find and interpret the rate of change of the population of bacteria with respect to time for the following numbers of
The total profit (in tens of dollars) from selling x self-help books is(a) Find the average profit function.(b) Find the marginal average profit function.(c) Find the average profit and the marginal average profit for a sales level of 8 books. Interpret your results.(d) Find the average profit and
Find the derivative of each of the following functions by first rewriting the function using the rules of logarithms.ƒ(t) = ln(3t2 + 7t - 4)5/3
In Exercises, find the equation of the tangent line to the graph of the given function at the given value of x.ƒ(x) = (x3 + 7)2/3; x = 1
In Exercises, find the equation of the tangent line to the graph of the given function at the given value of x.ƒ(x) = √x2 + 16; x = 3
Find the derivative of each of the following functions by first rewriting the function using the rules of logarithms.h(x) = ln(2x3 + 5x2)3/2
Consider the following table of values of the functions f and g and their derivatives at various points.Find the following using the table above.(a) Dx (g [ƒ(x)]) at x = 1(b) Dx (g [ƒ(x]) at x = 2 X f(x) f'(x) g(x) g'(x) 1 2 -6 2 2/7 2 4 -7 3 3/7 3 1 -8 4 4/7 4 3 -9 1 5/7
If g′(5) = 12 and h′(5) = -3, find ƒ′(5) for ƒ(x) = 3g(x) - 2h(x) + 3.
The total profit (in dollars) from selling x watches is P(x) = 0.52x2 - 0.0002x3. Find and interpret the following.(a) P(100) (b) P′(100)(c) P̅(100)(d) P̅′(100)
In Exercises, find all values x = a where the function is discontinuous. For each point of discontinuity, give (a) f (a) if it exists, (b)(c)(d) (e) Identifywhich conditions for continuity are not met. Be sure to note when the limit doesn’t exist. lim f(x), X a
In Exercises, find all values x = a where the function is discontinuous. For each point of discontinuity, give (a) f (a) if it exists, (b)(c)(d) (e) Identifywhich conditions for continuity are not met. Be sure to note when the limit doesn’t exist. lim f(x), X a
In Exercises, find all values x = a where the function is discontinuous. For each point of discontinuity, give (a) f (a) if it exists, (b)(c)(d) (e) Identifywhich conditions for continuity are not met. Be sure to note when the limit doesn’t exist. lim f(x), X a
In Exercises, find all values x = a where the function is discontinuous. For each point of discontinuity, give (a) f (a) if it exists, (b)(c)(d) (e) Identifywhich conditions for continuity are not met. Be sure to note when the limit doesn’t exist. lim f(x), X a
Decide whether each limit exists. If a limit exists, estimate its value.(a) (b) lim f(x) x-3
In Exercises, find all values x = a where the function is discontinuous. For each point of discontinuity, give (a) f (a) if it exists, (b)(c)(d) (e) Identifywhich conditions for continuity are not met. Be sure to note when the limit doesn’t exist. lim f(x), X a
In one research study, the population of a certain shellfish in an area at time t was closely approximated by the following graph.(a) Sketch a graph of the growth rate of the population.(b) The derivative of y(t) has a horizontal asymptote as t approaches infinity. What value does y′(t) approach?
In Exercises, find all values x = a where the function is discontinuous. For each point of discontinuity, give (a) f (a) if it exists, (b)(c)(d) (e) Identifywhich conditions for continuity are not met. Be sure to note when the limit doesn’t exist. lim f(x), X a
Decide whether each limit exists. If a limit exists, estimate its value.(a)(b) lim f(x) x-0
Decide whether each limit exists. If a limit exists, estimate its value.(a)(b) lim g(x) x-3
Decide whether each limit exists. If a limit exists, estimate its value.(a)(b) lim F(x) x-2
In Exercises, for all values of x = a where the function is discontinuous, determine if the discontinuity is removable or nonremovable.Exercise 9In Exercises, find all values x = a where the function is discontinuous. For each point of discontinuity, give (a) f (a) if it
The graph below shows the relationship between the speed of a bird in flight and the required power expended by flight muscles.(a) Sketch the graph of the rate of change of the power as a function of the speed.(b) What does the point at Vmp correspond to on your graph of P′? Describe the meaning
When the price of an essential commodity rises rapidly, consumption drops slowly at first. If the price continues to rise, however, a “tipping” point may be reached, at which consumption takes a sudden substantial drop. Suppose the accompanying graph shows the consumption of gasoline, G(t), in
A group of MIT professors created a function for transforming student test scores of 0 to 100 into grades of 0 to 5. As the graph below illustrates, most of the scores transformed into grades of 3, 4, or 5. Transformed grades 5- + نیا -0 0 20 40 60 Student scores 80 100
When the price of an essential commodity (such as gasoline) rises rapidly, consumption drops slowly at first. If the price continues to rise, however, a “tipping” point may be reached, at which consumption takes a sudden substantial drop. Suppose the accompanying graph shows the consumption of
Officials in California tend to raise the sales tax in years in which the state faces a budget deficit and then cut the tax when the state has a surplus. The graph below shows the California state sales tax in recent years. Let T(x) represent the sales tax per dollar spent in year x. Find the
(a) For the function ƒ(x) = -4x2 + 11x, find the value of ƒ′(3), as well as the approximation using(b) Repeat part (a) using h = 0.01.(c) Repeat part (a) using the function ƒ(x) = -2/x and h = 0.1.(d) Repeat part (c) using h = 0.01.(e) Repeat part (a) using the function ƒ(x) = 2x and h =
The table gives actual and projected year-end assets in Social Security trust funds, in trillions of current dollars, where Year represents the number of years since 2000. The polynomial function defined by ƒ(x) = 0.0000329x3 - 0.00450x2 + 0.0613x + 2.34 models the data quite well.(a) To verify
In Exercises, find the derivative of the function at the given point.(a) Approximate the definition of the derivative with small values of h.(b) Use a graphing calculator to zoom in on the function until it appears to be a straight line, and then find the slope of that line.ƒ(x) = x1/x; a = 2
Suppose the profit (in cents) from selling x lb of potatoes is given byP(x) = 15x + 25x2.Find the average rate of change in profit from selling each of the following amounts.(a) 6 lb to 7 lb (b) 6 lb to 6.5 lb (c) 6 lb to 6.1 lbFind the marginal profit (that is, the instantaneous rate of
In Exercises, use the properties of limits to help decide whether each limit exists. If a limit exists, find its value.(a) (b) x 1 if x < 3 Let f(x) = 2 - if 3 ≤ x ≤ 5. x + 3 if x > 5
Explain why the following rules can be used to find(a) If the degree of p(x) is less than the degree of q(x), the limit is 0.(b) If the degree of p(x) is equal to the degree of q(x), the limit is A/B, where A and B are the leading coefficients of p(x) and q(x), respectively.(c) If the degree of
In 2007, the United States Postal Service unveiled the first Forever Stamp. A Forever Stamp does not show a monetary value, and it may be used used to mail a one-ounce letter even if the postal rate changes in the future. The graph below shows the cost of a Forever Stamp, where C(t) is the cost in
We saw in the previous chapter that if a function s(t) gives the position of an object at time t, the derivative gives the velocity, that is, v(t) = s′(t). For each position function in Exercises , find (a) v(t) and (b) The velocity when t = 0, t = 5, and t = 10.s(t) = 11t2 + 4t + 2
We saw in the previous chapter that if a function s(t) gives the position of an object at time t, the derivative gives the velocity, that is, v(t) = s′(t). For each position function in Exercises , find (a) v(t) and (b) The velocity when t = 0, t = 5, and t = 10.s(t) = 18t2 - 13t + 8
From the data printed in the following table from the Minneapolis Star Tribune, a dog’s age when compared to a human’s age can be modeled using either a linear formula or a quadratic formula as follows:y1 = 4.13x + 14.63y2 = -0.033x2 + 4.647x + 13.347,where y1 and y2 represent a dog’s human
We saw in the previous chapter that if a function s(t) gives the position of an object at time t, the derivative gives the velocity, that is, v(t) = s′(t). For each position function in Exercises , find (a) v(t) and (b) The velocity when t = 0, t = 5, and t = 10.s(t) = 4t3 + 8t2 + t
We saw in the previous chapter that if a function s(t) gives the position of an object at time t, the derivative gives the velocity, that is, v(t) = s′(t). For each position function in Exercises , find (a) v(t) and (b) The velocity when t = 0, t = 5, and t = 10.s(t) = -3t3 + 4t2 - 10t
If a rock is dropped from a 144-ft building, its position (in feet above the ground) is given by s(t) = -16t2 + 144, where t is the time in seconds since it was dropped.(a) What is its velocity 1 second after being dropped? 2 seconds after being dropped?(b) When will it hit the ground?(c) What is
A ball is thrown vertically upward from the ground at a velocity of 64 ft per second. Its distance from the ground at t seconds is given by s(t) = -16t2 + 64t.(a) How fast is the ball moving 2 seconds after being thrown? 3 seconds after being thrown?(b) How long after the ball is thrown does it
Researchers who have been studying the alarming rate at which the level of the Dead Sea has been dropping have shown that the density d(x) (in g per cm3) of the Dead Sea brine during evaporation can be estimated by the function d(x) = 1.66 - 0.90x + 0.47x2, where x is the fraction of the remaining
The probability (as a percent) of scoring 3 or more on the Calculus AB Advanced Placement Examination can be very closely predicted as a function of a student’s PSAT/ NMSQT Score x by the functionP(x) = -0.00209x3 + 0.3387x2 - 15.15x + 208.6.Find the probability of scoring 3 or more for each of
A charter flight charges a fare of $200 per person plus $4 per person for each unsold seat on the plane. The plane holds 100 passengers. Let x represent the number of unsold seats.(a) Find an expression for the total revenue received for the flight R(x). (Multiply the number of people flying, 100 -
Suppose that a quantity can be described by the exponential growth equation y(t) = y0ekt.(a) Show that the quantity 1/t in (y(t)/y0) is a constant.(b) For the shingles data in Exercise 54 of the previous section, reproduced below, let t = 0 correspond to age 7. Calculate the expression in part (a)
The derivative value ƒ′(a) equals the slope of the tangent line to the graph of y = ƒ(x) at x = a.Determine whether each statement is true or false, and explain why.
A 500-mg dose of a drug is administered by rapid injection to a patient. The half-life of the drug is 9 hours.(a) Find a model for the amount of drug in the bloodstream t hours after the drug is administered.(b) Find the average rate of change of drug in the bloodstream between t = 0 and t = 2.
Determine whether each of the following statements is true or false, and explain why.The limit of a product is the product of the limits when each of the limits exists.
The graph of a derivative will intercept the x-axis at values of x where the function has a horizontal tangent line.Determine whether each statement is true or false, and explain why.
A drug is given to a patient by IV infusion at a drip rate of 350 mg/hr. The half-life of this drug is 3 hours.(a) Find a model of the amount of drug in the bloodstream t hours after the IV infusion begins.(b) Find the average rate of change of drug in the bloodstream between t = 0 and t = 3.
Determine whether each of the following statements is true or false, and explain why.The limit of a function may not exist at a point even though the function is defined there.
If a function is positive at x = a, then its derivative is also positive at x = a.Determine whether each statement is true or false, and explain why.
A drug with a half-life of 9 hours is found to be effective when the amount of drug in the bloodstream is 250 mg. A 250-mg loading dose is given by rapid injection followed by an IV infusion. What should the rate of infusion be to maintain this level of drug in the bloodstream?
Determine whether each of the following statements is true or false, and explain why.If a rational function has a polynomial in the denominator of higher degree than the polynomial in the numerator, then the limit at infinity must equal zero.
For each of the graphs in Exercises, determine the following: (a) The x-values where f′(x) = 0; (b) The x-values where f′(x) is undefined; (c) The open intervals where f′(x) > 0; and (d) The open intervals where f′(x) < 0. For part (e), Use your results to
If the tangent line at x = a is vertical, then ƒ′(a) does not exist.Determine whether each statement is true or false, and explain why.
Determine whether each of the following statements is true or false, and explain why.If the limit of a function exists at a point, then the function is continuous there.
Explain how to graph the derivative of a function given the graph of the function.
Determine whether each of the following statements is true or false, and explain why.A polynomial function is continuous everywhere.
Explain how to graph a function given the graph of the derivative function.
For each of the graphs in Exercises, determine the following: (a) The x-values where f′(x) = 0; (b) The x-values where f′(x) is undefined; (c) The open intervals where f′(x) > 0; and (d) The open intervals where f′(x) < 0. For part (e), Use your results to
Use the table feature on a graphing calculator or a spreadsheet to develop a table that shows how much of the drug is present in a patient’s system at the end of each 1/2-hour time interval for 10 hours for the model found in Exercise 3. Are your results surprising?Data from Exercise 3:A drug
Determine whether each of the following statements is true or false, and explain why.A rational function is continuous everywhere.
In Exercises, choose the best answer for each limit.(a) Is ∞.(b) Is -∞.(c) Does not exist.(d) Is 1. If lim f(x) = -∞ and lim f(x) = ∞, then lim f(x) x-1 x-1+ x-1
For each of the graphs in Exercises, determine the following: (a) The x-values where f′(x) = 0; (b) The x-values where f′(x) is undefined; (c) The open intervals where f′(x) > 0; and (d) The open intervals where f′(x) < 0. For part (e), Use your results to
Determine whether each of the following statements is true or false, and explain why.The derivative gives the average rate of change of a function.
For each of the graphs in Exercises, determine the following: (a) The x-values where f′(x) = 0; (b) The x-values where f′(x) is undefined; (c) The open intervals where f′(x) > 0; and (d) The open intervals where f′(x) < 0. For part (e), Use your results to
Determine whether each of the following statements is true or false, and explain why.The derivative gives the instantaneous rate of change of a function.
Each graphing calculator window shows the graph of a function f(x) and its derivative function f′(x). Decide which is the graph of the function and which is the graph of the derivative. -3 Y₁ 5 L -15 Y₂ 3
Determine whether each of the following statements is true or false, and explain why.The instantaneous rate of change is a limit.
Each graphing calculator window shows the graph of a function f(x) and its derivative function f′(x). Decide which is the graph of the function and which is the graph of the derivative. -4- (Y₂) 10 -10 Y₁ 4

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