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mathematics
calculus with applications

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

For each function in Exercises, find (a) The critical numbers;(b) The open intervals where the function is increasing; and(c) The open intervals where it is decreasing.ƒ(x) = x2/3
For each function in Exercises, find (a) The critical numbers;(b) The open intervals where the function is increasing; and(c) The open intervals where it is decreasing.ƒ(x) = (x + 1)4/5
Find the x-value of all points where the functions defined as follows have any relative extrema. Find the value(s) of any relative extrema.h(x) = (x + 1)3/5
Find the x-value of all points where the functions defined as follows have any relative extrema. Find the value(s) of any relative extrema. f(x) = x² - 2x + 1 x - 3
For each function in Exercises, find (a) The critical numbers;(b) The open intervals where the function is increasing; and(c) The open intervals where it is decreasing.g(x) = x3 + 3x2 + 3x + 1
Find the x-value of all points where the functions defined as follows have any relative extrema. Find the value(s) of any relative extrema. f(x) = x² - 6x + 9 x + 2
For each function in Exercises, find (a) The critical numbers;(b) The open intervals where the function is increasing; and(c) The open intervals where it is decreasing.g(x) = -x3 + 6x2 - 12x + 8
For each function in Exercises, find (a) The critical numbers;(b) The open intervals where the function is increasing; and(c) The open intervals where it is decreasing.h(x) = 5 - x1/3
For each function in Exercises, find (a) The critical numbers;(b) The open intervals where the function is increasing; and(c) The open intervals where it is decreasing.h(x) = (x - 1)3/5
For each function in Exercises, find (a) The critical numbers;(b) The open intervals where the function is increasing; and(c) The open intervals where it is decreasing.y = x - 4 ln(3x - 9)
Find the x-value of all points where the functions defined as follows have any relative extrema. Find the value(s) of any relative extrema. f(x) 2 zX In x
For each function in Exercises, find (a) The critical numbers;(b) The open intervals where the function is increasing; and(c) The open intervals where it is decreasing. f(x) = In 5x2+4 2 x² + 1
Find the x-value of all points where the functions defined as follows have any relative extrema. Find the value(s) of any relative extrema.ƒ(x) = x2ex - 3
Find the x-value of all points where the functions defined as follows have any relative extrema. Find the value(s) of any relative extrema. f(x) 2.x X
Find the x-value of all points where the functions defined as follows have any relative extrema. Find the value(s) of any relative extrema.ƒ(x) = 3xex + 2
For each function in Exercises, find (a) The critical numbers;(b) The open intervals where the function is increasing; and(c) The open intervals where it is decreasing.ƒ(x) = xe-3x
For each function in Exercises, find (a) The critical numbers;(b) The open intervals where the function is increasing; and(c) The open intervals where it is decreasing.ƒ(x) = x22-x
Find the x-value of all points where the functions defined as follows have any relative extrema. Find the value(s) of any relative extrema.ƒ(x) = 2x + ln x
For each function in Exercises, find (a) The critical numbers;(b) The open intervals where the function is increasing; and(c) The open intervals where it is decreasing.ƒ(x) = xex2-3x
For each function in Exercises, find (a) The critical numbers;(b) The open intervals where the function is increasing; and(c) The open intervals where it is decreasing.ƒ(x) = x2-x2
Find the x-value of all points where the functions defined as follows have any relative extrema. Find the value(s) of any relative extrema.ƒ(x) = x + 8-x
For each function in Exercises, find (a) The critical numbers;(b) The open intervals where the function is increasing; and(c) The open intervals where it is decreasing.y = x2/3 - x5/3
Use the derivative to find the vertex of each parabola.y = -2x2 + 12x - 5
(a) For the function in Exercise 19, find the average of the critical numbers.(b) For the function in Exercise 19, use a graphing calculator to find the roots of the function, and then find the average of those roots.(c) Compare your answers to parts (a) and (b). What do you notice?(d) Repeat part
For each function in Exercises, find (a) The critical numbers;(b) The open intervals where the function is increasing; and(c) The open intervals where it is decreasing.y = x1/3 + x4/3
Use the derivative to find the vertex of each parabola.y = ax2 + bx + c
Use the techniques of this chapter to find the vertex and intervals where ƒ is increasing and decreasing, given ƒ(x) = ax2 + bx + c, where we assume a > 0.
In each figure, the graph of the derivative of y = f(x) is shown. Find the locations of all relative extrema of f(x). f'(x) 10 - of -10 5 X
In Exercises, graph each function on a graphing calculator, and then use the graph to find all relative extrema (to three decimal places). Then confirm your answer by finding the derivative and using the calculator to solve the equation f′(x) = 0.ƒ(x) = x5 - x4 + 4x3 - 30x2 + 5x + 6
Repeat Exercise 45 under the assumption a < 0.Data from Exercise 45Use the techniques of this chapter to find the vertex and intervals where ƒ is increasing and decreasing, given ƒ(x) = ax2 + bx + c, where we assume a > 0.
In Exercises, graph each function on a graphing calculator, and then use the graph to find all relative extrema (to three decimal places). Then confirm your answer by finding the derivative and using the calculator to solve the equation f′(x) = 0.ƒ(x) = -x5 - x4 + 2x3 - 25x2 + 9x + 12
Where is the function defined by ƒ(x) = ex increasing? Decreasing? Where is the tangent line horizontal?
Graph ƒ(x) = 2 x + 1| + 4|x - 5| -20 with a graphing calculator in the window [-10, 10] by [-15, 30]. Use the graph and the function to determine the x-values of all extrema.
Repeat Exercise 47 with the function defined by ƒ(x) = ln x.Data from Exercise 47Where is the function defined by ƒ(x) = ex increasing? Decreasing? Where is the tangent line horizontal?
A friend looks at the graph of y = x2 and observes that if you start at the origin, the graph increases whether you go to the right or the left, so the graph is increasing everywhere. Explain why this reasoning is incorrect.
For each of the following functions, use a graphing calculator to find the open intervals where f(x) is (a) Increasing, or (b) Decreasing.f(x) = e0.001x - ln x
In Exercises, find (a) The number, q, of units that produces maximum profit; (b) The price, p, per unit that produces maximum profit; and (c) The maximum profit, P.C(q) = 80 + 18q; p = 70 - 2q
A county realty group estimates that the number of housing starts per year over the next three years will bewhere r is the mortgage rate (in percent).(a) Where is H(r) increasing? Interpret your results.(b) Where is H(r) decreasing? Interpret your results. H(r) 300 1 + 0.03r²⁹
For each of the following functions, use a graphing calculator to find the open intervals where f(x) is (a) Increasing, or (b) Decreasing.f(x) = ln(x2 + 1) - x0.3
In Exercises, find (a) The number, q, of units that produces maximum profit; (b) The price, p, per unit that produces maximum profit; and (c) The maximum profit, P.C(q) = 25q + 5000; p = 90 - 0.02q
The annual unemployment rates of the U.S. civilian noninstitutional population for 2000–2019 are shown in the graph. When is the function increasing? Decreasing? Constant? Unemployment rate (%) ∞ + + 2000 '03 '06 '09 Year '12 '15 '18
In Exercises, find (a) The number, q, of units that produces maximum profit; (b) The price, p, per unit that produces maximum profit; and (c) The maximum profit, P.C(q) = 100 + 20qe-0.01q; p = 40e-0.01q
Suppose the total cost C(x) (in dollars) to manufacture a quantity x of weed killer (in hundreds of liters) is given by C(x) = x3 - 2x2 + 8x + 50, where x 7 0.(a) Where is C(x) decreasing? Interpret your results.(b) Where is C(x) increasing? Interpret your results.
In Exercises, find (a) The number, q, of units that produces maximum profit; (b) The price, p, per unit that produces maximum profit; and (c) The maximum profit, P.C(q) = 21.047q + 3; p = 50 - 5 ln(q + 10)
A manufacturer sells video games with the following cost and revenue functions (in dollars), where x is the number of games sold, for 0 ≤ x ≤ 3300.C(x) = 0.32x2 - 0.00004x3R(x) = 0.848x2 - 0.0002x3Determine the interval(s) on which the profit function is increasing. Interpret your results.
On July 17, 2019, the power used in New York state (in thousands of megawatts) could be approximated by the function P(t) = -0.006785t3 + 0.1858t2 - 0.6549t + 20.62, where t is the number of hours since midnight, for 0 ≤ t ≤ 24. Find any relative extrema for power usage, as well as when they
A phone manufacturer has determined that the profit P(x) (in thousands of dollars) is related to the quantity x of phones produced (in hundreds) per month by P(x) = -(x - 4)ex - 4, 0 < x ≤ 3.9.(a) At what production levels is the profit increasing? Interpret your results.(b) At what levels is
The total profit P(x) (in thousands of dollars) from the sale of x units of a certain prescription drug is given by P(x) = ln(-x3 + 3x2 + 72x + 1) for x in 30, 104.(a) Find the number of units that should be sold in order to maximize the total profit.(b) What is the maximum profit?
The projected year-end assets in the Social Security trust funds, in trillions of dollars, where t represents the number of years since 2000, can be approximated by A(t) = 0.0000329t3 - 0.00450t2 + 0.0613t + 2.34, where 0 ≤ t ≤ 50.(a) Where is A(t) increasing? Interpret your results.(b) Where
The percent of concentration of a drug in the bloodstream t hours after the drug is administered is given byfor t ≥ 0.(a) On what time intervals is the concentration of the drug increasing? Interpret your results.(b) On what intervals is it decreasing? Interpret your results. K(t) = 4t 31² + 27
The annual unemployment rates of the U.S. civilian noninstitutional population for 2000–2019 are shown in the graph. Identify the years where relative extrema occur, and estimate the unemployment rate at each of these years. Unemployment rate (%) 2000 '04 80. Year '12 '16 '20
The demand equation for video games at one store is p = D(q) = 200e-0.1q, where p is the price (in dollars) and q is the quantity of games sold daily. Find the values of q and p that maximize revenue.
Suppose a certain drug is administered to a patient, with the percent of concentration of the drug in the bloodstream t hours later given by(a) On what time intervals is the concentration of the drug increasing? Interpret your results.(b) On what intervals is it decreasing? Interpret your results.
The demand equation for one type of computer networking system is p = D(q) = 500qe-0.0016q2, where p is the price (in dollars) and q is the quantity of servers sold per month. Find the values of q and p that maximize revenue.
In Exercise 61 in the section on Polynomial and Rational Functions, we gave the function defined by A(x) = 0.003631x3 - 0.03746x2 + 0.1012x + 0.009 as the approximate blood alcohol concentration in a 170-lb woman x hours after drinking 2 oz of alcohol on an empty stomach, for x in the interval [0,
Suppose that the cost function for a product is given by C(x) = 0.002x3 + 9x + 6912. Find the production level (i.e., value of x) that will produce the minimum average cost per unit C(x).
Researchers have found that the radius R to which a plant will grow is affected by the number of plants N in the area, as described by the equationwhere D is the density of the plants in an area and N0 is the number of plants in the area initially.(a) Find dR/dN.(b) Based on the sign of your answer
The number of people P(t) (in hundreds) infected t days after an epidemic begins is approximated byWhen will the number of people infected start to decline? P(t) = 10 In (0.19t + 1) 0.19t + 1
In the summer the activity level of a certain type of lizard varies according to the time of day. A biologist has determined that the activity level is given by the function a(t) = 0.008t3 - 0.288t2 + 2.304t + 7, where t is the number of hours after 12 noon. When is the activity level highest? When
The aortic pressure-diameter relation in a particular patient who underwent cardiac catheterization can be modeled by the polynomial D(p) = 0.000002p3 - 0.0008p2 + 0.1141p + 16.683, 55 ≤ p ≤ 130, where D(p) is the aortic diameter (in millimeters) and p is the aortic pressure (in mm Hg).
The standard normal probability function is used to describe many different populations. Its graph is the well-known normal curve. This function is defined byGive the intervals where the function is increasing and decreasing. f(x) = 1 V2T -xx²/2
The average individual daily milk consumption for herds of Charolais, Angus, and Hereford calves can be described by the function M(t) = 6.281t0.242e-0.025t, 1 ≤ t ≤ 26, where M(t) is the milk consumption (in kilograms) and t is the age of the calf (in weeks).(a) Find the time in which the
The figure shows estimated totals of nuclear weapons inventory for the United States and the former Soviet Union (and its successor states) from 1945 to 2019.(a) On what intervals were the total inventories of both countries increasing?(b) On what intervals were the total inventories of both
The metabolic rate of a person who has just eaten a meal tends to go up and then, after some time has passed, returns to a resting metabolic rate. This phenomenon is known as the thermic effect of food. Researchers have indicated that the thermic effect of food for one particular person is F(t) =
A group of researchers found that people prefer training films of moderate length; shorter films contain too little information, while longer films are boring. For a training film on the care of exotic birds, the researchers determined that the ratings people gave for the film could be approximated
The mathematical relationship between the age of a captive female moose and its mass can be described by the function M(t) = 369(0.93)tt0.36, t ≤ 12, where M(t) is the mass of the moose (in kilograms) and t is the age (in years) of the moose. Find the age at which the mass of a female moose is
Researchers have developed the following function that can be used to accurately predict the weight of Holstein cows (females) of various ages: W1(t) = 619(1 - 0.905e-0.002t)1.2386, where W1(t) is the weight of the Holstein cow (in kilograms) that is t days old. Where is this function increasing?
As we saw in the last section, the metabolic rate after a person eats a meal tends to go up and then, after some time has passed, returns to a resting metabolic rate. This phenomenon is known as the thermic effect of food and can be described for a particular individual as F(t) = -10.28 +
Social psychologists have found that as the discrepancy between the views of a speaker and those of an audience increases, the attitude change in the audience also increases to a point but decreases when the discrepancy becomes too large, particularly if the communicator is viewed by the audience
After a great deal of experimentation, two Atlantic Institute of Technology senior physics majors determined that when a bottle of French champagne is shaken several times, held upright, and uncorked, its cork travels according to s(t) = -16t2 + 40t + 3, where s is its height (in feet) above the
As a mathematics professor loads more weight in the back of his Subaru, the mileage goes down. Let x be the amount of weight (in pounds) that he adds, and let y = ƒ(x) be the mileage (in mpg).(a) Is ƒ′(x) positive or negative? Explain.(b) What are the units of ƒ′(x)?
In Exercises, use the quotient rule to find the derivative of each function. k(x) = x² + 7x - 2 x² - 2 +२
In Exercises, use the quotient rule to find the derivative of each function. y = 4x - 3 Vx
In Exercises, find the derivative of each function.h(x) = (x2 - 1)3
Use the rules for derivatives to find the derivative of each function defined as follows. s(t) = t³ - 2t (4t - 3)4
In Exercises, use the quotient rule to find the derivative of each function. h(z) = = 2.2 23.2 + 5
In Exercises, find the derivative of each function.y = (2x3 + 9x)5
Find the derivative of each function.y = (ln 4)1ln |3x|)
Use the rules for derivatives to find the derivative of each function defined as follows.p(t) = t2(t2 + 1)5/2
In Exercises, find the derivative of each function.y = (3x4 + 1)4(x3 + 4)
In Exercises, find the slope and the equation of the tangent line to the graph of the given function at the given value of x.y = x4 - 5x3 + 2; x = 2
Find the derivative of each function.y = log |3x|
Use the rules for derivatives to find the derivative of each function defined as follows.y = ln(5x + 3)
In Exercises, find the equation of the tangent line to the graph of the given function at the given point.ƒ(x) = (2x + 3)(√x2 - 6) at (1, -16)
Use the rules for derivatives to find the derivative of each function defined as follows.
At what points on the graph of ƒ(x) = 6x2 + 4x - 9 is the slope of the tangent line -2?
In your own words, explain how to form the composition of two functions.
Find the marginal average cost function of each function defined as follows.C(x) = √3x + 2
Find the marginal average cost function of each function defined as follows.C(x) = (x2 + 3)3
Find the marginal average cost function of each function defined as follows.C(x) = (4x + 3)4
In Exercises, find the derivative of each function. y 6 4 Vx
In Exercises, use the quotient rule to find the derivative of each function. g(x) = x² - 4x + 2 x² + 3 2
In Exercises, use the product rule to find the derivative of each function.y = (7x - 6)2
Use the rules for derivatives to find the derivative of each function defined as follows. y = -3V814 - 1 -
In Exercises, find the derivative of each function.y = 5x4 + 9x3 + 12x2 - 7x
In Exercises, find the derivative of each function.y = -100√x - 11x2/3
Find the derivative of each function. y 3x² In x
Use the rules for derivatives to find the derivative of each function defined as follows.k(x) = (5x3 - 1)6
Write each function as the composition of two functions. (There may be more than one way to do this.)y = e5x-3
In Exercises, find the derivative of each function. y -2 Vx

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