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mathematics
calculus with applications
Calculus For Business, Economics And The Social And Life Sciences 11th Brief Edition Laurence Hoffmann, Gerald Bradley, David Sobecki, Michael Price - Solutions
Each ounce of Food I contains 3 g of carbohydrate and 2 g of protein, and each ounce of Food II contains 5 g of carbohydrate and 3 g of protein. Suppose x ounces of Food I are mixed with y ounces of Food II. The foods are combined to produce a blend that contains exactly 73 g of carbohydrate and 46
The accompanying graph shows how the growth rate R(T) of a bacterial colony changes with temperature T.*a. Over what range of values of T does the growth rate R(T) double?b. What can be said about the growth rate for 25 c. What happens when the temperature reaches roughly 45ºC? Does it make sense
Show that the equationmust have at least one solution on the interval 0 ≤ x ≤ 1. = x/A √x x² + 2x - 1
A rectangular poster contains 25 square centimeters of print surrounded by margins of 2 centimeters on each side and 4 centimeters on the top and bottom. Express the total area of the poster (printing plus margins) as a function of the width of the printed portion.
An open box with a square base is to be built for $48. The sides of the box will cost $3 per square meter, and the base will cost $4 per square meter. Express the volume of the box as a function of the length of its base.
The hero of a popular spy story has escaped from the headquarters of an international diamond smuggling ring in the tiny Mediterranean country of Azusa. Our hero, driving a stolen milk truck at 72 kilometers per hour, has a 40-minute head start on his pursuers, who are chasing him in a Ferrari
In some animal species, the intake of food is affected by the amount of vigilance maintained by the animal while feeding. In essence, it is hard to eat heartily while watching for predators that may eat you. In one model,† if the animal is foraging on plants that offer a bite of size S, the
The accompanying graph represents a function f (x) that oscillates between 1 and −1 more and more frequently as x approaches 0 from either the right or the left. Doesexist? If so, what is its value? lim f(x) x-0
Two jets bound for Los Angeles leave New York 30 minutes apart. The first travels at 550 miles per hour, while the second travels at 650 miles per hour. At what time will the second plane pass the first?
Show that two nonvertical lines are parallel if and only if they have the same slope.
Explain why there must have been some time in your life when your weight in pounds was the same as your height in inches.
At age 15, Michaela is twice as tall as her 5-yearold brother Juan, but on Juan’s 21st birthday, they find that he is 6 inches taller. Explain why there must have been a time when Michaela and Juan were exactly the same height.
A wire is stretched horizontally, as shown in the accompanying figure. An experiment is conducted in which different weights are attached at the center and the corresponding vertical displacements are measured. When too much weight is added, the wire snaps. Based on the data in the following table,
A publisher estimates that the cost of producing between 1,000 and 10,000 copies of a certain textbook is $50 per copy; between 10,001 and 20,000, the cost is $40 per copy; and between 20,001 and 50,000, the cost is $35 per copy.a. Find a function F(N) that gives the total cost of producing N
A manufacturer buys $20,000 worth of machinery that depreciates linearly so that its trade-in value after 10 years will be $1,000.a. Express the value of the machinery as a function of its age, and draw the graph.b. Compute the value of the machinery after 4 years.c. When does the machinery become
For tax purposes, the book value of certain assets is determined by depreciating the original value of the asset linearly over a fixed period of time. Suppose an asset originally worth V dollars is linearly depreciated over a period of N years, at the end of which it has a scrap value of S
Dr. Adams owns $1,500 worth of medical books which, for tax purposes, are assumed to depreciate linearly to zero over a 10-year period. That is, the value of the books decreases at a constant rate so that it is equal to zero at the end of 10 years. Express the value of the doctor’s books as a
It has been observed that the number of chirps made by a cricket each minute depends on the temperature. Crickets won’t chirp if the temperature is 38ºF or less, and observations yield the following data:a. Express T as a linear function of C.b. How many chirps would you expect to hear if the
Membership in a swimming club costs $250 for the 12-week summer season. If a member joins after the start of the season, the fee is prorated; that is, it is reduced linearly.a. Express the membership fee as a function of the number of weeks that have elapsed by the time the membership is purchased
To encourage motorists to form car pools, the transit authority in a certain metropolitan area has been offering a special reduced rate at toll bridges for vehicles containing four or more persons. When the program began 30 days ago, 157 vehicles qualified for the reduced rate during the morning
Since the beginning of the month, a local reservoir has been losing water at a constant rate. On the 12th of the month the reservoir held 200 million gallons of water, and on the 21st it held only 164 million gallons.a. Express the amount of water in the reservoir as a function of time, and draw
A certain stock had an initial public offering (IPO) price of $10 per share and is traded 24 hours a day. Sketch the graph of the share price over a 2-year period for each of the following cases:a. The price increases steadily to $50 a share over the first 18 months and then decreases steadily to
Students at a state college may preregister for their fall classes by mail during the summer. Those who do not preregister must register in person in September. The registrar can process 35 students per hour during the September registration period. Suppose that after 4 hours in September, a total
The average height H in centimeters of a child of age A years can be estimated by the linear function H = 6.5A + 50. Use this formula to answer these questions.a. What is the average height of a 7-year-old child?b. How old must a child be to have an average height of 150 cm?c. What is the average
The average scores of incoming students at an eastern liberal arts college in the SAT mathematics examination have been declining at a constant rate in recent years. In 2005, the average SAT score was 575, while in 2010 it was 545.a. Express the average SAT score as a function of time.b. If the
In certain parts of the world, the number of deaths N per week have been observed to be linearly related to the average concentration x of sulfur dioxide in the air. Suppose there are 97 deaths when x = 100 mg/m3 and 110 deaths when x = 500 mg/m3.a. What is the functional relationship between N and
Ethyl alcohol is metabolized by the human body at a constant rate (independent of concentration). Suppose the rate is 10 milliliters per hour.a. How much time is required to eliminate the effects of a liter of beer containing 3% ethyl alcohol?b. Express the time T required to metabolize the effects
The following table gives the length of year L (in earth years) of each planet in the solar system, along with the mean (average) distance D of the planet from the sun, in astronomical units (1 astronomical unit is the mean distance of the earth from the sun).a. Plot the point (D, L) for each
In Aesop’s fable about the race between the tortoise and the hare, the tortoise trudges along at a slow, constant rate from start to finish. The hare starts out running steadily at a much more rapid pace, but halfway to the finish line, stops to take a nap. Finally, the hare wakes up, sees the
Graphon the same set of coordinate axes using [−10, 10]1 by [−10, 10]1 for a starting range. Adjust the range settings until both lines are displayed. Are the two lines parallel? y 54 270 x 63 and y 19 139 695 X 346 14
Show that if a nonvertical line L1 with slope m1 is perpendicular to a line L2 with slope m2, then m2 = −1/m1. L₁ A (a, b) K 10 (a, c) B L₂ -X
Graphon the same set of coordinate axes using [−10, 10]1 by [−10, 10]1. Are the two lines parallel? y 25 -x + 7 13 and y 2 144 45 -x + 630 229
Aria owns a rare book that doubles in value every 10 years. In 1900, the book was worth $100.a. How much was the book worth in 1930? In 2000? How much should Aria expect her book to be worth in the year 2020?b. Is the value of the book a linear function of its age? Answer the question by
A rental company rents a piece of equipment for a $60.00 flat fee plus an hourly fee of $5.00 per hour.a. Make a chart showing the number of hours the equipment is rented and the cost for renting the equipment for 2 hours, 5 hours, 10 hours, and t hours of time.b. Write an algebraic expression
In Exercises 1 through 14, compute the indicated values of the given function.f(x) = 3x2 + 5x − 2; f(0), f(−2), f(1)
In Exercises 7 through 10, find the distance between the given points.(7, −3) and (5, 3)
In Exercises 19 through 24, determine the domain of the given function. g(x) = x²2² +5 x + 2
In Exercises 13 through 28, sketch the graph of the given function. Include all x and y intercepts.f(x) = 2 − 3x
In Exercises 13 through 28, sketch the graph of the given function. Include all x and y intercepts.f(x) = x(2x + 5)
In Exercises 25 through 32, find the composite function f(g(x)). f(u) = = 1 -2, 8(x) = x − 1 - u
In Exercises 33 through 38, find the difference quotient, f(x + h) − f(x)/h. f(x) = = X x + 1
In Exercises 1 through 14, compute the indicated values of the given function. 1 g(x) = x + g(-1), g(1), g(2) X
In Exercises 1 through 14, compute the indicated values of the given function. f(x) = X x² + 1 ;ƒ(2), f(0), f(-1)
In Exercises 1 through 14, compute the indicated values of the given function.f(x) = 3x + 5; f(0), f(−1), f(2)
In Exercises 1 through 6, plot the given points in a rectangular coordinate plane.(4, 3)
In Exercises 1 through 14, compute the indicated values of the given function.f(x) = −7x + 1; f(0), f(1), f(−2)
In Exercises 1 through 14, compute the indicated values of the given function. h(t) = √² + 2t + 4; h(2), h(0), h(-4)
In Exercises 1 through 6, plot the given points in a rectangular coordinate plane.(−2, 7)
In Exercises 1 through 14, compute the indicated values of the given function. g(u) = (u + 1)3/2; g(0), g(-1), g(8)
In Exercises 1 through 14, compute the indicated values of the given function. f(t) 1 V3 - 2t =f(1), f(-3), f(0)
In Exercises 1 through 6, plot the given points in a rectangular coordinate plane.(5, −1)
In Exercises 1 through 14, compute the indicated values of the given function. f(t) = (2t - 1)-3/2; f(1), f(5), f(13)
In Exercises 1 through 14, compute the indicated values of the given function.h(t) = (2t + 1)3; h(−1), h(0), h(1)
In Exercises 7 through 10, find the distance between the given points. (0.2) and 0, 13 5'8
In Exercises 1 through 6, plot the given points in a rectangular coordinate plane.(−1, −8)
In Exercises 1 through 6, plot the given points in a rectangular coordinate plane.(0, −2)
In Exercises 1 through 6, plot the given points in a rectangular coordinate plane.(3, 0)
In Exercises 1 through 14, compute the indicated values of the given function. f(x) = x - x - 2); f(1), f(2), f(3)
In Exercises 7 through 10, find the distance between the given points.(3, −1) and (7, 1)
In Exercises 7 through 10, find the distance between the given points.(4, 5) and (−2, −1)
In Exercises 11 and 12, classify each function as a polynomial, a power function, or a rational function. If the function is not one of these types, classify it as “different.” a. f(x) = x¹.4 b. f(x) = -2x³ c. f(x) = d. f(x) = 3x² + 8 (3x - 5)(4 - x)² 3x² x + 1 - 4x + 7
In Exercises 1 through 14, compute the indicated values of the given function. h(x) -2x + 4 if x ≤ 1 √x² + 1 if x > 1 h(3), h(1), h(0), h(−3)
In Exercises 13 through 28, sketch the graph of the given function. Include all x and y intercepts. f(x) = V1- x
In Exercises 1 through 14, compute the indicated values of the given function. g(x) = 4 + x|; g(-2), g(0), g(2)
In Exercises 11 and 12, classify each function as a polynomial, a power function, or a rational function. If the function is not one of these types, classify it as “different.” a. f(x) = -2 + 3x² + 5x4 b. f(x)=√x + 3x (x − 3)(x + 7) c. f(x) = = d. f(x) = -5x³ - 2x² + 3 3 (2x + 9) ²3.
In Exercises 1 through 14, compute the indicated values of the given function. (3 f(t)t + 1 Vt ift < -5 if -5 ≤ t ≤ 5; f(-6), f(-5), f(16) ift > 5
In Exercises 15 through 18, determine whether or not the given function has the set of all real numbers as its domain. g(x) X 1+x²2²
In Exercises 15 through 18, determine whether or not the given function has the set of all real numbers as its domain. f(x) x + 1 2 x² - 1
In Exercises 15 through 18, determine whether or not the given function has the set of all real numbers as its domain. f(t) = V1 - t
In Exercises 15 through 18, determine whether or not the given function has the set of all real numbers as its domain. h(t) = V² + 1
In Exercises 19 through 24, determine the domain of the given function. f(x) = √2x + 6
In Exercises 13 through 28, sketch the graph of the given function. Include all x and y intercepts.f(x) = x
In Exercises 13 through 28, sketch the graph of the given function. Include all x and y intercepts.f(x) = x2
In Exercises 19 through 24, determine the domain of the given function. f(x) = x³ 3x² + 2x + 5 -
In Exercises 13 through 28, sketch the graph of the given function. Include all x and y intercepts.f(x) = √x
In Exercises 13 through 28, sketch the graph of the given function. Include all x and y intercepts. f(x) [x² + x3 if x < 1 1 - 2x if x ≥ 1
In Exercises 19 through 24, determine the domain of the given function. f(t) t + 1 t²-1-2
In Exercises 13 through 28, sketch the graph of the given function. Include all x and y intercepts.f(x) = 2x − 1
In Exercises 19 through 24, determine the domain of the given function. h(s) = √²-4
In Exercises 13 through 28, sketch the graph of the given function. Include all x and y intercepts. f(x) = 2x 1 if x < 2 (3 if x ≥ 2
In Exercises 13 through 28, sketch the graph of the given function. Include all x and y intercepts.f(x) =−x2 − 2x + 15
In Exercises 25 through 32, find the composite function f(g(x)). f(u) = (u - 1)³ + 2u², g(x) = x + 1
In Exercises 13 through 28, sketch the graph of the given function. Include all x and y intercepts. f(x) = fx-1 x + 1 ifx≤0 if x > 0
In Exercises 13 through 28, sketch the graph of the given function. Include all x and y intercepts.f(x) = (x − 1)(x + 2)
In Exercises 13 through 28, sketch the graph of the given function. Include all x and y intercepts. f(x) (9-x x² + x2 if x ≤ 2 ifx > 2
In Exercises 13 through 28, sketch the graph of the given function. Include all x and y intercepts.f(x) = x2 + 2x − 8
In Exercises 25 through 32, find the composite function f(g(x)). f(u) = (2u+10)², g(x) = x - 5
In Exercises 19 through 24, determine the domain of the given function.
In Exercises 25 through 32, find the composite function f(g(x)). 7 = x + ₂x = (x)8° n 1 f(u)
In Exercises 13 through 28, sketch the graph of the given function. Include all x and y intercepts.f(x) = x3
In Exercises 13 through 28, sketch the graph of the given function. Include all x and y intercepts.f(x) = −x3 + 1
In Exercises 25 through 32, find the composite function f(g(x)).f(u) = 3u2 + 2u − 6, g(x) = x + 2
In Exercises 25 through 32, find the composite function f(g(x)). f(u) = u², g(x) = 1 x-1
In Exercises 25 through 32, find the composite function f(g(x)).f(u) = u2 + 4, g(x) = x − 1
In Exercises 25 through 32, find the composite function f(g(x)). f(u) = Vu+ 1, g(x) = x² - 1
In Exercises 29 through 34, find the points of intersection (if any) of the given pair of curves and draw the graphs.y = x2 and y = 3x − 2
In Exercises 29 through 34, find the points of intersection (if any) of the given pair of curves and draw the graphs.y = 3x + 5 and y = −x + 3
In Exercises 29 through 34, find the points of intersection (if any) of the given pair of curves and draw the graphs.2x − 3y = −8 and 3x − 5y = −13
In Exercises 35 through 38, the graph of a function f(x) is given. In each case find:(a) The y intercept.(b) All x intercepts.(c) The largest value of f(x) and the value(s) of x for which it occurs.(d) The smallest value of f(x) and the value(s) of x for which it occurs. -2 4 2 y 5 + 2 T - -2 X
In Exercises 29 through 34, find the points of intersection (if any) of the given pair of curves and draw the graphs.y = x2 − x and y = x − 1
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