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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
A decibel, named for Alexander Graham Bell, is the smallest increase of the loudness of sound that is detectable by the human ear. In physics, it is shown that when two sounds of intensity I1 and I2 (watts/cm3) occur, the difference in loudness is D decibels, whereWhen sound is rated in relation to
Suppose that t hours after an antibiotic is administered orally, its concentration in the patient’s bloodstream is given by a surge function of the form C(t) = Ate−kt, where A and k are positive constants and C is measured in micrograms per milliliter of blood. Blood samples are taken
Money is deposited into a bank offering interest at an annual rate of 6% compounded continuously. Find the percentage rate of change of the balance with respect to time.
Jane McGee, the mathematics editor at a major publishing house, estimates that if x thousand complimentary copies are distributed to professors, the first-year sales of a certain new text will be f(x) = 20 − 15e−0.2x thousand copies. Currently, Jane is planning to distribute 10,000
Program a computer or use a calculator to evaluateOn the basis of these calculations, what can you conjecture about the behavior ofas n decreases without bound? 1 + n for n = -1,000, -2,000, . . . , -50,000.
Show that the reflection of the point (a, b) in the line y = x is (b, a). y (b, a) 0 (a, b) y = x -X
According to a logistic model based on the assumption that the earth can support no more than 40 billion people, the world’s population (in billions) t years after 1960 is given by a function of the formwhere C and k are positive constants. Find the function of this form that is consistent with
A country exports electronic components E and textiles T. Suppose at a particular time t = t0, the revenue (in billions of dollars) derived from each of these goods is and that E is growing at 9%, while T is declining at 2%. At what relative rate is total export revenue R = E + T changing at this
The national income I(t) of a particular country is increasing by 2.3% per year, while the population P(t) of the country is decreasing at the annual rate of 1.75%. The per capita income C is defined to bea. Find the derivative of ln C(t).b. Use the result of part (a) to determine the percentage
Public health records indicate that t weeks after the outbreak of a certain strain of influenza, approximatelythousand people had caught the disease. At what rate was the disease spreading at the end of the second week? At what time is the disease spreading most rapidly? Q(t) = 80 4 + 76e-1.2t
Program a computer or use a calculator to evaluate n ( ₁ + 1)² n for n = 1,000, 2,000, . . . , 50,000.
The effect of temperature on the reaction rate of a chemical reaction is given by the Arrhenius equationwhere k is the rate constant, T (in Kelvin) is the temperature, and R is the gas constant. The quantities A and E0 are fixed once the reaction is specified. Let k1 and k2 be the reaction rate
Program a computer or use a calculator to estimate lim 1 + n→ +∞ 3 n 2n
Show that if q(p) units of a quantity are demanded when the price is p, the price elasticity of demand is given by the ratio of derivatives E(p) - (In q)' (In p)'
Program a computer or use a calculator to estimate lim 2 n→ +∞ 5 1/3 2n,
Sketch the graph of y = logb x for 0 x in the line y = x. Then answer these questions:a. Is the graph of y = logb x rising or falling for x > 0?b. Is the graph concave upward or concave downward for x > 0?c. What are the intercepts of the graph? Does the graph have any horizontal or vertical
It is sometimes useful for actuaries to be able to project mortality rates within a given population. A formula sometimes used for computing the mortality rate D(t) for women in the age group 25–29 iswhere t is the number of years after a fixed base year and D0 is the mortality rate when t = 0.a.
In each case, use one of the laws of exponents to prove the indicated law of logarithms.a. The quotient rule: ln u/v = ln u − ln vb. The power rule: ln ur = r ln u
The acidity of a solution is measured by its pH value, which is defined by pH = −log10 [H3O+], where [H3O+] is the hydronium ion concentration (moles/liter) of the solution. On average, milk has a pH value that is three times the pH value of a lime, which in turn has half the pH value of an
In a model developed by John Helms,* the water evaporation E(T) for a ponderosa pine is given by where T (degrees Celsius) is the surrounding air temperature.a. What is the rate of evaporation when T = 30ºC?b. What is the percentage rate of evaporation? At what temperature does the percentage rate
The demand for a certain commodity is D(p) = 3,000e−0.01p units per month when the market price is p dollars per unit.a. At what rate is the consumer expenditure E(p) = pD(p) changing with respect to price p?b. At what price does consumer expenditure stop increasing and begin to decrease?c. At
A Cro-Magnon cave painting at Lascaux, France, is approximately 15,000 years old. Approximately what ratio of 14C to 12C would you expect to find in a fossil from the same period as the painting?
Show that if y is a power function of x, so that y = Cxk where C and k are constants, then ln y is a linear function of ln x.
A certain industrial machine depreciates so that its value after t years becomes Q(t) = 20,000e−0.4t dollars.a. At what rate is the value of the machine changing with respect to time after 5 years?b. At what percentage rate is the value of the machine changing with respect to time after t years?
“Lucy,” the famous prehuman whose skeleton was discovered in Africa, has been found to be approximately 3.8 million years old.Approximately what percentage of original 14C would you expect to find if you tried to apply carbon dating to Lucy? Why would this be a problem if you were actually
Use the graphing utility of your calculator to graph y = 10x, y = x, and y = log10 x on the same coordinate axes (use [−5, 5]1 by [−5, 5]1). How are these graphs related?
It is projected that t years from now, the population of a certain country will be P(t) = 50e0.02t million.a. At what rate will the population be changing with respect to time 10 years from now?b. At what percentage rate will the population be changing with respect to time t years from now? Does
It is projected that t years from now, the population of a certain town will be approximately P(t) thousand people, whereAt what rate will the population be changing 10 years from now? At what percentage rate will the population be changing at that time? P(t) = 100 1 + e -0.2t
In Exercises 83 through 86 solve for x.x = ln(3.42 × 10−8.1)
The radioactive isotope gallium-67 (67Ga), used in the diagnosis of malignant tumors, has a half-life of 46.5 hours. If we start with 100 milligrams of the isotope, how many milligrams will be left after 24 hours? When will there be only 25 milligrams left? Answer these questions by first using a
According to the Ebbinghaus model (recall Exercise 60, Section 4.1), the fraction F(t) of subject matter you will remember from this course t months after the final exam can be estimated by the formula F(t) = B + (1 − B)e−kt, where B is the fraction of the material you will never forget and k
An international agency determines that the number of individuals of an endangered species that remain in the wild t years after a protection policy is instituted may be modeled bya. At what rate is the population changing at time t? When is the population increasing? When is it decreasing?b. When
A population model developed by the U.S. Census Bureau uses the formulato estimate the population of the United States (in millions) for every 10th year from the base year 1790. Thus, for instance, t = 0 corresponds to 1790, t = 1 to 1800, t = 10 to 1890, and so on. The model excludes Alaska and
In Exercises 83 through 86 solve for x. 3.500e0.31x ,-3.5x 1+257e-1.1x e
In Exercises 83 through 86 solve for x.e0.113x + 4.72 = 7.031 − x
Use a graphing utility to graph y = 2−x, y = 3−x, y = 5−x, and y = (0.5)−x on the same set of axes. How does a change in base affect the graph of the exponential function?
Use a numerical differentiation utility to find f'(c), where c = 0.65 andThen use a graphing utility to sketch the graph of f(x) and to draw the tangent line at the point where x = c. f(x) = In √√x+1 (1 + 3x)4.
In Exercises 1 through 4, find the area of the shaded region. y 0 y = x(x² - 4) X
The agency in Exercise 85 studies a second endangered species but fails to receive funding to develop a policy of protection. The population of the species is modeled bya. At what rate is the population changing at time t? When is the population increasing? When is it decreasing?b. When is the rate
Use a graphing utility to draw the graphs ofon the same set of axes. How do these graphs differ? y = √3¹, y = √3*, and y = 3-*
In Exercises 83 through 86 solve for x.ln(x + 3) − ln x = 5 ln(x2 − 4)
Two plants grow in such a way that t days after planting, they are P1(t) and P2(t) centimeters tall, respectively, wherea. At what rate is the first plant growing at time t = 10 days? Is the rate of growth of the second plant increasing or decreasing at this time?b. At what time do the two plants
For base b > 0, b ≠ 1, show thata. By using the fact that bx = ex ln b.b. By using logarithmic differentiation. d dx -(b) = (In b)b*
Evaluate each of these definite integrals. a. b. C. 3/2 1₁²(x²¹²2 + 3) dx Se e³-x dx 3 d. 10 X x + 1 dx x + 3 So V²²² +6² +4² dx 6x
In an experiment to test learning, a subject is confronted by a series of tasks, and it is found that t minutes after the experiment begins, the number of tasks successfully completed isa. For what values of t is the learning function R(t) increasing? For what values is it decreasing?b. When is the
Find these indefinite integrals (antiderivatives). a. for³ - 2. f³² - 2x + 4 dx b. X [√x(x² - 1) dx C. d. e. f. X √3x + 5e-2x) dx (3 + 2x²)³/2 In Vx X √xe¹+² - dx el + x² dx dx
Let a and b be any positive numbers other than 1.a. Show that (loga b)(logb a) = 1.b. Show that loga x = logbx log, a for any x > 0.
Solve this equation with three decimal place accuracy: log5 (x + 5) - log2x = 2 log10(x² + 2x)
Use a graphing utility to draw the graphs ofon the same axes. Do these graphs intersect? y = ln(1 + x²) and_ y = X
Make a table for the quantitieswith n = 8, 9, 12, 20, 25, 31, 37, 38, 43, 50, 100, and 1,000. Which of the two quantities seems to be larger? Do you think this inequality holds for all n ≥ 8? (√n)√n+1 and (√n + 1)Vñ,
Repeat Exercise 91 with the function f(x) = (3.7x2 − 2x + 1)e−3x + 2 and c = −2.17.Data from Exercises 91Use a numerical differentiation utility to find f'(c), where c = 0.65 andThen use a graphing utility to sketch the graph of f(x) and to draw the tangent line at the point where x = c. f(x)
A quantity grows so thatFind the percentage rate of change of Q with respect to t. ekt t Q(t) = Qo-
In Exercises 1 through 20, find the indicated indefinite integral. (x³ + √x - 9) dx
In Exercises 1 and 2, the table gives the coordinates (x, f (x)) of points on the graph of a function f over the interval a ≤ x ≤ b. In each case, estimate the value of the indicated definite integral by forming a Riemann sum using left endpoints. Sof(x) dx
Use a graphing utility to draw the graphs of y = 3x and y = 4 − ln √x on the same axes. Then use TRACE and ZOOM to find all points of intersection of the two graphs.
For the consumers’ demand functions D(q) in Exercises 1 through 6:(a) Find the total amount of money consumers are willing to spend to obtain q0 units of the commodity.(b) Sketch the demand curve and interpret the consumer willingness to spend in part (a) as an area. D(q) 300 (0.19 + 1)² dollars
In Exercises 1 through 4, find the area of the shaded region. y 0 |y=x³ y = √x -X
In Exercises 1 through 30, find the indicated integral. Check your answers by differentiation. 1- -3 dx
In Exercises 1 and 2, fill in the table by specifying the substitution you would choose to find each of the four given integrals. Integral a. (3x + 4)5/2² dx 4 b. 3-x √3 c. [₁e²-1² di dt d. dx [1(2 + 19³ d Substitution u
In Exercises 1 through 20, find the indicated indefinite integral. p2/3. X +5+ √x dx Vx) d
A cool drink is removed from a refrigerator on a hot summer day and placed in a room whose temperature is 30 Celsius. According to Newton’s law of cooling, the temperature of the drink t minutes later is given by a function of the form f(t) = 30 − Ae−kt. Show that the rate of change of the
In Exercises 1 and 2, the table gives the coordinates (x, f (x)) of points on the graph of a function f over the interval a ≤ x ≤ b. In each case, estimate the value of the indicated definite integral by forming a Riemann sum using left endpoints. Sof(x) dx
In Exercises 1 through 30, find the indicated integral. Check your answers by differentiation. dx
In Exercises 1 and 2, fill in the table by specifying the substitution you would choose to find each of the four given integrals. a. · b. [re-x dx C. Integral d. 3 dx (2x - 5)4 é e + 1 dt 1+ 3 √/₁² + 6t + 5 dt Substitution u
For the consumers’ demand functions D(q) in Exercises 1 through 6:(a) Find the total amount of money consumers are willing to spend to obtain q0 units of the commodity.(b) Sketch the demand curve and interpret the consumer willingness to spend in part (a) as an area.D(q) = 2(64 − q2) dollars
In Exercises 1 through 6, an initial population P0 is given along with a renewal rate R, and a survival function S(t). In each case, use the given information to find the population at the end of the indicated term T.P0 = 50,000; R(t) = 40; S(t) = e−0.1t, t in months; term T = 5 months
In Exercises 1 through 4, find the area of the shaded region. 0 y = x y= 2 x + 1 -X
In Exercises 1 through 20, find the indicated indefinite integral. fax4. (x² - 5e-2x) dx
In Exercises 3 through 36, find the indicated integral and check your answer by differentiation. fizr. 2x + 6)³ dx
In Exercises 3 through 8, estimate the value of the definite integralby computing the Riemann sum of f on the interval a ≤ x ≤ b for n = 8 subintervals, using left endpoints. Then find the actual value of the integral using the fundamental theorem of calculus.f(x) = 4 − x over 0 ≤ x ≤ 4
In each case, find the area of the specified region.a. The region bounded by the curve , y = x + √x the x axis, and the lines x = 1 and x = 4.b. The region bounded by the curve y = x2 − 3x and the line y = x + 5.
For the consumers’ demand functions D(q) in Exercises 1 through 6:(a) Find the total amount of money consumers are willing to spend to obtain q0 units of the commodity.(b) Sketch the demand curve and interpret the consumer willingness to spend in part (a) as an area. D(q) = 400 0.5g + 2 dollars
For the consumers’ demand functions D(q) in Exercises 1 through 6:(a) Find the total amount of money consumers are willing to spend to obtain q0 units of the commodity.(b) Sketch the demand curve and interpret the consumer willingness to spend in part (a) as an area. D(q) = 300 4q+ 3 dollars per
In Exercises 3 through 8, estimate the value of the definite integralby computing the Riemann sum of f on the interval a ≤ x ≤ b for n = 8 subintervals, using left endpoints. Then find the actual value of the integral using the fundamental theorem of calculus.f(x) = 3 − 2x over −1 ≤ x ≤
In Exercises 1 through 30, find the indicated integral. Check your answers by differentiation. fx³ dx
Find the average value of the functionover the interval 1 ≤ x ≤ 2. f(x) = x 2 - X
In Exercises 1 through 20, find the indicated indefinite integral. 5 [(2√³s + ²) ds
In Exercises 1 through 6, an initial population P0 is given along with a renewal rate R, and a survival function S(t). In each case, use the given information to find the population at the end of the indicated term T.P0 = 100,000; R(t) = 300; S(t) = e−0.02t, t in days; term T = 10 days
In Exercises 1 through 4, find the area of the shaded region. y y = x² + 1 y = 2r-2 X
In Exercises 3 through 36, find the indicated integral and check your answer by differentiation. e5x+3 dx
In Exercises 3 through 8, estimate the value of the definite integralby computing the Riemann sum of f on the interval a ≤ x ≤ b for n = 8 subintervals, using left endpoints. Then find the actual value of the integral using the fundamental theorem of calculus.f(x) = x2 over 1 ≤ x ≤ 2 Sof(x)
In Exercises 1 through 6, an initial population P0 is given along with a renewal rate R, and a survival function S(t). In each case, use the given information to find the population at the end of the indicated term T.P0 = 500,000; R(t) = 800; S(t) = e−0.011t, t in years; term T = 3 years
In Exercises 1 through 30, find the indicated integral. Check your answers by differentiation. [vi Vt dt
In Exercises 1 through 20, find the indicated indefinite integral. [(5x³² = 3) di dx
In Exercises 3 through 36, find the indicated integral and check your answer by differentiation. Sva √4x - 1 dx
In Exercises 1 through 30, find the indicated integral. Check your answers by differentiation. 14 X dx
The government of a certain country estimates that t years from now, imports will be increasing at the rate I'(t) and exports at the rate E'(t), both in billions of dollars per year, whereThe trade deficit is D(t) = I(t) − E(t). By how much will the trade deficit for this country change over the
In Exercises 3 through 36, find the indicated integral and check your answer by differentiation. 1 3x + 5 dx
In Exercises 3 through 8, estimate the value of the definite integralby computing the Riemann sum of f on the interval a ≤ x ≤ b for n = 8 subintervals, using left endpoints. Then find the actual value of the integral using the fundamental theorem of calculus.f(x) = 1 − x2 over 1 ≤ x ≤ 4
In Exercises 1 through 30, find the indicated integral. Check your answers by differentiation. √3 3et dx
In Exercises 1 through 6, an initial population P0 is given along with a renewal rate R, and a survival function S(t). In each case, use the given information to find the population at the end of the indicated term T.P0 = 800,000; R(t) = 500; S(t) = e−0.005t, t in months; term T = 5 months
In Exercises 1 through 20, find the indicated indefinite integral. П(зе -Х - Х "). + + 2e3x 2x dx
In Exercises 1 through 6, an initial population P0 is given along with a renewal rate R, and a survival function S(t). In each case, use the given information to find the population at the end of the indicated term T.P0 = 500,000; R(t) = 100e0.01t; S(t) = e−0.013t, t in years; term T = 8 years
The marginal revenue of producing q units of a certain commodity is R'(q) = q(10 − q) hundred dollars per unit. How much additional revenue is generated as the level of production is increased from 4 to 9 units?
For the consumers’ demand functions D(q) in Exercises 1 through 6:(a) Find the total amount of money consumers are willing to spend to obtain q0 units of the commodity.(b) Sketch the demand curve and interpret the consumer willingness to spend in part (a) as an area.D(q) = 40e−0.05q dollars per
In Exercises 3 through 8, estimate the value of the definite integralby computing the Riemann sum of f on the interval a ≤ x ≤ b for n = 8 subintervals, using left endpoints. Then find the actual value of the integral using the fundamental theorem of calculus. Sof(x) dx a
In Exercises 3 through 36, find the indicated integral and check your answer by differentiation. xPx-Pf XD
In Exercises 5 through 18, sketch the given region R and then find its area.R is the region bounded by the lines y = x, y = −x, and x = 1.
In Exercises 1 through 6, an initial population P0 is given along with a renewal rate R, and a survival function S(t). In each case, use the given information to find the population at the end of the indicated term T.P0 = 300,000; R(t) = 150e0.012t; S(t) = e−0.02t, t in months; term T = 20 months
In Exercises 3 through 8, estimate the value of the definite integralby computing the Riemann sum of f on the interval a ≤ x ≤ b for n = 8 subintervals, using left endpoints. Then find the actual value of the integral using the fundamental theorem of calculus. Sof(x) dx a
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