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mathematics
calculus with applications
Calculus With Applications 12th Edition Margaret L. Lial - Solutions
How long will it take for $1 to triple at an annual inflation rate of 8% compounded continuously?
Find the interest rate needed for $6000 to grow to $8000 in 3 years with continuous compounding.
The U.S. consumer price index (CPI, or cost of living index) has risen over the years, as shown in the table below, using an index in which the average over the years 1982 to 1984 is set to 100. In Exercise 69 of Section 1.1, we modeled this increase with a linear function. In this exercise, we
Xavier 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?
A concert promoter finds she can sell 1000 tickets at $50 each. She will not sell the tickets for less than $50, but she finds that for every $1 increase in the ticket price above $50, she will sell 10 fewer tickets.(a) Express n, the number of tickets sold, as a function of p, the price.(b)
A certain viral infection causes a fever that typically lasts 6 days. A model of the fever (in°F) on day x, 1 ≤ x ≤ 6, isAccording to the model, on what day should the maximum fever occur? What is the maximum fever? F(x) = 2 -x² + 3 कर 14 3 -x + 96.
Suppose the cost in dollars to produce x hundreds of nails is given byC(x) = x2 + 4x + 7.(a) Sketch a graph of C(x).(b) Find a formula for C(x + 1) - C(x), the cost to produce an additional hundred nails when x hundred are already produced. (This quantity is approximately equal to the marginal
Researchers have found that the 95th percentile (the value at which 95% of the data are at or below) for respiratory rates (in breaths per minute) during the first 3 years of infancy is given byy = 101.82411-0.0125995x+0.00013401x2for awake infants andy = 101.72858-0.0139928x+0.00017646x2for
The intensity of light (in appropriate units) passing through water decreases exponentially with the depth it penetrates beneath the surface according to the function I(x) = 10e-0.3x, where x is the depth in meters. A certain water plant requires light of an intensity of 1 unit. What is the
When glucose is infused into a person’s bloodstream at a constant rate of c grams per minute, the glucose is converted and removed from the bloodstream at a rate proportional to the amount present. The amount of glucose in grams in the bloodstream at time t (in minutes) is given bywhere a is a
Geologists sometimes measure the age of rocks by using “atomic clocks.” By measuring the amounts of potassium-40 and argon-40 in a rock, the age t of the specimen (in years) is found with the formulawhere A and K, respectively, are the numbers of atoms of argon- 40 and potassium-40 in the
The following table contains the average distance D from the sun for the eight planets and their period P of revolution around the sun in years.The distances are given in astronomical units (A.U.); 1 A.U. is the average distance from Earth to the sun. For example, since Jupiter’s distance is 5.2
In late March of 2020, it was estimated that the number of cases of COVID-19 in New York state was doubling every 3 days. At the time, the number of cases was estimated to be 26,000.(a) Assuming that this trend continues, find a function of the form y = y0(2t/k), without doing any calculations,
The production of an oil well has decreased exponentially from 128,000 barrels per year 5 years ago to 100,000 barrels per year at present.(a) Letting t = 0 represent the present time, write an exponential equation for production y in terms of time t in years.(b) Find the time it will take for
Let ƒ(x) = 2x2 + 5 and g(x) = 3x2 + 4x - 1. Find the following.(a) ƒ(-3) (b) g(2) (c) ƒ(3m)(d) g(-k) (e) ƒ(x + h) (f) g(x + h)(g)ƒ(x + h) - ƒ(x) / h(h)g(x + h) - g(x) / h
Graph the following by hand.y = log2 (x - 1)
Graph the following by hand.y =(1/2)x-1
Graph the following by hand.y = (1/5)2x-3
Graph the following by hand.y = 4-x + 3
Nuclear energy derived from radioactive isotopes can be used to supply power to space vehicles. The output of the radioactive power supply for a certain satellite is given by the function y = 40e-0.004t, where y is in watts and t is the time in days.(a) How much power will be available at the end
Graph the following by hand.y = 4x
25 g of polonium-210 is decaying exponentially. After 50 days 19.5 g of polonium-210 is left.(a) Write a function in the form y = y0ekt giving the number of grams of polonium-210 after t days.(b) Write the function from part (a) in the form y = y0at/50.(c) Use your answer from part (a) to find the
Suppose $26,000 is borrowed for 4 years at 6% interest. Find the interest paid over this period if the interest is compounded as follows.(a) Annually(b) Semiannually(c) Quarterly(d) Monthly(e) Continuously
Graph the following by hand.ƒ(x) = 6x / x + 2
Graph the following by hand.ƒ(x) = 4x - 2 / 3x + 1
Graph the following by hand. ƒ(x) = 2 /3x - 6
The half-life of plutonium-241 is approximately 14.4 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?
The graph in the next column shows how the risk of chromosomal abnormality in a child rises with the age of the mother.(a) Read from the graph the risk of chromosomal abnormality (per 1000) at ages 20, 35, 42, and 49.(b) Assuming the graph to be of the form y = Cekt, find k using t = 20 and t =
Graph the following by hand. ƒ(x) = 8 / x
Find the half-life of each radioactive substance, where t is measured in years. See Example 2.Krypton-85: A(t) = A0e-0.0644t.Data from Example 2.Carbon-14 is a radioactive form of carbon that is found in all living plants and animals. After a plant or animal dies, the carbon-14 disintegrates.
The following table shows the number of drug overdose deaths involving nonprescription synthetic narcotics (primarily fentanyl) in recent years.(a) Letting t represent the years since 2010, find a function of the form y = y0ekt that fits the data at 2011 and 2016.(b) If you have a graphing
Graph the following by hand.y = -(x + 2)3 - 2
Find the half-life of each radioactive substance, where t is measured in years. See Example 2.Cobalt-60: A(t) = A0e-0.131t.Data from Example 2.Carbon-14 is a radioactive form of carbon that is found in all living plants and animals. After a plant or animal dies, the carbon-14 disintegrates.
Graph the following by hand. y = -(x - 1)4 + 4
Graph the following by hand.ƒ(x) = 1 - x4
Graph the following by hand.ƒ(x) = x3 - 3
An article on cancer treatment contains the following statement: A 37% 5-year survival rate for women with ovarian cancer yields an estimated annual mortality rate of 0.1989. The authors of this article assume that the number of survivors is described by the exponential decay function given at the
Graph the following by hand.y = 3x2 - 9x + 2
The growth of bacteria in food products makes it necessary to time-date some products (such as milk) so that they will be sold and consumed before the bacteria count is too high. Suppose for a certain product that the number of bacteria present is given by ƒ(t) = 500e0.1t, under certain storage
Graph the following by hand.y = -x2 + 4x + 2
Graph the following by hand.y = -(1/4)x2 + x +2
Under optimal conditions, a population of Escherichia coli bacteria will double in 17 minutes. Suppose a culture starts with 20,000 bacteria.(a) Write a function in the form y = y02t/k giving the number of bacteria after t minutes.(b) Rewrite the function from part (a) in the form y = y0at.(c)
When a person swallows giardia cysts, stomach acids and pancreatic enzymes cause the cysts to release trophozoites, which divide every 12 hours.(a) Suppose the number of trophozoites at time t = 0 is y0. Write a function in the form y = y0ekt giving the number after t hours.(b) Write the function
Each of the following is the graph of a polynomial function. Give the possible values for the degree of the polynomial, and give the sign ( + or - ) for the leading coefficient. 4 2 0 --4 -6 1 X
Each of the following is the graph of a polynomial function. Give the possible values for the degree of the polynomial, and give the sign ( + or - ) for the leading coefficient. -3 -2 -1 10 5 -10 -20 0 2 3 4
Graph the following by hand.y = 2x2 + 3x - 1
Each of the following is the graph of a polynomial function. Give the possible values for the degree of the polynomial, and give the sign ( + or - ) for the leading coefficient. -3 30 20 10 -110 2 4 x
Find the domain of each function defined as follows. y = 3x X 4
Find the domain of each function defined as follows. = y x = 2 - 2x + 3
Find the domain of each function defined as follows.y = ln(x2 - 16)
The population of the world in the year 1600 was about 500 million, and in the year 2020 was about 7795 million.(a) Assuming that the population of the world grows exponentially, find the equation in the form P0ekt for the population P(t) in millions in the year t.(b) Use your answer from part (a)
Find the domain of each function defined as follows.y = ln(x + 7)
Sales of a new model of digital camera are approximated by S(x) = 5000 - 4000e-x, where x represents the number of years that the digital camera has been on the market, and S(x) represents sales in thousands of dollars.(a) Find the sales in year 0.(b) When will sales reach $4,500,000?(c) Find the
Andre 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 compounded(a) Quarterly.(b) Continuously.
Sales of a new model of compact disc player are approximated by the function S(x) = 1000 - 800e-x, where S(x) is in appropriate units and x represents the number of years the disc player has been on the market.(a) Find the sales during year 0.(b) In how many years will sales reach 500
List the ordered pairs obtained from the following if the domain of x for each exercise is {-3, -2, -1, 0, 1, 2, 3}. Graph each set of ordered pairs. Give the range. y X x² + 1
Mina, 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 effective rate
Solve equation.e-x = (e4)x+3
A couple wants to have $40,000 in 5 years for a down payment on a new house.(a) How much should they deposit today, at 6.4% compounded quarterly, to have the required amount in 5 years?(b) How much interest will be earned?(c) If they can deposit only $20,000 now, how much more will they need to
Solve equation.(e3)-2x = e-x+5
List the ordered pairs obtained from the following if the domain of x for each exercise is {-3, -2, -1, 0, 1, 2, 3}. Graph each set of ordered pairs. Give the range.y = (2x - 1)(x + 1)
A company must pay a $307,000 settlement in 3 years.(a) What amount must be deposited now at 6% compounded semiannually to have enough money for the settlement?(b) How much interest will be earned?(c) Suppose the company can deposit only $200,000 now. How much more will be needed in 3 years?(d)
Solve equation.16x+3 = 642x-5
Determine whether each of the following statements is true or false, and explain why.Describe in words what a logarithm is.
Frank must make a balloon payment of $20,000 in 4 years. Find the present value of the payment if it includes annual interest of 6.5% compounded monthly.
Solve equation.25x = 125x+2
Determine whether each of the following statements is true or false, and explain why.What can you tell about the graph of a polynomial function of degree n before you plot any points?
Melissa deposits $7500 of lottery winnings in an account paying 6% interest compounded monthly. What effective rate does the account earn?
Solve equation.4x = 8x+1
Determine whether each of the following statements is true or false, and explain why.How do you find a vertical asymptote? A horizontal asymptote?
A firm deposits some funds in a special account at 6.2% compounded quarterly. What effective rate will they earn?
Determine whether each of the following statements is true or false, and explain why.What is a function? A linear function? A quadratic function? A rational function?
Vicki bought a television set with money borrowed from the bank at 9% interest compounded semiannually. What effective interest rate did she pay?
Determine whether each of the following statements is true or false, and explain why.The half-life of a radioactive substance is the time required for half of the initial quantity to decay.
Find the present value of each amount.$25,000 if interest is 4.6% compounded continuously for 8 years.
Determine whether each of the following statements is true or false, and explain why.The function g(x) = ex grows faster than the function ƒ(x) = ln x.
Find the present value of each amount.$7300 if interest is 5% compounded continuously for 3 years.
Determine whether each of the following statements is true or false, and explain why.ln 4/ln8 = ln 4 - ln 8
Find the present value of each amount.$45,678.93 if interest is 7.2% compounded monthly for 11 months.
Determine whether each of the following statements is true or false, and explain why.eln(-2) = -2
For Exercises, match the correct graph A–F to the function without using your calculator. Notice that there are more functions than graphs; some of the functions are equivalent. After you have answered all of them, use a graphing calculator to check your answers. Each graph in this group is
Find the present value of each amount.$10,000 if interest is 6% compounde.d quarterly for 8 years.
Determine whether each of the following statements is true or false, and explain why.eln 2 = 2
Find the effective rate corresponding to each nominal rate of interest.5% compounded continuously.
Determine whether each of the following statements is true or false, and explain why.log10 0 = 1
Find the effective rate corresponding to each nominal rate of interest.8% compounded continuously.
Determine whether each of the following statements is true or false, and explain why.(ln 3)4 = 4 ln 3
Determine whether each of the following statements is true or false, and explain why.The amount of money after two years if $2000 is invested in an account that is compounded monthly with an annual rate of 4% is A = 2000(1+ 24 4 12, dollars.
Find the effective rate corresponding to each nominal rate of interest.6% compounded quarterly.
Determine whether each of the following statements is true or false, and explain why.1n(5 + 7) = 1n 5 + 1n 7
Find the effective rate corresponding to each nominal rate of interest.4% compounded quarterly.
Determine whether each of the following statements is true or false, and explain why.log11 = 0
Show that if a radioactive substance has a half-life of T, then the corresponding exponential decay function can be written as A(t0 = A0 (1/2)(t/T).
Show that if a radioactive substance has a half-life of T, then the corresponding constant k in the exponential decay function is given by k = -(ln 2)/T.
Determine whether each of the following statements is true or false, and explain why.The domain of the function ƒ(x) = 1/x2 -4 includes all real numbers except x = 2.
What is meant by the half-life of a quantity?
Determine whether each of the following statements is true or false, and explain why.Since 3-2 =1/9 we can conclude that log 3 1/9 = -2
In the exponential growth or decay function, explain the circumstances that cause k to be positive or negative.
Determine whether each of the following statements is true or false, and explain why.The function ƒ(x) = 1/x - 6 has a vertical asymptote at y = 6.
In the exponential growth or decay function y = y0 ekt, what does y0 represent? What does k represent?
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