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mathematics
calculus with applications
Calculus With Applications 12th Edition Margaret L. Lial - Solutions
The differential equation s″(t) = -B2s(t) approximately describes the motion of a pendulum, known as simple harmonic motion. Verify that s(t) = A cos(Bt + C) satisfies this differential equation.
Before Gerardus Mercator designed his map of the world in 1569, sailors who traveled in a fixed compass direction could follow a straight line on a map only over short distances. Over long distances, such a course would be a curve on existing maps, which tried to make area on the map proportional
Use l’Hospital’s rule, where applicable, to find each limit. lim x 0 x3 - 4x2 + 6x 3.x
Find the monthly house payment necessary to amortize each of the loans in Exercises. Then find the unpaid balance after 5 years for each loan. Assume that interest is compounded monthly.$249,560 at 7.75% for 25 years
In Exercises, first get a common denominator; then find the limits that exist. 12ex lim x-0 x3 12 x3 12 X 6 X
In his will the late Mr. Hudspeth said that each child in his family could have an annuity of $2000 at the end of each year for 9 years or the equivalent present value. If money can be deposited at 8% compounded annually, what is the present value?
In the NCAA Men’s Basketball Tournament, 64 teams are initially paired off. Before the recent changes, a champion was crowned through a series of single-elimination games.(a) Write a geometric sequence whose sum determines the number of games that must be played to determine the champion team.(b)
1. Find values for S and R that satisfy S = 0.3R and 0.2 (100 + S) = R. Show that these solutions give the same stipend and rent as found by summing the infinite series.2. Suppose that instead of a stipend increase of $100, the state cuts Mr. Jones’s stipend by $50. Assuming that Mr. Jones is
Use l’Hospital’s rule where applicable to find each limit. In(x - 1) lim x 2 x - 2
Use l’Hospital’s rule where applicable to find each limit. Xex lim x0 et 1
Identify the geometric series that converge. Give the sum of each convergent series. 36 +3+ 1 1 + 4 48 +
Find the present value of each ordinary annuity.Payments of $50,000 are made quarterly for 10 years at 8% compounded quarterly.
Use the formula for the sum of the first n terms of a geometric sequence to evaluate the following sums. i=0 3 -(2)
Mariah wants to buy a car that she estimates will cost $24,000 in 5 years. How much money must she deposit at the end of each quarter at 5% interest compounded quarterly in order to have enough in 5 years to pay for her car?
Find i2 and i3, the next two approximations to the monthly interest rate.P = $600, M = $57, n = 12, i1 = 0.02
Suppose a train leaves a station at noon traveling 100 mph. Two hours later, on an adjacent track, a second train leaves the station heading in the same direction traveling 125 mph. Determine when both trains are the same distance from the station.(a) Solve this problem using algebra.(b) Solve this
Use l’Hospital’s rule where applicable to find each limit. lim x ln(e* - 1) x->0t
In Exercises, the nth term of a sequence is given. Calculate the first five partial sums. an 1 2n - 1
Use the formula for the sum of the first n terms of a geometric sequence to evaluate the following sums. Σ (-3) 4 3 4 i=0
Find a polynomial of degree 3 such that ƒ(0) = 3, ƒ′(0) = 6, ƒ′′(0) = 12, and ƒ′′′(0) = 24.
Wahid has invested $12,000 in a certificate of deposit that has a 4.75% annual interest rate. Determine the doubling time for this investment using the doubling-time formula. How does this compare with the estimate given by the rule of 70?
Use l’Hospital’s rule where applicable to find each limit. lim X-00 (In x)² X
In Exercises, the nth term of a sequence is given. Calculate the first five partial sums. An 1 (n + 2)(n + 3)
Use the formula for the sum of the first n terms of a geometric sequence to evaluate the following sums. 5 i=0 3 -(-2) 2 mla
Harv’s Meats will need to buy a new deboner machine in 4 years. At that time Harv expects the machine to cost $12,000. To accumulate enough money to pay for the machine, Harv decides to deposit a sum of money at the end of each 6-month period in an account paying 4% compounded semiannually. How
Find i2 and i3, the next two approximations to the monthly interest rate.P = $15,000, M = $337, n = 60, i1 = 0.01
In the fifth century b.c., the Greek philosopher Zeno posed a paradox involving a race between Achilles (the fastest runner at the time) and a tortoise. The tortoise was given a head start, but once the race began, Achilles quickly reached the point where the tortoise had started. By then the
(a) Generalize the result of Example 2 to show that if x is small compared with a,(b) Use the result of part (a) to approximate 3√66 to 5 decimal places, and compare with the actual value. (a + x) ¹/n all + xa¹/n na
Infant mortality is an example of a relatively rare event that can be described by the Poisson distribution, for which the probability of x occurrences is given by(a) Verify that ƒ describes a probability distribution by showing that(b) Calculate the expected value for ƒ, given by(c) In 2019, the
Find a polynomial of degree 4 such that ƒ(0) = 1, ƒ′(0) = 1, ƒ′′(0) = 2, ƒ′″(0) = 6, and ƒ(4)(0) = 24. Generalize this result to a polynomial of degree n, assuming that ƒ(n)(0) = n!
Use l’Hospital’s rule where applicable to find each limit. Vx lim x→ In (In x)
It is anticipated that a bank stock in which Kinaya has invested $15,000 will achieve an annual interest rate of 6%. Determine the doubling time for this investment using the doubling-time formula. How does this compare with the estimate given by the rule of 72?
Use the Taylor series given in the text to find the Taylor series for the functions defined as follows. Give the interval of convergence of each series. f(x) = 4 3 - x
A firm must pay off $40,000 worth of bonds in 7 years.(a) Find the amount of each annual payment to be made into a sinking fund, if the money earns 6% compounded annually.(b) What annual payment should be made if the firm can get interest of 8% compounded annually?
Use the formula for the sum of the first n terms of a geometric sequence to evaluate the following sums. 8 i=0 64 (1) 2
A famous story about the outstanding mathematician John von Neumann (1903–1957) concerns the following problem: Two bicyclists start 20 miles apart and head toward each other, each going 10 miles per hour. At the same time, a fly traveling 15 miles per hour leaves the front wheel of one bicycle,
In the year 2019, the proportion of U.S. major league baseball players who were foreign born was 251 out of 882. Suppose we begin to randomly select major league players until we find one who is foreign born. Such an experiment can be described by the geometric distribution, for which the
In sports such as squash, played using English scoring, a player can score a point only when serving. If the serving player loses that rally, there is no score, and the serve goes to the other player. Suppose that the probability that Player A wins any rally is x, where 0 < x < 1, and that
Use l’Hospital’s rule where applicable to find each limit. lim 500-X √x x3 X
Use the formula for the sum of the first n terms of a geometric sequence to evaluate the following sums. 6 ³¹ (²³) 3 Σ 281 i=0
Use the Taylor series given in the text to find the Taylor series for the functions defined as follows. Give the interval of convergence of each series. f(x) = 2x 1 + 3x
With Individual Retirement Accounts (IRAs), a worker whose income does not exceed certain limits can deposit up to a certain amount annually, with taxes deferred on the principal and interest. To attract depositors, banks have been advertising the amount that would accumulate by retirement. Suppose
A book on management science gives the equationto determine N, the time until a particular part can be expected to need replacing. (λ and k are constants for a particular machine.) To find a useful approximate value for N when λN is near 0, go through the following steps.(a) Find a Taylor
In the Milton Bradley game Trouble™, each player takes turns pressing a “popper” that contains a single die. To begin moving a game piece around the board a player must first pop a 6 on the die. The number of tries required to get a 6 can be described by the geometric distribution.(a) Using
In Exercises, use a Taylor polynomial of degree 2 at x = 0 to approximate the desired value. Compare your answers with the results obtained by direct substitution.The profit (in thousands of dollars) when x thousand tons of apples are sold is Find P(0.3). P(x) = 20 + x² 50 + x
Use l’Hospital’s rule where applicable to find each limit. lim In(ex + 1) 5x
Let D represent duration, a term in finance that measures the length of time an investor must wait to receive half of the value of a cash flow stream totaling S dollars. Let r be the rate of interest and V the value of the investment. The value of S can be calculated by two formulas that are
Use the Taylor series given in the text to find the Taylor series for the functions defined as follows. Give the interval of convergence of each series. f(x) = = x² 2 x + 1
Use the Taylor series given in the text to find the Taylor series for the functions defined as follows. Give the interval of convergence of each series. f(x) = 3x³ 2 x
Use l’Hospital’s rule where applicable to find each limit. lim 00
With Individual Retirement Accounts (IRAs), a worker whose income does not exceed certain limits can deposit up to a certain amount annually, with taxes deferred on the principal and interest. To attract depositors, banks have been advertising the amount that would accumulate by retirement. Suppose
Use l’Hospital’s rule where applicable to find each limit. lim x5-0.001x x-00
A certain machine annually loses 20% of the value it had at the beginning of that year. If its initial value is $12,000, find its value at the following times.(a) The end of the fifth year(b) The end of the eighth year
With Individual Retirement Accounts (IRAs), a worker whose income does not exceed certain limits can deposit up to a certain amount annually, with taxes deferred on the principal and interest. To attract depositors, banks have been advertising the amount that would accumulate by retirement. Suppose
Use l’Hospital’s rule where applicable to find each limit. lim x3 x³ + 1 で XU[ z*
Use the Taylor series given in the text to find the Taylor series for the functions defined as follows. Give the interval of convergence of each series. f(x) = ln(1 − 2x)
With Individual Retirement Accounts (IRAs), a worker whose income does not exceed certain limits can deposit up to a certain amount annually, with taxes deferred on the principal and interest. To attract depositors, banks have been advertising the amount that would accumulate by retirement. Suppose
Use the Taylor series given in the text to find the Taylor series for the functions defined as follows. Give the interval of convergence of each series. 1 f(x) = In 1 + x = In (1 3
Each year a machine loses 30% of the value it had at the beginning of the year. Find the value of the machine at the end of 6 years if it cost $200,000 new.
In Exercises, use a Taylor polynomial of degree 2 at x = 0 to approximate the desired value. Compare your answers with the results obtained by direct substitution.The profit (in thousands of dollars) from the sale of x thousand packages of note paper is P(x) = ln(100 + 3x). Find P(0.6) if ln 100 is
In Exercises, first get a common denominator; then find the limits that exist. lim x-0 (주)
In Exercises, use a Taylor polynomial of degree 2 at x = 0 to approximate the desired value. Compare your answers with the results obtained by direct substitution.Revenue from selling agricultural products often increases at a slower and slower rate as more of the products are sold. Suppose the
Rylee sells some land in Nevada. She will be paid a lump sum of $60,000 in 7 years. Until then, the buyer pays 8% interest, compounded quarterly.(a) Find the amount of each quarterly interest payment.(b) The buyer sets up a sinking fund so that enough money will be present to pay off the $60,000.
A company is offering a job with a salary of $30,000 for the first year. Each year after that, the salary increases by 5%. Assume that this raise continues every year.(a) Determine the salary for the first three years.(b) Find the common ratio, r, and the general term, an.(c) Use the general term
In Exercises, use a Taylor polynomial of degree 2 at x = 0 to approximate the desired value. Compare your answers with the results obtained by direct substitution.The amount (in milliliters) of a certain drug in the bloodstream x minutes after being administered isFind A(0.05). A(x) = = 6x 1 + 10x
In Exercises, use a Taylor polynomial of degree 2 at x = 0 to approximate the desired value. Compare your answers with the results obtained by direct substitution.For a certain electronic part, the cost to make the part declines as more parts are made. Suppose that the cost (in dollars) to
In 1995, Oseola McCarty donated $150,000 to the University of Southern Mississippi to establish a scholarship fund. What is unusual about her is that the entire amount came from what she was able to save each month from her work as a washer woman, a job she began in 1916 at the age of 8, when she
Drake bought a hot tub from a friend. He agreed to pay a lump sum of $4000 after 5 years. Until then, he pays 6% interest, compounded semiannually.(a) Find the amount of each semiannual interest payment.(b) Drake sets up a sinking fund so that enough money will be present to pay off the $4000. He
An oil well produced $4,000,000 of income its first year. Each year thereafter, the well produced 3/4 as much income as the previous year. What is the total amount of income produced by the well in 8 years?
Use the Taylor series given in the text to find the Taylor series for the functions defined as follows. Give the interval of convergence of each series.ƒ(x) = e-2x2
What lump sum deposited today at 5% compounded annually for 8 years will provide the same amount as $1000 deposited at the end of each year for 8 years at 6% compounded annually?
In Exercises, first get a common denominator; then find the limits that exist. X lim x→1 x - 1 1 In x,
Use the Taylor series given in the text to find the Taylor series for the functions defined as follows. Give the interval of convergence of each series.ƒ(x) = e-5x
Suppose you could save $1 on January 1, $2 on January 2, $4 on January 3, and so on. What amount would you save on January 31? What would be the total amount of your savings during January?
In Exercises, first get a common denominator; then find the limits that exist. 2 lim x-0 X In(1 + 2x) z.t
The population of a certain colony of bacteria increases by 5% each hour. After 7 hours, what is the percent increase in the population over the initial population?
The electric potential at a distance z from an electrically charged disk of radius R was given aswhere k1 is a constant.(a) Suppose z is much larger than R. By writingasand using the Taylor polynomial of degree 1 for √1 + x, show that the potential can be approximated bythe result given in the
According to a text on species survival, the probability P that a certain species survives is given by P = 1 - e-2k, where k is a constant. Use a Taylor polynomial to show that if k is small, P is approximately 2k.
Use the Taylor series given in the text to find the Taylor series for the functions defined as follows. Give the interval of convergence of each series.ƒ(x) = 2x3e-3x
Use l’Hospital’s rule, where applicable, to find each limit. lim x-2 x³x² - x - 2 x2 x² - 4
The half-life of a radioactive substance is the time it takes for half the substance to decay. Suppose the half-life of a substance is 3 years and that 1015 molecules of the substance are present initially. How many molecules will be unchanged after 15 years?
Explain what is wrong with the following calculation using l’Hospital’s rule. x² lim x0x² + 3 2x lim x-0 2x 1
Use the Taylor series given in the text to find the Taylor series for the functions defined as follows. Give the interval of convergence of each series.ƒ(x) = x6e-x
In the “Million Dollar Lottery,” a winner is paid a million dollars at the rate of $50,000 per year for 20 years. Assume that these payments form an ordinary annuity and that the lottery managers can invest money at 6% compounded annually. Find the lump sum that the management must put away to
Find the following limit, which is the first one given by l’Hospital in his calculus text Analysis of Infinitely Small Quantities for the Understanding of Curves, published in 1696. lim x→a √2a³x - x² - ava²x - Vax³ a
A bicycle wheel rotates 400 times per minute. If the rider removes his or her feet from the pedals, the wheel will start to slow down. Each minute, it will rotate only 3/4 as many times as in the preceding minute. How many times will the wheel rotate in the fifth minute after the rider’s feet are
In most states, the winnings of million-dollar lottery jackpots are divided into equal payments given annually for 20 years. (In Colorado, the results are distributed over 25 years.) This means that the present value of the jackpot is worth less than the stated prize, with the actual value
A piece of paper is 0.008 in. thick.(a) Suppose the paper is folded in half, so that its thickness doubles, for 12 times in a row. How thick is the final stack of paper?(b) Suppose it were physically possible to fold the paper 50 times in a row. How thick would the final stack of paper be?
Maximiliano wants to save the $3000 that he earned over the summer. He could invest the money at 5% simple interest, and then after t years he would have As = 3000(1 + 0.05t) dollars. But he could also invest the money at 5% compounded semiannually, and then after t years he would have Ac =
Serenity buys a new car costing $22,000. She agrees to make payments at the end of each month for 4 years. If she pays 9% interest, compounded monthly, what is the amount of each payment? Find the total amount of interest Serenity will pay.
Use l’Hospital’s rule, where applicable, to find each limit. lim x--5 x³ - 3x² + 4x - 1 x² - 25
If P dollars are invested at a yearly interest rate r per year, compounded m times per year for t years, then the compound amount A isWhen m →∞, then the interest is compounded continuously. Use the following steps to derive a formula for the compound amount when interest is compounded
Some game shows sponsor tournaments where in each game, three individuals play against each other, yielding one winner and two losers. The winners of three such games then play each other, until the final game of three players produces a tournament winner. Suppose 81 people begin such a
Find the monthly house payment necessary to amortize each of the loans in Exercises. Then find the unpaid balance after 5 years for each loan. Assume that interest is compounded monthly.$170,892 at 8.11% for 30 years
Use l’Hospital’s rule, where applicable, to find each limit. lim x-0 In(3x + 1) X
A study of population dynamics in historic China defines b as the ratio of bandits to civilians (peasants and rulers) and s(b) as a measure of how successful the peasants are in fighting the bandits.To prove results on how the population coped with various pressures and how easily erratic cycles
Find the monthly house payment necessary to amortize each of the loans in Exercises. Then find the unpaid balance after 5 years for each loan. Assume that interest is compounded monthly.$353,700 at 7.95% for 30 years
Use l’Hospital’s rule, where applicable, to find each limit. lim 5ex - 5 2 x0x³8x² + 7x
Use l’Hospital’s rule, where applicable, to find each limit. lim x-0 √5 + x - √5 X
Use l’Hospital’s rule, where applicable, to find each limit. ,2x -xe² 2x lim x-0 e²x 1
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