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mathematics
calculus with applications
Calculus With Applications 12th Edition Margaret L. Lial - Solutions
Use l’Hospital’s rule, where applicable, to find each limit. √x - 4 lim x 16 x 16
Use l’Hospital’s rule, where applicable, to find each limit. lim x-0 √5 + x - √5 V5 - x 2.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.$196,511 at 7.57% for 25 years
Use l’Hospital’s rule, where applicable, to find each limit. lim x-0 1 + 2x - (1 + x) ¹/² x3
Use l’Hospital’s rule, where applicable, to find each limit. lim x²e-√x x →∞
When Ms. Thompson died, she left $25,000 to her husband, which he deposited at 6% compounded annually. He wants to make annual withdrawals from the account so that the money (principal and interest) is gone in exactly 8 years.(a) Find the amount of each withdrawal.(b) Find the amount of each
The trustees of a college have accepted a gift of $150,000. The donor has directed the trustees to deposit the money in an account paying 6% per year, compounded semiannually. The trustees may withdraw an equal amount of money at the end of each 6-month period; the money must last 5 years.(a) Find
Use l’Hospital’s rule, where applicable, to find each limit. Vx lim x→∞ In(x³ + 1)
An insurance firm pays $4000 for new office signs for their branch. It amortizes the loan for the signs in 4 annual payments at 8% compounded annually. Prepare an amortization schedule for the signs.
In Exercises, first get a common denominator; then find the limits that exist. lim x-0 -13 + x²
Certain large semitrailer trucks cost $72,000 each. Ace Trucking buys such a truck and agrees to pay for it with a loan that will be amortized with 9 semiannual payments at 9.5% compounded semiannually. Prepare an amortization schedule for this truck.
In Exercises, first get a common denominator; then find the limits that exist. - lim x-0 e³x 2 X' 1 x² 3 X
In Exercises, first get a common denominator; then find the limits that exist. lim x0 2 ( ++)
A printer manufacturer charges $1048 for a high-speed printer. A firm of tax accountants buys 8 of these machines. They make a down payment of $1200 and agree to amortize the balance with monthly payments at 10.5% compounded monthly for 4 years. Prepare an amortization schedule showing the first
In Exercises, first get a common denominator; then find the limits that exist. lim x-0 In(1 - 4x) x² + 4 X
When Aaliyah opened her law office, she bought $14,000 worth of law books and $7200 worth of office furniture. She paid $1200 down and agreed to amortize the balance with semiannual payments for 5 years at 8% compounded semiannually. Prepare an amortization schedule for this purchase.
Use Newton’s method to find a solution to the nearest hundredth for each equation in the given interval.x3 - 8x2 + 18x - 12 = 0; [4, 5]
Use Newton’s method to find a solution to the nearest hundredth for each equation in the given interval.3x3 - 4x2 - 4x - 7 = 0; [2, 3]
Use Newton’s method to find a solution to the nearest hundredth for each equation in the given interval.x4 + 3x3 - 4x2 - 21x - 21 = 0; [2, 3]
Use Newton’s method to find a solution to the nearest hundredth for each equation in the given interval.x4 + x3 - 14x2 - 15x - 15 = 0; [3, 4]
Use Newton’s method to approximate each radical to the nearest thousandth.√37.6
Use Newton’s method to approximate each radical to the nearest thousandth.√51.7
Use Newton’s method to approximate each radical to the nearest thousandth.3√94.7
Use Newton’s method to approximate each radical to the nearest thousandth.4√102.6
A mine produced $750,000 of income during its first year. Each year thereafter, income increased by 18%. Find the total income produced in the first 8 years of the mine’s life.
In 4 years, Jaxson must pay a pledge of $5000 to his school’s building fund. He wants to set up a sinking fund to accumulate that amount. What should each semiannual payment into the fund be at 8% compounded semiannually?
Kamila deposits $491 at the end of each quarter for 9 years. If the account pays 9.4% compounded quarterly, find the final amount in the account.
Darrell deposits $1526.38 at the end of each 6-month period in an account paying 7.6% compounded semiannually. How much will be in the account after 5 years?
Charlotte borrows $20,000 from the bank to help her expand her business. She agrees to repay the money in equal payments at the end of each year for 9 years. Interest is at 8.9% compounded annually. Find the amount of each payment.
Samir wants to expand his pharmacy. To do this, he takes out a bank loan of $49,275 and agrees to repay it at 12.2% compounded monthly over 48 months. Find the amount of each payment necessary to amortize this loan.
Find the monthly house payments for the following mortgages.$177,110 at 8.45% for 30 years
Find the monthly house payments for the following mortgages.$156,890 at 7.74% for 25 years
Giancarlo has invested $14,000 in a certificate of deposit that has a 3.25% 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?
It is anticipated that a bank stock in which Rivka has invested $16,000 will achieve an annual interest rate of 9%. 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?
At a summer picnic, the number of bacteria in a bowl of potato salad doubles every 20 minutes. Assume that there are 1000 bacteria at the beginning of the picnic. How many bacteria are present after 2 hours, assuming that no one has eaten any of the potato salad?
The number of reported crimes in a city was about 22,700 in a recent year. Due to the creation of a neighborhood crime program, the city hopes the number of crimes decreases each year by 8%. Let xn denote the number of crimes in the city n years after the neighborhood crime program began. Find a
Find the sum of the first five terms of each geometric sequence.a1 = -5, r = 4
Use the method in Example 4 (with five terms of the appropriate Taylor series) to approximate the areas of the following regions.The region bounded by ƒ(x) = 1/(1 - x3), x = 0, x = 1/2, and the x-axisExample 4The standard normal curve of statistics is given byFind the area bounded by this curve
Use Taylor polynomials of degree 4 at x = 0, found in Exercises above, to approximate the quantities in Exercises. Round answers to 4 decimal places. 15.88
For the functions defined as follows, find the Taylor polynomials of degree 4 at 0.ƒ(x) = ln(1 + 2x)
Find the present value of each ordinary annuity.Payments of $1280 are made annually for 9 years at 7% compounded annually.
Use Newton’s method to find a solution for each equation in the given intervals. Find all solutions to the nearest hundredth.2 1n x + x - 3 = 0; [1, 4]
Find Taylor polynomials of degree 4 at 0 for the functions defined as follows.ƒ(x) = ln(1 + x)2/3
Use Newton’s method to find each root to the nearest thousandth.√250
It is impossible for the sum of an infinite number of positive values to equal a finite number.Determine whether each statement is true or false, and explain why.
The rule of 72 is used to estimate the doubling time for a quantity that increases at an annual rate of 6%.Determine whether each statement is true or false, and explain why.
The repeating decimal 0.18181818 . . . can be expressed as the infinite geometric seriesDetermine the rational number whose decimal expression is 0.18181818 . . . . 3 (13) +18(7) + 0.18(7) + 100 100 0.18 0.18[
Use l’Hospital’s rule where applicable to find each limit. ex lim x-0 ex - 1 + x 1- x
Use l’Hospital’s rule where applicable to find each limit. lim x-3 √x² +7-4 x² - 9
In Exercises, find the Taylor series for the functions defined as follows. Give the interval of convergence for each series.ƒ(x) = ln(1 - 5x2)
The following classical formulas for computing the value of π were developed by François Viète (1540–1603) and Gottfried von Leibniz (1646–1716), respectively:and(a) Use the product of the first three terms of Viète’s formula and the sum of the first four terms of Leibniz’s formula to
Use Taylor polynomials of degree 4 at x = 0, found in Exercises above, to approximate the quantities in Exercises. Round answers to 4 decimal places.e0.06
For each sequence that is geometric, find r and an.7/4, -7/12, 7/36, -7/108,...
Use Taylor polynomials of degree 4 at x = 0, found in Exercises above, to approximate the quantities in Exercises. Round to 4 decimal places.5e0.04
Find the lump sum deposited today that will yield the same total amount as payments of $10,000 at the end of each year for 15 years, at the following interest rates. Interest is compounded annually.5%
Use Newton’s method to find each root to the nearest thousandth.√300
Use Taylor polynomials of degree 4 at x = 0, found in Exercises above, to approximate the quantities in Exercises. Round answers to 4 decimal places.e1.02
Use l’Hospital’s rule where applicable to find each limit. lim x→5 √x² + 11 - 6 x² - 25
Use Taylor polynomials of degree 4 at x = 0, found in Exercises above, to approximate the quantities in Exercises. Round to 4 decimal places.√1.03
By properties of logarithms,Use this to find a Taylor series for ln [(1 + x)/(1 - x)]. In " (1+x)_ = - In(1 + x) − ln(1 − x). -
Find the sum of the first five terms of each geometric sequence.3, 6, 12, 24,...
Find the lump sum deposited today that will yield the same total amount as payments of $10,000 at the end of each year for 15 years, at the following interest rates. Interest is compounded annually.6%
Use Newton’s method to find each root to the nearest thousandth.3√9
A sugar factory receives an order for 1000 units of sugar. The production manager thus orders production of 1000 units of sugar. He forgets, however, that the production of sugar requires some sugar (to prime the machines, for example), and so he ends up with only 900 units of sugar. He then orders
Use Taylor polynomials of degree 4 at x = 0, found in Exercises above, to approximate the quantities in Exercises. Round answers to 4 decimal places.e-0.07
Use l’Hospital’s rule where applicable to find each limit. lim x->0 1 + 1 -X 3 (1 + x) ¹/3 x²
Use Taylor polynomials of degree 4 at x = 0, found in Exercises above, to approximate the quantities in Exercises. Round to 4 decimal places.3√26.94
Use the Taylor series for ex to suggest thatfor all x close to zero. et ≈ 1 + x + x² 2
Find the sum of the first five terms of each geometric sequence.5, 20, 80, 320,...
Find the lump sum deposited today that will yield the same total amount as payments of $10,000 at the end of each year for 15 years, at the following interest rates. Interest is compounded annually.8%
Use Newton’s method to find each root to the nearest thousandth.3√15
The government claims to be able to stimulate the economy substantially by giving each taxpayer a $200 tax rebate. They reason that 90% of this amount, or (0.90)($200) = $180, will be spent. An additional 90% of this $180 will then be spent, and so on.(a) If the government claim is true, how much
Use l’Hospital’s rule where applicable to find each limit. 2e5x lim x-0 25x² - 10x - 2 5x³
Use Taylor polynomials of degree 4 at x = 0, found in Exercises above, to approximate the quantities in Exercises. Round answers to 4 decimal places.√8.92
Use Taylor polynomials of degree 4 at x = 0, found in Exercises above, to approximate the quantities in Exercises. Round to 4 decimal places.ln 2.05
Use the Taylor series for e-x to suggest thatfor all x close to zero. ex≈ 1 x + - x² 2
Find the sum of the first five terms of each geometric sequence.12, -6, 3, -3/2,...
Find the payments necessary to amortize each loan.$2500, 16% compounded quarterly, 6 quarterly payments
Use Newton’s method to find each root to the nearest thousandth.3√100
We computed the present value of a continuous flow of money. Suppose that instead of a continuous flow, an amount C is deposited each year, and the annual interest rate is r. Then the present value of the cash flow over n years is(a) Show that the present value can be simplified to(b) Show that the
Suppose a charity is raffling off a prize with a fair market value of $4000. According to the IRS, the winnings are subject to a 25% withholding tax on the fair market value of the prize. To encourage ticket sales, the charity has decided to pay the withholdings for the winner. The charity thought
Use Taylor polynomials of degree 4 at x = 0, found in Exercises above, to approximate the quantities in Exercises. Round answers to 4 decimal places. V-1.05
Use Taylor polynomials of degree 4 at x = 0, found in Exercises above, to approximate the quantities in Exercises. Round answers to 4 decimal places.√16.3
Use l’Hospital’s rule where applicable to find each limit. lim x-0 V1 + x - V1 - x X
Use Taylor polynomials of degree 4 at x = 0, found in Exercises above, to approximate the quantities in Exercises. Round to 4 decimal places.ln 3.06
Find the sum of the first five terms of each geometric sequence.18, -3, 1/2, -1/12,...
Find the payments necessary to amortize each loan.$1000, 8% compounded annually, 9 annual payments
Use Newton’s method to find each root to the nearest thousandth.3√121
Use the Taylor series for ex to show that ex ≥ 1 + x for all x.
Use Taylor polynomials of degree 4 at x = 0, found in Exercises above, to approximate the quantities in Exercises. Round to 4 decimal places.0.922/3
Use Taylor polynomials of degree 4 at x = 0, found in Exercises above, to approximate the quantities in Exercises. Round answers to 4 decimal places. 7.91
Use l’Hospital’s rule where applicable to find each limit. lim x-0 √3x - √3 + x V3 X
Find the sum of the first five terms of each geometric sequence.a1 = 3, r = -2
Find the payments necessary to amortize each loan.$90,000, 8% compounded annually, 12 annual payments
In modeling the number of claims filed by an individual under an automobile policy during a three-year period, an actuary makes the simplifying assumption that for all integerswhere pn represents the probability that the policyholder files n claims during the period. Under this assumption, what is
Use Newton’s method to find the critical points for the functions defined as follows. Approximate them to the nearest hundredth. Decide whether each critical point leads to a relative maximum or a relative minimum.ƒ(x) = x3 - 3x2 - 18x + 4
Use l’Hospital’s rule where applicable to find each limit. lim x 0 √x² Vi - 5x + 4 X
An insurance company determines it cannot write medical malpractice insurance profitably and stops selling the coverage. In spite of this action, the company will have to pay claims for many years on existing medical malpractice policies. The company pays 60 for medical malpractice claims the year
Use the method in Example 4 (with five terms of the appropriate Taylor series) to approximate the areas of the following regions.The region bounded by ƒ(x) = ex2, x = 0, x = 1/3, and the x-axisExample 4The standard normal curve of statistics is given byFind the area bounded by this curve and the
Use the Taylor series for e-x to show that e-x ≥ 1 - x for all x.
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