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mathematics
linear algebra

Questions and Answers of Linear Algebra

a) Find the vertex. b) Find the axis of symmetry. c) Determine whether there is a maximum or minimum value and find that value. 1. g(x) = x2 - 6 2. f (x) = -x2 + 6x - 8 3. G(x) = -2x2 - 4x - 7
Solve using any method. a. (ex + e-x) / (ex - e-x) = 3 b. ln 1ln x2 = 2 c. √ln x = ln√x
Given that a = (log125 5) log5 125, find the value of log3 a.
Given that a = log8 225 and b = log2 15, express a as a function of b.
Given that f (x) = ex - e-x, find f -1(x) if it exists.
The Houston-Sugar Land-Baytown metropolitan area is the fifth largest metropolitan area in the United States. In 2010, the population of this area was 5.9 million, and the exponential growth rate was
In February 2006, in the Valley of the Kings in Egypt, a team of archaeologists uncovered the first tomb since King Tut's tomb was found in 1922. The tomb contained five wooden sarcophagi that
Complete the following table.
The number N of farms in the United States has declined continually since 1950. In 1950, there were 5.6 million farms, and in 2008 that number had decreased to 2.2 million (Sources: U.S. Department
Photovoltaic solar power has become more affordable in recent years. Solar-power plants project that the average cost per watt of electricity generated by solar panels will decrease exponentially as
The market value of the 1964 - 1965 Lamborghini 350 GT has had a recent upswing. The car's value increased from $66,000 in 1999 to $220,000 in 2009 (Source: "1964-1965 Lamborghini 350 GT," by David
The number of cable TV networks has grown exponentially from 28 in 1980 to 565 in 2010 (Source: National Cable & Telecommunications Association). a) Using the data for 1980 and 2010, find the value
In 1909, the Pittsburgh Pirates shortstop Honus Wagner forced the American Tobacco Company to withdraw his baseball card that was packaged with cigarettes. Fewer than 60 of the Wagner cards still
In a town whose population is 3500, a disease creates an epidemic. The number of people N infected t days after the disease has begun is given by the function N(t) = 3500 / 1 + 19.9e-0.6t a) Graph
A lake is stocked with 640 fish of a new variety. The size of the lake, the availability of food, and the number of other fish restrict the growth of that type of fish in the lake to a limiting value
Suppose that a body with temperature T1 is placed in surroundings with temperature T0 different from that of T1. The body will either cool or warm to temperature T1t2 after time t, in minutes,
Under ideal conditions, a population of rabbits has an exponential growth rate of 11.7% per day. Consider an initial population of 100 rabbits. a) Find the exponential growth function. b) Graph the
In Exercises, determine which, if any, of these functions might be used as a model for the data in the scatter plot.a) Quadratic, f (x) = ax2 + bx + cb) Polynomial, not quadraticc) Exponential, f (x)
In 1900, 0.2% of the U.S. population, or 122,000 people, were ages 85 and older. This number grew to 5,751,000 in 2010. The table on the following page lists data regarding the percentage of the U.S.
Complete the following table.
In an economics class, students were given a final exam at the end of the course. Then they were retested with an equivalent test at subsequent time intervals. Their scores after time x, in months,
After door-to door canvassing, census workers determine the number of unoccupied homes. The table below lists the number of homes, in millions that were unoccupied in four census years. Year,
The debt among older Americans has been exponentially increasing in recent years. The table below lists, for selected years, the average debt per household, where the primary head is age 55 or
The average price of an average equality acre of farmland has increased over $2000 in the last ten years. The table below lists the price of an acre of average-quality farmland in selected
A company introduced a new software product on a trial run in a city. They advertised the product on television and found the following data regarding the percent P of people who bought the product
United States exports increased from $12 billion in 1950 to $1.550 trillion, or $1550 billion, in 2009 (Sources: www.dlc.org; U.S. Department of Commerce). Assuming that the exponential growth model
Following the birth of a child, a parent wants to make an initial investment P0 that will grow to $50,000 for the child's education at age 18. Interest is compounded continuously at 7%. What should
The supply function and the demand function for the sale of a certain type of DVD player are given by S(p) = 150e0.004p and D(p) = 480e-0.003p, Where S(p) is the number of DVD players that the
A beam of light enters a medium such as water or smog with initial intensity I0. Its intensity decreases depending on the thickness (or concentration) of the medium. The intensity I at a depth (or
Given that y = axb, take the natural logarithm on both sides. Let Y = ln y and X = ln x. Consider Y as a function of X. What kind of function is Y?
Given that y = aex, take the natural logarithm on both sides. Let Y = ln y. Consider Y as a function of x. What kind of function is Y?
The population of Haiti has a growth rate of 0.787% per year. In 2010, the population was 10,032,619. The land area of Haiti is 32,961,561,600 square yards. (Source: Statistical Abstract of the
In May 2010, a 1932 painting, Nude, Green Leaves, and Bust, by Pablo Picasso, sold at a New York City art auction for $106.5 million to an anonymous buyer. This is a record price for any work of art
Suppose that $10,000 is invested at an interest rate of 5.4% per year, compounded continuously. a) Find the exponential function that describes the amount in the account after time t, in years. b)
Complete the following table.
In 1970, Amos Flora of Flora, Indiana, discovered teeth and jawbones while dredging a creek. Scientists discovered that the bones were from a mastodon and that they had lost 77.2% of their carbon-14.
Explain how the graph of f (x) = ln x can be used to obtain the graph of g(x) = e x-2.
Atmospheric pressure P at an altitude a is given by P = P0e-0.00005a, Where P0 is the pressure at sea level, approximately 14.7 lb/in2 (pounds per square inch). Explain how a barometer, or some
Explain the errors, if any, in the following loga ab3 = (loga a) (loga b3) = 3 loga b.
In Exercises, given the function: a) Sketch the graph and determine whether the function is one-to-one. b) If it is one-to-one, find a formula for the inverse. 1. f (x) = 2 - 3x 2. f (x) = (x + 2) /
For the function f, use composition of functions to show that f -1 is as given. a. f (x) = 6x - 5, f -1(x) = (x + 5) / 6 b. f (x) = (x + 1) / x, f -1(x) = 1/(x - 1)
Find the inverse of the given one-to-one function f. Give the domain and the range of f and of f-1 and then graph both f and f-1 on the same set of axes a. f (x) = 2 - 5x b. f (x) = (x - 3) / (x + 2)
In Exercises, match the equation with one of the figures (a)-(f), which followa.b. c. d. e. f. 1. f (x) = ex-3 2. f (x) = log3 x 3. y = -log3 (x + 1) 4. y = (1/2)x
Find each of the following. Do not use a calculator. a. log5 125 b. log 100,000 c. ln e d. ln 1 e. log 101/4
Find each of the following using a calculator. Round to four decimal places. a. log 11 b. log 0.234 c. ln 3 d. ln 0.027 e. log 1-32
Express as a single logarithm and, if possible, simplify. a. 3 logb x - 4 logb y + ½ logb z b. ln (x3 - 8) - ln (x2 + 2x + 4) + ln (x + 2)
Express in terms of sums and differences of logarithms. a. ln 4√(wr2) b. log 3√(M2/N)
Given that loga 2 = 0.301, loga 5 = 0.699, and loga 6 = 0.778, find each of the following. a. loga 3 b. loga 50
Find the inverse of the relation {(1.3, -2.7), (8, -3), (-5, 3), (6, -3), (7, -5)}.
Following the birth of triplets, the grandparents deposit $30,000 in a college trust fund that earns 4.2% interest, compounded quarterly. a) Find a function for the amount in the account after t
How old is a skeleton that has lost 27% of its carbon-14?
What is the loudness, in decibels, of a sound whose intensity is 1000 I0?
The average walking speed w, in feet per second, of a person living in a city of population P, in thousands, is given by the function w(P) = 0.37 ln P + 0.05. a) The population of Wichita, Kansas, is
Cash Social Security distributions were $35 million, or $0.035 billion, in 1940. This amount has increased exponentially to $683 billion in 2009. (Source: Social Security Administration) Assuming the
The population of Papua New Guinea was 6.188 million in 2011, and the exponential growth rate was 1.985% per year (Source: CIA World Fact book)a) Find the exponential growth function. b) Estimate
Graph the function and determine whether the function is one-to-one using the horizontal-line test. a. f (x) = - | x | + 3 b. f (x) = x2 + 1 c. f (x) = 2x - 3 / 4
The cost of mailing a letter increased 40¢ from 1958 to 2011. The table below lists the cost of a first-class stamp in selected years. Year, x First-Class Postage Cost C (in cents) 1958, 0
Using only a graphing calculator, determine whether the following functions are inverses of each other: f (x) = (4 + 3x) / (x - 2) , g(x) = (x + 4) / (x - 3).
a) Use a graphing calculator to graph f (x) = 5e-x ln x in the viewing window [-1, 10, -5, 5]. b) Estimate the relative maximum and minimum values of the function.
Which of the following is the horizontal asymptote of the graph of f (x) = ex-3 + 2? a. y = -2 b. y = -3 c. y = 3 d. y = 2
Find each of the following. Do not use a calculator. a. log 0.00001 b. ln e c. ln 1 d. log4 5√4
Express as a single logarithm: 2 loga x - loga y + ½ loga z
Express ln 5√x2y in terms of sums and differences of logarithms.
Suppose $1000 is invested at interest rate k, compounded continuously, and grows to $1144.54 in 3 years. a) Find the interest rate. b) Find the exponential growth function. c) Find the balance after
In Exercises, given the function: a) Sketch the graph and determine whether the function is one-to-one. b) If it is one-to-one, find a formula for the inverse. 1. f (x) = x3 + 1 2. f (x) = 1 - x
Use composition of functions to show that f -1 is as given: f (x) = -4x + 3, f -1(x) = (3 - x) / 4.
Find the inverse of the one-to-one function f (x) = 1/(x - 4). Give the domain and the range of f and of f -1 and then graph both f and f -1 on the same set of axes.
Refer to Exercise 79. If there is a wet spring, then supplies of food increase to 1080 units of F1 and 810 units of 2. In this case, what is the maximum total number of these animals that the forest
Graph the system of inequalities. a. y ≥ x2 - 2, y ≤ 2 - x2 b. y < x + 1, y ≥ x2
Dawson Handcrafted Furniture produces chairs and sofas. Each chair requires 20 ft of wood, 1 lb of foam rubber, and 2 yd2 of fabric. Each sofa requires 100 ft of wood, 50 lb of foam rubber, and 20
Decompose into partial fractions. Check your answers using a graphing calculator. a. (x + 7) / ((x - 3) (x + 2)) b. 2x / ((x + 1) (x - 1)
Find the zeros of the polynomial function. a. f(x) = x3 + x2 + 9x + 9 b. f(x) = x3 - 3x2 + x - 3 c. f(x) = x3 + x2 - 3x - 2
Decompose into partial fractions. a. (9x3 - 24x2 + 48x) / (x - 2)4 (x + 1) [Let the expression equal A/(x + 1) + P(x) / (x - 2)4 and find P(x)]. b. x / (x4 - a4)
Computer warehouse.com charges $3 for shipping orders up to 10 lb, $5 for orders from 10 lb up to 15 lb, and $7.50 for orders of 15 lb or more. One day shipping charges for 150 orders totaled $680.
Solve the system of equations using Gaussian elimination or Gauss-Jordan elimination.\ 2x + y = 5, 3x + 2y = 6
Solve the system of equations using Gaussian elimination or Gauss-Jordan elimination. 3x + 2y - 3z = -2, 2x + 3y + 2z = -2, x + 4y + 4z = 1
Explain in your own words when using the elimination method for solving a system of equations is preferable to using the substitution method.
In Exercises match the system of equations with one of the graphs (a)-(f), which follow.1. x + y = -2, y = x - 8 2. x - y = -5, x = -4y 3. x - 2y = -1, 4x - 3y = 6 4. 2x - y = 1, x + 2y = -7 2x - 3y
Solve using the substitution method. Use a graphing calculator to check your answer. a. x + y = 9, 2x - 3y = -2 b. 3x - y = 5, x + y = 1/2 c. x - 2y = 7, x = y + 4
Solve using the elimination method. Also determine whether each system is consistent or inconsistent and whether the equations are dependent or independent. Use a graphing calculator to check your
It is estimated that in 2030, there will be a total of 4.072 million knee replacement and hip replacement surgeries, with knee replacements outnumbering hip replacements by 2.928 million (Source:
The two most common names for streets in the United States are Second Street and Main Street, with 15,684 streets bearing one of these names. There are 260 more streets named Second Street than Main
In 2009, the Better Business Bureau received the most complaints from consumers about cell-phone providers and cable/satellite TV providers. These two industries accounted for a total of 70,093
It is projected that in 2018, the average total of medical bills for an obese adult will be $2460 higher than the average for an adult with a healthy weight. Together, the average amount of medical
About 250,000 Amish live in the United States. Of that number, 73,950 live in Wisconsin and Ohio. The number of Amish in Ohio is 12,510 more than three times the number in Wisconsin. (Source: Young
Using the calorie count of one eighth of a 9-in. pie, we find that the number of calories in a piece of pecan pie is 221 less than twice the number of calories in a piece of lemon meringue pie. If
In 2010, the number of refugees from Africa and Near East/South Asia was 49,087. The number from Near East/South Asia was 4133 less than three times the number of refugees from Africa. (Source: U.S.
A mail-order garden equipment business shipped 120 packages one day. Customers were charged $3.50 for each standard-delivery package and $7.50 for each express-delivery package. Total shipping
One evening 1500 concert tickets were sold for the Fairmont Summer Jazz Festival. Tickets cost $25 for a covered pavilion seat and $15 for a lawn seat. Total receipts were $28,500. How many of each
Quinn inherited $15,000 and invested it in two municipal bonds that pay 4% and 5% simple interest. The annual interest is $690. Find the amount invested at each rate.
Turi's Tee Shirt Shack sold 36 shirts one day. Each short-sleeved tee shirt costs $12, and each long-sleeved tee shirt costs $18. Total receipts for the day were $522. How many of each kind of shirt
The owner of The Daily Grind coffee shop mixes French roast coffee worth $9.00 per pound with Kenyan coffee worth $7.50 per pound in order to get 10 lb of a mixture worth $8.40 per pound. How much of
Bentley's Office Solutions offers its sales representatives a choice between being paid a commission of 8% of sales and being paid a monthly salary of $1500 plus a commission of 1% of sales. For what
A one-cup serving of spaghetti with meatballs contains 260 Cal (calories) and 32 g of carbohydrates. A one-cup serving of chopped iceberg lettuce contains 5 Cal and 1 g of carbohydrates. (Source:
One serving of tomato soup contains 100 Cal and 18 g of carbohydrates. One slice of whole wheat bread contains 70 Cal and 13 g of carbohydrates. (Source: Home and Garden Bulletin No. 72, U.S.
A Leisure Time Cruises riverboat travels 46 km downstream in 2 hr. It travels 51 km upstream in 3 hr. Find the speed of the boat and the speed of the stream.
A DC10 airplane travels 3000 km with a tailwind in 3 hr. It travels 3000 km with a headwind in 4 hr. Find the speed of the plane and the speed of the wind.
Two private airplanes travel toward each other from cities that are 780 km apart at speeds of 190 km/h and 200 km / h. They leave at the same time. In how many hours will they meet?
Christopher's boat travels 45 mi downstream in 3 hr. The return trip upstream takes 5 hr. Find the speed of the boat in still water and the speed of the current.
The supply and demand for a video-game system are related to price by the equations y = 70 + 2x, y = 175 - 5x, Respectively, where y is the price, in dollars, and x is the number of units, in

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