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mathematics
calculus

Questions and Answers of Calculus

Let f(x) = 1 + x + x2 and h ( 0.(a) Is f(2 + 1) = f(2) + f(1)?(b) Find f(x + h).(c) Does f(x + h) = f(x) + f(h)?(d) Does f(x + h) = f(x) + h?(e) FindAnd simplify?
Let f (x) = 3x2 - 6x and h ( 0.(a) Is f(3 + 2) = f (3) + 2?(b) Find f (x + h).(c) Does f(x + h) = f(x) + h?(d) Does f(x + h) = f(x) + f(h)?(e) FindAnd simplify?
If f (x) = x - 2x2 and h ( 0, find the following and simplify.(a) f(x + h)(b)
If f (x) = 2x2 - x + 3 and h ( 0, find the following and simplify.(a) f(x + h)(b)
1. If y = f (x) has the graph in Problem 5, find the following.Graph in problem 5,(a) f(9)(b) f(5)2. Suppose y = g(x) has the graph in Problem 8.(a) Find g(0).(b) How many x-values in the domain of
The graph of y = x2 = 4x is shown below.(a) What are the coordinates of the point Q? Do they satisfy the equation? (b) What are the coordinates of R? Do they satisfy the equation? (c) If the
The graph of y = 2x2 is shown below.(a) Does the point (1, 1) lie on the graph? Do the coordinates satisfy the equation? (b) What are the coordinates of point R? Do they satisfy the equation? (c) If
State the domain and range of each of the functions in Problems. 1. y = x2 + 4 2. y = x2 - 1 3. y = (x + 4 4. y = (x2 + 1
In following questions are the relations defined by the tables functions? Explain why or why not and give the domain and range.1.2.
In following question a function and its graph are given. In each problem, find the domain.1.2. 3. f(x) = 4 + (49 - x2 4. f(x) = - 2 - (9 - x2
For f(x) and g(x) given in Problems following questions, find(a) (f + g) (x)(b) (f - g) (x)(c) (f ( g) (x)(d) (f / g) (x)1. f(x) = 3x g(x) = x32. f(x) = (x g(x) = 1/x3. f(x) = (2x g(x) = x24. f(x) =
For f(x) and g(x) given in following questions, find (a). (f * g) (x) (b). (g * f) (x) (c). f(f(x)) (d). f2(x) = (f * f) (x) 1. f(x) = (x - 1)3 g(x) = 1 - 2x 2. f(x) = 3x g(x) = x3 - 1 3. f (x) =
A couple seeking to buy a home decides that a monthly payment of $800 fits their budget. The amount they can borrow, A, is a function of the time t, in years, it will take to repay the debt. If we
When a debt is refinanced, sometimes the term of the loan (that is, the time it takes to repay the debt) is shortened. Suppose the current interest rate is 7%, and a couple's current debt is
The figure below gives the monthly Social Security benefits for persons whose benefit would be $1000 at a full retirement age of 66 as a function of the age these beneficiaries start receiving
The graph shows the Dow Jones Industrial Average on a particularly tumultuous day soon after the U.S. credit rating was downgraded from AAA to AA+ in 2011. If t represents the number of hours after
As the U.S. population diversifies, the number of White, non-Hispanic individuals (the traditional "White majority") accounts for a smaller percent of the total. The following figure shows data from
Determine whether each graph represents y as a function of x. Explain your answer.1.2. 3. 4.
The number (in millions) of women in the workforce, given as a function f of the year for selected years from 1920 and projected to 2016, is shown in the figure.(a) How many women were in the labor
The equationgives the relation between temperature readings in Celsius and Fahrenheit.(a) Is C a function of F?(b) What is the mathematical domain of this function?(c) If we consider this equation as
The profit from the production and sale of a product is P(x) = 47x - 0.01x2 - 8000, where x represents the number of units produced and sold. Give(a) The profit from the production and sale of 2000
The total cost of producing a product is given byC(x) = 300x + 0.1x2 + 1200where x represents the number of units produced. Give(a) The total cost of producing 10 units(b) The value of C(100)(c) The
If a test that has reliability r is lengthened by a factor n (n ( 1), the reliability R of the new test is given byIf the reliability is r0.6, the equation becomes (a) Find R(1). (b) Find R(2); that
Suppose that the cost C (in dollars) of removing p percent of the particulate pollution from the smokestacks of an industrial plant is given by(a) Find the domain of this function. In parts (b)-(e),
If a box with a square cross section is to be sent by a delivery service, there are restrictions on its size such that its volume is given by V = x2(108 - 4x), where x is the length of each side of
Suppose that the profit from the production and sale of x units of a product is given byIn addition, suppose that for a certain month the number of units produced on day t of the month isx = q(t) =
For many species of fish, the weight W is a function of the length L that can be expressed byW = W(L) = kL3 k = constantSuppose that for a particular species k0.02, that for this species the length
Suppose that a company's revenue R = f (C) is a function f of the number of customers C. Suppose also that the amount spent on advertising A affects the number of customers so that C = g(A) is a
Two of the processes (functions) used by a manufacturer of factory-built homes are sanding (denote this as function s) and painting (denote this as function p). Write a sentence of explanation for
A farmer wishes to fence the perimeter of a rectangular lot with an area of 1600 square feet. If the lot is x feet long, express the number of feet L of fence needed as a function of x?
1. A shipping crate has a square base with sides of length x feet, and it is half as tall as it is wide. If the material for the bottom and sides of the box costs $2.00 per square foot and the
Determine if the equation represents y as a function of x. 1. y = 3x3 2. y = 6x2 3. y2 = 3x 4. y2 = 10x2
Find the intercepts and graph the functions.1. 3x + 4y = 122. 6x - 5y = 903. 5x - 8y = 604. 2x - y + 17 = 0
In following questions for each given graph, determine whether each line has a slope that is positive, negative, 0, or undefined.1.2.
Find the slope and y-intercept and then graph each equation.1.2.3. y = 34. y = -2
Write the slope-intercept form of the equation of the line that has the given slope and y-intercept. Then graph the line. 1. Slope 1/2 and y-intercept -3 2. Slope 4 and y-intercept 2 3. Slope -2 and
Write the equation of each line described or shown.1. Through (3, 2) and (-1, -6)2. Through (-4, 2) and (2, 4)3. Through (7, 3) and (-6, 2)4. Through (10, 2) and (8, 7)
Determine whether the following pairs of equations represent parallel lines, perpendicular lines, or neither of these.1. 3x + 2y = 6; 2x - 3y = 62. 5x - 2y = 8; 10x - 4y = 83. 6x - 4y = 12; 3x - 2y =
A $360,000 building is depreciated by its owner. The value y of the building after x months of use is y = 360,000 - 1500x.(a) Graph this function for x ( 0.(b) How long is it until the building is
Find the slope of the line passing through the given pair of points. 1. (22, 11) and (15, -17) 2. (-6, -12) and (-18, -24) 3. (3, -1) and (-1, 1) 4. (-5, 6) and (1, -3)
Using Social Security Administration data for selected years from 1950 and projected to 2050, the U.S. population (in millions) can be described byp(t) = 2.53t + 162.2where t is the number of years
The percent of the U.S. population with Internet service can be described by y = 1.36x + 55.98 where x is the number of years past 1990. (a) Find the slope and y-intercept of this equation. (b) What
The percent p of high school students who occasionally smoke cigarettes can be described by p = 38.56 - 1.53t where t is the number of years past 1995. (a) Find the slope and p-intercept of this
When x is the number of years past 1990, the world forest area (that is, natural forest or planted stands) as a percent of land area is given byF(x) = - 0.065x + 31.39(a) Find the slope and
Two models for measuring the effects of high temperature and humidity are the Summer Simmer Index and the Apparent Temperature.* For an outside temperature of 100 oF, these indices relate the
According to the U.S. Census Bureau, the relation between the average annual earnings of males and females with various levels of educational attainment can be modeled by the functionF = 0.78M -
1. The U.S. gross domestic product (GDP), y, for the years from 2011 and projected to 2021 is given byt = 977.8x + 13,643.2where x is the number of years past 1990 and y is in billions of dollars.(a)
The size of the U.S. civilian workforce for the years from 1950 and projected to 2050 can be approximated by a linear equation determined by the line connecting the points (1950, 62.2) and (2050,
1. According to USA Today Research, most pensions for state legislators are based on the product of the legislator's years of service, compensation, and a multiplier (usually between 1% and 5%). When
The following table gives the number of drinks and the resulting blood alcohol percent for a 180-pound man legally considered driving under the influence (DUI).(a) Is the average rate of change of
It has been estimated that a certain stream can support 85,000 fish if it is pollution-free. It has further been estimated that for each ton of pollutants in the stream, 1700 fewer fish can be
Each day, a young person should sleep 8 hours plus 1/4 hour for each year that the person is under 18 years of age. Assuming the relation is linear, write the equation relating hours of sleep y and
The R-value of insulation is a measure of its ability to resist heat transfer. For fiberglass insulation, 31/2 inches is rated at R-11 and 6 inches is rated at R-19. Assuming this relationship is
Suppose the cost of a business property is $960,000 and a company wants to use a straight-line depreciation schedule for a period of 240 months. If y is the value of this property after x months,
The Seven Countries Study, conducted by Ancel Keys, was a long-term study of the relationship of cholesterol to coronary heart disease (CHD) mortality in men. The relationship was approximated by the
Use a graphing utility with the standard viewing window to graph each function. 1. y = x2 + 4x + 1 2. y = 4 - 3x - x2 3. y = x3 - 3x 4. y = x3 - 6x2
Show the correct way to enter each function in a calculator.1.2.
Graph each equation with a graphing utility using(a) The specified window.(b) The standard viewing window (for comparison).1. y = 0.01x3 + 0.3x2 - 72x + 150x-min = -100, x-max = 80;y-min = -2000,
(a) Algebraically find the x- and y-intercepts of the graph and(b) Graph each equation with a graphing utility, using a window that includes the intercepts.1. y = 0.001x - 0.032. y = 50,000 - 100x
The graph of a function is shown in the standard viewing window. Experiment with the viewing window to obtain a complete graph.1. y = -0.15(x - 10.2)2 + 102. y = x2 - x - 42 3. 4.
Graph the equations in the standard viewing window on a graphing utility. Identify any linear functions. 1. 4x - y = 8 2. 5x + y = 8 3. 4x2 + 2y = 5 4. 6x2 + 3y = 8
Use VALUE or TABLE to evaluate the function.1. If f(x) = x3 - 3x2 + 2, find f(-2) and f(3/4).2. Iffind f(3) and f (-4)?
Graph each function using a window that gives a complete graph.1.2. 3. 4.
Use the x-intercept method to find one solution of each equation. 1. 0 = 6x - 21 2. 12x + 28 = 0 3. 10 = x2 - 3x 4. 6x2 + 4x = 4
Use a graphing utility to (a) approximate two x-intercepts of the graph of the function (to four decimal places) and (b) give the approximate zeros of the function. 1. y = x2 - 7x - 9 2. y = 2x2 - 4x
With U.S. Census Bureau data, the model that relates the average annual earnings (in thousands of dollars) of females, F, and males, M, with various levels of educational attainment was found to be F
By using U.S. Bureau of Labor Statistics data from 1950 and projected to 2050, the number of women in the workforce (in millions) can be described by W = 0.79x + 20.86, where x equals the number of
Suppose that the consumer expenditure E (in dollars) depends on the market price p per unit (in dollars) according to E = 10,000p - 100p2 (a) Graph this equation with a graphing calculator and the
The height above ground (in feet) of a ball thrown vertically into the air is given by S = 112t - 16t2 where t is the time in seconds since the ball was thrown.(a) Graph this equation with a graphing
An advertising agency has found that when it promotes a new product in a city of 350,000 people, the rate of change R of the number of people who are aware of the product is related to the number of
In a study using 50 foreign-language vocabulary words, the learning rate L (in words per minute) was found to depend on the number of words already learned, x, according to the equation L = 20 -
Using data from 1920 and projected to 2030, the percent P of men 65 years of age or older in the labor force can be modeled by P = 0.000078x3 - 0.0107x2 - 0.182x + 64.68 where x is the number of
Using data from 2010 and projected to 2016, the number of millions of U.S. cell phone users can be described by Y = - 0.187x2 + 5.71x + 232 where x is the number of years past 2010. (a) Use a table
The Millcreek watershed area was heavily strip mined for coal. Because of the resulting pollution, the streams cannot support fish. Suppose the cost C of obtaining stream water that contains p
Suppose the cost C of removing p percent of the particulate pollution from the exhaust gases at an industrial site is given bywhere 0 < p < 100.(a) Use the restriction on p and experiment with
The dollars per capita of federal tax burden T can be described by T = 2.11x3- 93.01x2 + 1373.9x + 394 where x is the number of years past 1990. (a) Graph this function for values of x that
Use a graphing utility with the specified window to graph each equation.1. y = x3 - 12x - 1x-min = -5, x-max = 5;y-min = -20, y-max = 202. y = x4 - 4x3 + 63. x-min = - 4, x-max = 6;4. y-min= - 40,
Decide whether the system of equations in each problem has one solution, no solution, or an infinite number of solutions. If the system has one solution, estimate it?1.2. 3. 4.
Solve each system by elimination or by any convenient method?1.2.3.4.
Use a graphing calculator or Excel to find the solution of each system of equations in Problems?1.2.3.4.
Use the left-to-right elimination method to solve the systems in Problems?1.2.3.4.5.
When U.S. citizens travel abroad, they need some understanding of Celsius temperature readings-does one need mittens if the temperature will be 20° C? A quick conversion formula often used by tour
Using U.S. Census Bureau data (and projections to 2050), the percent of the U.S. population that is Black B(x) and Hispanic H(x) can be described byB(x) = 0.057x + 12.3 and H(x) = 0.224x + 9.01where
A concert promoter needs to make $42,000 from the sale of 1800 tickets. The promoter charges $20 for some tickets and $30 for the others. Let x represent the number of $20 tickets and y represent the
A woman has $500,000 invested in two rental properties. One yields an annual return of 10% on her investment, and the other returns 12% per year on her investment. Her total annual return from the
One safe investment pays 10% per year, and a riskier investment pays 18% per year. A woman who has $145,600 to invest would like to have an income of $20,000 per year from her investments. How much
A bank lent $237,000 to a company for the development of two products. If the loan for product A was for $69,000 more than that for product B, how much was lent for each product?
A woman has $470,000 invested in two rental properties. One yields 10% on the investment, and the other yields 12%. Her total income from them is $51,000. How much is her income from each property?
Mr. Jackson borrowed money from his bank and on his life insurance to start a business. His interest rate on the bank loan was 10%, and his rate on the insurance loan was 12%. If the total amount
Each ounce of substance A supplies 5% of the nutrition a patient needs. Substance B supplies 12% of the required nutrition per ounce. If digestive restrictions require that the ratio of substance A
A glass of skim milk supplies 0.1 mg of iron and 8.5 g of protein. A quarter pound of lean red meat provides 3.4 mg of iron and 22 g of protein. If a person on a special diet is to have 7.15 mg of
Bacteria of species A and species B are kept in a single test tube, where they are fed two nutrients. Each day the test tube is supplied with 10,600 units of the first nutrient and 19,650 units of
A biologist has a 40% solution and a 10% solution of the same plant nutrient. How many cubic centimeters of each solution should be mixed to obtain 25 cc of a 28% solution? Explain
A nurse has two solutions that contain different concentrations of a certain medication. One is a 20% concentration and the other is a 5% concentration? How many cubic centimeters of each should he
Medication A is given every 4 hours and medication B is given twice each day. The total intake of the two medications is restricted to 50.6 mg per day, for a certain patient. If the ratio of the
A concert promoter needs to take in $760,000 on the sale of 16,000 tickets. If the promoter charges $40 for some tickets and $60 for others, how many of each type must be sold to yield the $760,000?
Solve the systems of equations by using graphing?1.2. 3. 4.
A nut wholesaler sells a mix of peanuts and cashews. He charges $2.80 per pound for peanuts and $5.30 per pound for cashews. If the mix is to sell for $3.30 per pound, how many pounds each of peanuts
How many cubic centimeters of a 20% solution of a nutrient must be added to 100 cc of a 2% solution of the same nutrient to make a 10% solution of the nutrient?
How many gallons of washer fluid that is 13.5% antifreeze must a manufacturer add to 200 gallons of washer fluid that is 11% antifreeze to yield washer fluid that is 13% antifreeze?

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