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mathematics
calculus

Questions and Answers of Calculus

If a ball is thrown upward at 96 feet per second from the top of a building that is 100 feet high, the height of the ball is given by S = 100 + 96t - 16t2 feet where t is the number of seconds after
Solve each equation by factoring.1. x2 - 4x = 122. x2 = 11x = 103. 9 - 4x2 = 04. 25x2 - 16 = 0
A tennis ball is thrown into the air from the top of a hotel that is 350 feet above the ground. The height of the ball from the ground is given by D(t) = - 16t2 + 10t + 350 feet where t is the time,
The amount of airborne particulate pollution p from a power plant depends on the wind speed s, among other things, with the relationship between p and s approximated by p = 25 - 0.01s2 (a) Find the
The sensitivity S to a drug is related to the dosage size by S = 100x - x2 where x is the dosage size in milliliters. (a) What dosage(s) will yield 0 sensitivity? (b) Explain what your answer in part
The time t, in seconds, that it takes a Corvette to accelerate to x mph can be described by t = 0.001(0.732x2 + 15.417x + 607.738) How fast is the Corvette going after 8.99 seconds? Give your answer
Social Security benefits are paid from a trust fund. As mentioned in the Application Preview, the trust fund balance, B, in trillions of 2012 dollars, t years past the year 2000 is described by B = -
For the years from 2001 to 2012, the percent p of high school seniors who have tried marijuana can be considered as a function of time t according to p = f(t) = 0.17t2 - 2.61t + 52.64 where t is the
Two projectiles are shot into the air over a lake. The paths of the projectiles are given by(a) y = -0.0013x2 + x + 10 and(b)where y is the height and x is the horizontal distance traveled (both in
The Ace Jewelry Store sold a necklace for $144. If the percent profit (based on cost) equals the cost of the necklace to the store, how much did the store pay for it? Usewhere P is profit and C is
The global spending on travel and tourism (in billions of dollars) can be described by the equation t = 0.787x2 - 11.0x + 290 where x equals the number of years past 1990. Find the year after 1990 in
Using data from 2006 and with projections to 2021, total annual expenditures for national health care (in billions of dollars) can be described by E = 7.94x2 + 33.2x + 2190 where x is the number of
Because of friction from the walls of an artery, the velocity of a blood corpuscle in an artery is greatest at the center of the artery and decreases as the distance r from the center increases. The
The model for body-heat loss depends on the coefficient of convection K, which depends on wind speed v according to the equationK2 = 16v + 4where v is in miles per hour. Find the positive coefficient
A fissure in the earth appeared after an earthquake. To measure its vertical depth, a stone was dropped into it, and the sound of the stone's impact was heard 3.9 seconds later. The distance (in
(a) Find the vertex of the graph of the equation,(b) Determine if the vertex is a maximum or minimum point,(c) Determine what value of x gives the optimal value of the function, and(d) Determine the
For each function in Problems,(a) Tell how the graph of y = x2 is shifted, and(b) Graph the function.1. y = (x - 3)2 + 12. y = (x - 10)2 + 13. y = (x + 2)2 - 24. y = (x + 12)2 - 8
Graph each function with a graphing utility. Use the graph to find the vertex and zeros. Check your results algebraically?1.2. y = 0.1(x2 + 4x - 32)3. y = Â¼ x2 + 3x + 124. y = x2 - 2x + 5
Find the average rate of change of the function between the given values of x?1. y = - 5x - x2 between x = - 1 and x = 1.2. y = 8 + 3x + 0.5x2 between x = 2 and x = 4.
Find the vertex and zeros and use them to determine a range for a graphing utility that includes these values; graph the function with that range.1. y = 63 + 0.2x - 0.01x22. y = 0.2x2 + 16x + 1403. y
(a) Find the vertex of each function f(x). Use the vertex to set the window in which to graph y = f(x) and then(b) Graphically approximate the solutions to f(x) = 0.1. f(x) = 8x2 - 16x - 162. f(x) =
Complete the following.(a) Use the graph of y = f(x) to find an integer solution to f(x) = 0.(b) Use the solution from (a) to find a factor of f (x).(c) Factor f (x).(d) Solve f (x) = 0.1. f(x) = 3x2
The daily profit from the sale of a product is given by P = 16x - 0.1x2 - 100 dollars.(a) What level of production maximizes profit?(b) What is the maximum possible profit?
The daily profit from the sale of x units of a product is P = 80x - 0.4x2 - 200 dollars. (a) What level of production maximizes profit? (b) What is the maximum possible profit?
1. The yield in bushels from a grove of orange trees is given by Y = x(800 - x), where x is the number of orange trees per acre. How many trees will maximize the yield?2. One of the early results in
The sensitivity S to a drug is related to the dosage x (in milligrams) by S = 1000x - x2 Sketch the graph of this function and determine what dosage gives maximum sensitivity. Use the graph to
If 100 feet of fence is used to enclose a rectangular yard, then the resulting area is given by A = x(50 - x) where x feet is the width of the rectangle and 50 - x feet is the length. Graph this
1. The rate of photosynthesis R for a certain plant depends on the intensity of light x, in lumens, according toR = 270x - 90x2Sketch the graph of this function, and determine the intensity that
Two projectiles are shot into the air from the same location. The paths of the projectiles are parabolas and are given by(a) y = -0.0013x2 + x + 10 and(b)where x is the horizontal distance and y is
The speed at which water travels in a pipe can be measured by directing the flow through an elbow and measuring the height to which it spurts out the top. If the elbow height is 10 cm, the equation
The following figure shows the graph of a total cost function, with x equal to the number of units produced.(a) Is the average rate of change of cost greater from x = a to x = b or from x = b to x =
The following figure shows the graph of a total revenue function, with x equal to the number of units sold.(a) Is the average rate of change of revenue negative from x = a to x = b or from x = b to x
The owner of an apartment building can rent all 50 apartments if she charges $600 per month, but she rents one fewer apartment for each $20 increase in monthly rent.(a) Construct a table that gives
The owner of a skating rink rents the rink for parties at $600 if 50 or fewer skaters attend, so that the cost per person is $12 if 50 attend. For each 5 skaters above 50, she reduces the price per
The Pension Benefit Guaranty Corporation is the agency that insures pensions. The figure shows one study's projection for the agency's total resources, initially rising (from taking over the assets
When a stone is thrown upward, it follows a parabolic path given by a form of the equation y = ax2 + bx + c If y = 0 represents ground level, find the equation of a stone that is thrown from ground
Rising health care costs are a growing concern for everyone in the United States. With U.S. Centers for Medicare and Medicaid Services data from 2006 and projected to 2021, U.S. per capita health
Using U.S. Census Bureau data for selected years from 1970 and projected to 2040, the percent of the total workforce that is female is given by p(t) = -0.0036t2 + 0.38t + 38.62 where t is the number
Online sales, which make up about 5% of total retail sales in the United States, are expected to top $400 billion by 2017. Using data from 2000 and projected to 2017, the revenue, in billions of
Determine whether each function's vertex is a maximum point or a minimum point and find the coordinates of this point. Find the zeros, if any exist, and the y-intercept. Then sketch the graph of the
The total costs for a company are given by C(x) = 2000 + 40x + x2 and the total revenues are given by R(x) = 130x Find the break-even points?
If, in a monopoly market, the demand for a product is p = 1600 - x and the revenue is R = px, where x is the number of units sold, what price will maximize revenue?
1. The profit function for a certain commodity is P(x) = 110x - x2 - 1000. Find the level of production that yields maximum profit, and find the maximum profit?2. The profit function for a firm
(a) Graph the profit function P(x) = 80x - 0.1x2 - 7000. (b) Find the vertex of the graph. Is it a maximum point or a minimum point? (c) Is the average rate of change of this function from x = a <
(a) Graph the profit function P(x) = 50x - 0.2x2 - 2000. (b) Find the vertex of the graph. Is it a maximum point or a minimum point? (c) Is the average rate of change of this function from x = a <
(a) Form the profit function for the cost and revenue functions in Problem 3, and find the maximum profit.In problem 3,If a company has total costs C(x) = 15,000 + 35x + 0.1x2 and total revenues
(a) Form the profit function for the cost and revenue functions in Problem 4, and find the maximum profit.In problem 4,If total costs are C(x) = 1600 + 1500x and total revenues are R(x) = 1600x - x2,
Suppose a company has fixed costs of $28,000 and variable cost per unit of 2/5 x + 222 dollars, where x is the total number of units produced. Suppose further that the selling price of its product is
Suppose a company has fixed costs of $300 and variable cost per unit of 3/4x + 1460 dollars, where x is the total number of units produced. Suppose further that the selling price of its product is
The following table gives the total revenues of Cable-net Communications for selected years.Year ______________ Total Revenues (millions)2008 ............................... $63.132009
If a firm has the following cost and revenue functions, find the break-even points?
Assume that sales revenues for Continental Divide Mining can be modeled byR(t) = 0.031t2 + 0.776t + 0.179where t is the number of years past 2003.(a) Use the function to determine the year in which
Assume that costs and expenses for Continental Divide Mining can be modeled byC(t) = - 0.012t2 + 0.492t + 0.725where t is the number of years past 2003.(a) Use R(t) as given in Problem 20 and form
A supply function and a demand function are given.(a) Sketch the first-quadrant portions of those functions on the same set of axes.(b) Label the market equilibrium point.(c) Algebraically determine
If the supply function for a commodity is p = q2 + 8q + 16 and the demand function is p = - 3q2 + 6q + 436, find the equilibrium quantity and equilibrium price?
If the supply function for a commodity is p = q2 + 8q + 20 and the demand function is p = 100 - 4q - q2, find the equilibrium quantity and equilibrium price?
If the demand function for a commodity is given by the equation p2 + 4q = 1600 and the supply function is given by the equation 300 -p2 + 2q = 0, find the equilibrium quantity and equilibrium price?
If the supply and demand functions for a commodity are given by 4p - q = 42 and (p + 2)q = 2100, respectively, find the price that will result in market equilibrium?
If the supply and demand functions for a commodity are given by p - q = 10 and q(2p - 10) = 2100, what is the equilibrium price and what is the corresponding number of units supplied and demanded?
If a company has total costs C(x) = 15,000 + 35x + 0.1x2 and total revenues given by R(x) = 385x - 0.9x2, find the break-even points?
If the supply and demand functions for a certain product are given by the equations 2p - q + 6 = 0 and (p + q) (q + 10) = 3696, respectively, find the price and quantity that give market equilibrium?
The supply function for a product is 2p - q - 10 = 0, while the demand function for the same product is (p + 10)(q + 30) = 7200. Find the market equilibrium point?
The supply and demand for a product are given by 2p - q = 50 and pq = 100 + 20q, respectively. Find the market equilibrium point?
For the product in Problem 31, if a $22 tax is placed on production of the item, then the supplier passes this tax on by adding $22 to his selling price. Find the new equilibrium point for this
For the product in Problem 32, if a $12.50 tax is placed on production and passed through by the supplier, find the new equilibrium point. In problem 32, The supply and demand for a product are given
If total costs are C(x) = 1600 + 1500x and total revenues are R(x) = 1600x - x2, find the break-even points?
Given that P(x) = 11.5x - 0.1x2 - 150 and that production is restricted to fewer than 75 units, find the break-even points?
If the profit function for a firm is given by P(x) = -1100 + 120x - x2 and limitations on space require that production is less than 100 units, find the breakeven points?
Find the maximum revenue for the revenue function R(x) = 385x - 0.9x2?
1. Find the maximum revenue for the revenue function R(x) = 1600x - x2?2. If, in a monopoly market, the demand for a product is p = 175 - 0.50x and the revenue function is R = px, where x is the
Decide whether each function whose graph is shown is the graph of a cubic (third degree) or quartic (fourth-degree) function.1.(a)(b)2.(a)(b)Match each equation with the correct graph among those
Graph the functions given below 1. y = (x + 1) (x - 3) (x - 1) 2. y = x3 - 8x2 + 19x - 12 3. y = 1 - 2x / x 4. y = x / x - 1
(a) Graph each function with a graphing utility;(b) Classify each function as a polynomial function, a rational function, or a piecewise defined function;(c) Identify any asymptotes; and(d) Use the
If a box with a square cross section is to be sent by the postal service, there are restrictions on its size such that its volume is limited by V = x2(108 - 4x), where x is the length of each side of
Fixed costs FC are business costs that remain constant regardless of the number of units produced. Some items that might contribute to fixed costs are rent and utilities. FC = 2000 Is an equation
Global spending on travel and tourism (in billions of dollars) can be modeled by f(x) = 15.875x1.18 where x is the number of years after 1980. (a) Does the graph of this function bend upward or bend
For fiddler crabs, data gathered by Thompson show that the relationship between the weight C of the claw and the weight W of the body is given by C = 0.11W1.54 where W and C are in grams. (a) Does
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) Is C(p) undefined at any p-value? If so, what
If the weekly total cost of producing 27" Toshiba television sets is given by C(x) = 50,000 + 105x, where x is the number of sets produced per week, then the average cost per unit is given by(a)
If 100 feet of fence is to be used to enclose a rectangular yard, then the resulting area of the fenced yard is given by A = x(50 - x) Where x is the width of the rectangle. (a) If A = A(x), find
The monthly charge for water in a small town is given byWhere x is water usage in hundreds of gallons and f(x) is in dollars. Find the monthly charge for each of the following usages: (a) 30
By using data for selected years from 1950 and projected to 2040, the gross domestic product (GDP), in billions of dollars, of the United States is modeled byWhere x is the number of years past 1950
The monthly charge (in dollars) for x kilowatt hours (kWh) of electricity used by a commercial customer is given by the following function.Find the monthly charges for the following usages. (a) 5
The postage charged for first-class mail is a function of its weight. The U.S. Postal Service uses the following table to describe the rates. Weight Increment ______________________ Rate ($) First
In a given year the U.S. federal income tax owed by a married couple filing jointly can be found from the following table:(a) For incomes up to $137,300, write the piecewise defined function T with
The demand function for a product is given by P = 200 / 2 + 0.1x Where x is the number of units and p is the price in dollars. (a) Graph this demand function for 0 ( x ( 250, with x on the horizontal
In studying lynchings between 1899 and 1946, psychologist Brian Mullin found that the size of a lynch mob relative to the number of victims predicted the level of brutality. He developed a formula
A manufacturer estimates that the cost of a production run for a product isWhere x is the number of machines used. (a) Graph this total cost function for values of x ( 0. (b) Interpret any turning
Determine whether the scatter plot should be modeled by a linear, power, quadratic, cubic, or quartic function.1.2. 3. 4. 5. 6. 7. 8.
(a) Plot the given points below(b) Determine what type of function best models the data, and(c) Find the equation that is the best fit for the data.1.2. 3. 4.
The following table gives the projected population, in thousands, of U.S. females under the age of 18.(a) Create a linear function that models the projected population y, in thousands, as a function
The following table gives the millions of metric tons of carbon dioxide emissions in the United States for selected years from 2010 and projected to 2032.(a) Create a linear function that models
Disposable income is the amount left after taxes have been paid and is one measure of the health of the economy. The following table gives the total U.S disposable income (in billions of 2005
As the following table shows, projections indicate that the percent of U.S. adults with diabetes could dramatically increase.(a) Find a linear model that fits the data in the table, with x = 0 in
The table gives the wind chill temperature when the outside temperature is 20°F.(a) Use x as the wind speed and create a quadratic model for these data. (b) At what wind speed does the model
The developing economies' share of the global gross domestic product (GDP) from 2003 and projected to 2017 is shown in the following table.(a) Find the quadratic function that best models the
The table gives the gross domestic product (GDP), in billions of dollars, of the United States for selected years from 2000 and projected to 2070.(a) Create a scatter plot of the data, with y
As the baby boom generation ages and the proportion of the U.S. population over age 65 increases, the number of Americans with Alzheimer's disease and other dementia is projected to grow each year.
The following table gives the U.S. crude oil production, in billions of barrels, for selected years from 2010 and projected to 2030.(a) Find the linear function that models the data, with x equal to
Energy use per dollar of GDP indexed to 1980 means that energy use for any year is viewed as a percent of the use per dollar of GDP in 1980. The following data show the energy use per dollar of GDP,

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