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mathematics
calculus

Questions and Answers of Calculus

In Problems 1-4, graph each function. 1. y = log3 x 2. y = log4 x 3. y = ln x 4. y = log9 x
In Problems 1 and 2, use properties of logarithms or a definition to simplify each expression. Check each result with a change-of-base formula.1. (a) log3 27(b) log5 (1 / 5)2. (a) log4 16(b) log9 3
If f (x) = ln (x), find f (√e).
In Problems 1 and 2, evaluate each logarithm by using properties of logarithms and the following facts. loga x = 3.1 loga y = 1.8 loga z = 2.7 1. (a) loga (xy) (b) loga (x/z) (c) loga (x4) (d) loga
Write each expression in Problems 1-3 as the sum or difference of two logarithmic functions containing no exponents. 1. log (x / x + 1) 2. ln [(x + 1)(4x + 5)] 3. log7 (x 3√x + 4)
Use the properties of logarithms to write each expression in Problems 1-3 as a single logarithm. 1. ln x - ln y 2. log3 (x + 1) + log3 (x - 1) 3. log5 (x + 1) + 1/2 log5 x
In Problems 1-3, use a calculator to determine whether expression (a) is equivalent to expression (b). If they are equivalent, state what properties are being illustrated. If they are not equivalent,
(a) Use a graphing utility to graph f (x) = ln x and f (x - c) = ln (x - c) for c = - 4, - 2, 1, and 5 on the same axes. (b) For each c-value, identify the vertical asymptote and the domain of y = ln
In Problems 1-4, solve for x by writing the equation in exponential form. 1. log3 x = 4 2. log4 x = - 2 3. log16 x = 1/2 4. log25 x = 1 / 2
Use a graphing utility to graph f (x) = ln x and f (ax) = ln (ax) for a = - 2, - 1, - 0.5, 0.2, and 3. Explain the differences between the graphs (a) When a > 0 and when a < 0. (b) When |a| >1 and
In Problems 1 and 2, use a change-of-base formula to evaluate each logarithm. 1. (a) log2 17 (b) log5 (0.78) 2. (a) log3 12 (b) log8 (0.15)
In Problems 1-4, use a change-of-base formula to rewrite each logarithm. Then use a graphing utility to graph the function. 1. y = log5 x 2. y = log6 x 3. y = log13 x 4. y = log16 x
Prove logarithmic Property IV.
Prove logarithmic Property V.
The most devastating earthquake since 2000 was the Haiti quake in 2010 that measured 7.0 on the Richter scale and resulted in more than 222,500 deaths, 300,000 injuries, and 1.3 million displaced
In May of 2008, an earthquake measuring 6.8 on the Richter scale struck near the east coast of Honshu, Japan. In March of 2011, a quake measuring 9.0 struck that same region. How many times more
The world's strongest earthquake struck Chile in 1960 and measured 9.5 on the Richter scale. The 2010 Chilean quake at 8.8 was the world's sixth largest. Find the ratio of their intensities.
Graph the equation for loudness of sound in decibels. Use I / I0 as the independent variable. Use the fact that the loudness of sound (in decibels) perceived by the human ear depends on intensity
The background noise level of a relatively quiet room is about L1 = 32 decibels and of a heated argument can exceed L2 = 66 decibels. Find the ratio I2 / I1 of these associated intensities. Use the
Most common solutions have pH ranges between 1 and 14. What values of [H] are associated with these extremes? Use the following information. Chemists use the pH (hydrogen potential) of a solution to
Find the approximate pH of each of the following. (a) Blood: [H+] = 3.98 × 10-8 = 0.0000000398 (b) Beer: [H+] = 6.31 × 10-5 = 0.0000631 (c) Vinegar: [H+] = 6.3 × 10-3 = 0.0063 Use the following
Sometimes pH is defined as the logarithm of the reciprocal of the concentration of hydrogen ions. Write an equation that represents this sentence, and explain how it and the equation given in the
Find the approximate hydrogen ion concentration [H+] for each of the following. (a) Apples: pH = 3.0 (b) Eggs: pH = 7.79 (c) Water (neutral): pH = 7.0 Use the following information. Chemists use the
In problems 1 and 2, use the formula 2 = (1 + r / 100n)nt to find the doubling time t, in years, for an investment at r% compounded n times per year. Write each exponential statement in logarithmic
For selected years from 1970 and projected to 2050, the number, in millions, of women in the workforce is given byw(x) = 37.6 ln x - 81.2where x is the number of years past 1950(a) Graph this
In Example 6 we used data from 1920 to 2020 and found that the life span in the United States depended on the year of birth according to the equation L(x) = 11.027 + 14.304 ln x where x is the number
As the following table shows, projections indicate that the percent of U.S. adults with diabetes could dramatically increase.(a) Find a logarithmic model that fits the data in the table, with x = 0
The percent of all workers 16 years and older that are female, for selected years from 1970 and projected to 2040, is given in the following table.(a) Find the logarithmic function that models the
The percent of the U.S. population that uses the Internet, during selected years from 2000, is given in the following table.(a) Find the logarithmic function that models the percent as a function of
The table below gives the millions of White non-Hispanic individuals in the U.S. civilian non-institutional population 16 years and older for selected years from 1980 and projected to 2050.(a) Find a
In Problems 1-5, solve each equation. Give answers correct to 3 decimal places in Problems 1-5. 1. 83x = 32,768 2. 72x = 823,543 3. 0.13P = P(2-x) 4. 0.05A = A(1.06)-x 5. 25,000 = 10,000(1.05)2x
(a) Fix A = 100 and graph y = f (x) = 100 / 1 + ce-x for c = 0.25, 1, 9, and 49. (b) What effect does c have on the graphs? Let f (x) = A / 1 + ce-x. Use a graphing utility to make the requested
(a) Fix c = 1 and graph y = f (x) = A / 1 + ce-x for A = 50, 100, and 150. (b) What effect does A have on the graphs? Let f (x) = A / 1 + ce-x. Use a graphing utility to make the requested graphs.
The sales decay for a product is given by S = 50,000e-0.8x, where S is the monthly sales and x is the number of months that have passed since the end of a promotional campaign. (a) What will be the
The sales of a product decline after the end of an advertising campaign, with the sales decay given by S = 100,000e-0.5x, where S represents the weekly sales and x represents the number of weeks
The purchasing power P (in dollars) of an annual amount of A dollars after t years of 5% inflation decays according to P = Ae-0.05t (a) How long will it be before a pension of $60,000 per year has a
A statistical study shows that the fraction of television sets of a certain brand that are still in service after x years is given by f (x) = e-0.15x. (a) What fraction of the sets are still in
An initial amount of 100 g of the radioactive isotope thorium-234 decays according to Q(t) = 100e-0.02828t where t is in years. How long before half of the initial amount has disintegrated? This time
A breeder reactor converts stable uranium-238 into the isotope plutonium-239. The decay of this isotope is given by A(t) = A0e-0.00002876t where A(t) is the amount of the isotope at time t, in years,
If the population of a certain county was 100,000 in 1998 and 110,517 in 2008, and if the formula y = P0eht applies to the growth of the county's population, estimate the population of the county in
The population of a certain city grows according to the formula y = P0e0.03t. If the population was 250,000 in 2000, estimate the year in which the population reaches 350,000.
For the years from 2006 and projected to 2021, the national health care expenditures H, in billions of dollars, can be modeled by H = 2009e0.05194t where t is the number of years past 2005. When are
For selected years from 1900 to 2013, the national debt d, in billions of dollars, can be modeled by d = 1.60e0.0834t where t is the number of years past 1900. How long will it be before the debt is
The demand function for a certain commodity is given by p = 100e-q/2.(a) At what price per unit will the quantity demanded equal 6 units?(b) If the price is $1.83 per unit, how many units will be
The demand function for a product is given by p = 3000e-q/3. (a) At what price per unit will the quantity demanded equal 6 units? (b) If the price is $149.40 per unit, how many units will be
If the supply function for a product is given by p = 100eq/(q + 1), where q represents the number of hundreds of units, what will be the price when the producers are willing to supply 300 units?
The total cost function for x units of a product is given by C(x) =2500 ln (2x +1) + 1500 (a) Find the total cost of producing 80 items. (b) How many units can be produced before total costs reach
The total cost function for a product is C(x) =800 ln (x + 10) +1700 where x is the number of units produced. (a) Find the total cost of producing 100 units. (b) Producing how many units will give
The millions of White non-Hispanic individuals in the U.S. civilian non-institutional population 16 years and older for selected years from 1980 and projected to 2050 can be modeled by the function
Centers for Disease Control and Prevention data from 2010 and projected to 2050 indicate that adult diabetes in the United States could dramatically increase. With x0 in 2000, the percent of U.S.
If $8500 is invested at 11.5% compounded continuously, the future value S at any time t (in years) is given by S = 8500e0.115t (a) What is the amount after 18 months? (b) How long before the
If $1000 is invested at 10% compounded continuously, the future value S at any time t (in years) is given by S = 1000e0.1t. (a) What is the amount after 1 year? (b) How long before the investment
If $5000 is invested at 9% per year compounded monthly, the future value S at any time t (in months) is given by S = 5000(1.0075)t.(a) What is the amount after 1 year?(b) How long before the
If $10,000 is invested at 1% per month, the future value S at any time t (in months) is given by S = 10,000(1.01)t. (a) What is the amount after 1 year? (b) How long before the investment doubles?
(a) Use the models to predict the company's profit in 2020. (b) How long before the profit found in part (a) is predicted to double? An investment services company experienced dramatic growth in the
Use the models to find how long before the company's profit reaches $500 million. An investment services company experienced dramatic growth in the last two decades. The following models for the
Using Social Security Administration data for selected years from 2012 and projected to 2050, with a purchasing power of $1.00 in 2012, the function P(x) = 1.078(1.028-x) gives the purchasing power
The function that models the growth in the number of billions of cubic feet of shale-natural gas in China, with x as the number of years after 2010, is y = 0.0117(1.75x) Find the year when the number
Suppose the supply of x units of a product at price p dollars per unit is given by p = 10 + 5 ln (3x + 1) How many units would be supplied when the price is $50 each?
Say the demand function for a product is given by p = 100 ln (q + 1). (a) What will be the price if 19 units are demanded? (b) How many units, to the nearest unit, will be demanded if the price is
The president of a company predicts that sales will increase after she assumes office and that the number of monthly sales will follow the curve given by N = 3000(0.2)0.6t, where t represents the
Because of a new market opening, the number of employees of a firm is expected to increase according to the equation N = 1400(0.5)0.3t, where t represents the number of years after the new market
Suppose that the equation N = 500(0.02)0.7t represents the number of employees working t years after a company begins operations. (a) How many employees are there when the company opens (at t =
A firm predicts that sales will increase during a promotional campaign and that the number of daily sales will be given by N = 200(0.01)0.8t, where t represents the number of days after the campaign
The concentration y of a certain drug in the bloodstream t hours after an oral dosage (with 0 ≤ t ≤ 15) is given by the equation y = 100(1 - e-0.462t) (a) What is y after 1 hour (t = 1)? (b) How
Suppose that the number y of otters t years after otters were reintroduced into a wild and scenic river is given by y = 2500 - 2490e-0.1t (a) Find the population when the otters were reintroduced (at
On a college campus of 10,000 students, a single student returned to campus infected by a disease. The spread of the disease through the student body is given bywhere y is the total number infected
The number of people N(t) in a community who are reached by a particular rumor at time t is given by the equation(a) Find N(0). (b) What is the upper limit on the number of people affected? (c) How
Suppose that the market share y (as a percent) that a company expects t months after a new product is introduced is given by y = 40 - 40e-0.05t. (a) What is the market share after the first month
An advertising agency has found that when it promotes a new product in a certain market of 350,000, the number of people x who are aware of the product t days after the ad campaign is initiated is
Pollution levels in a lake have been modeled by the equationx = 0.05 + 0.18e-0.38twhere x is the volume of pollutants (in cubic kilometers) and t is the time (in years).(a) Find the initial pollution
Suppose that the length x (in centimeters) of an individual of a certain species of fish is given byx = 50 - 40e-0.05twhere t is its age in months.(a) Find the length after 1 year.(b) How long (to
With U.S. Bureau of Labor Statistics data since 1950 and projected to 2040, the total civilian labor force age 16 years and older (in millions) can be modeled bywhere t is the number of years after
When two chemicals, A and B, react to form another chemical C (such as in the digestive process), this is a special case of the law of mass action, which is fundamental to studying chemical reaction
The following table gives the total number of U.S adults with diabetes for selected years from 2010 and projected to 2050.(a) Find the logistic function that models these data. Use x equal to the
The following table gives the numbers of species in the United States that were endangered in various years from 1990 to 2012. (a) Find the logistic function that models these data. Use x as the
1. Write each statement in logarithmic form.(a) 2x = y(b) 3y = 2x2 Write each statement in exponential form.(a) log7 (1/49) = - 2(b) log4 x = -1
In Problems 1-5 evaluate each logarithm without using a calculator. 1. log5 1 2. log2 8 3. log25 5 4. log3 (1 / 3) 5. log3 38
In Problems 1-4, if logax = 1.2 and logay = 3.9, find each of the following by using the properties of logarithms. 1. loga (x / y) 2. loga √x 3. loga (xy) 4. loga (y4)
In Problems 1 and 2, use the properties of logarithms to write each expression as the sum or difference of two logarithmic functions containing no exponents. 1. log ( yz) 2. ln √x + 1 / x
1. Is it true that ln x + ln y = ln (x + y) for all positive values of x and y?2. If f (x) = ln x, find f (e-2).3. If f (x) = 2x + log (7x - 4), find f(2).4. If f (x) = ex + ln (x1), find f(0).
Graph the following functions. 1. y = ex 2. y = e-x 3. y = log2 x 4. y = 2x 5. y = 1/2 (4x)
If f (x) = ln (3ex - 5), find f(ln 2).
In Problems 1 and 2, use a change-of-base formula to evaluate each logarithm. 1. log9 2158 2. log12 (0.0195)
In Problems 1 and 2, rewrite each logarithm by using a change-of-base formula, then graph the function with a graphing utility. 1. y = log √3 x 2. f (x) = log11 (2x - 5)
In Problems 1-5, solve each equation. 1. 64x = 46,656 2. 8000 = 250(1.07)x 3. 11,000 = 45,000e-0.05x 4. 312 = 300 + 300e-0.08x 5. log3(4x - 5) = 3
By using Social Security Administration data for selected years from 2012 and projected to 2050, the U.S. consumer price index (CPI) can be modeled by the functionC(t) = 92.7e0.0271twhere t is the
The purchasing power P of a $60,000 pension after t years of 3% annual inflation is modeled by P(t) = 60,000(0.97)t (a) What is the purchasing power after 20 years? (b) Graph this function for t = 0
The following table shows the U.S. average annual wage in thousands of dollars for selected years from 2012 and projected to 2050.(a) Find an exponential function that models these data. Use x equal
The average poverty threshold for 1990-2011 for a single individual can be modeled by y = - 4199.9 + 4436.3 ln x where x is the number of years past 1980 and y is the annual income in dollars (a)
The following table gives the total U.S. population, in millions, for selected years from 1990 and projected to 2050.(a) Find a logarithmic function model for the data. Use x as the number of years
The stellar magnitude M of a star is related to its brightness B as seen from earth according towhere B0 is a standard level of brightness (the brightness of the star Vega). (a) Find the magnitude of
The sales decay for a product is given by S = 50,000e-0.1x, where S is the weekly sales (in dollars) and x is the number of weeks that have passed since the end of an advertising campaign. (a) What
1. The sales decay for a product is given by S = 50,000e-0.6x, where S is the monthly sales (in dollars) and x is the number of months that have passed since the end of an advertising campaign. What
If $5000 is invested at 13.5%, compounded continuously, then the future value S at any time t (in years) is given by S = 5000e0.135t. (a) What is the amount after 9 months? (b) How long will it be
Because of a new advertising campaign, a company predicts that sales will increase and that the yearly sales will be given by the equationN = 10,000(0.3)0.5twhere t represents the number of years
The spread of a highly contagious virus in a high school can be described by the logistic functionwhere x is the number of days after the virus is identified in the school and y is the total number
Find the requested value and tell what the other numbers represent.a. Find r: 250 = 1000(r)(4)b. Find t: 1500 = 5000(0.05)(t)c. Find P: 9600 = P + P(0.05)(4)d. Find P: 11,800 = P + P(0.03)(6)
If you borrow $1600 for 2 years at 14% annual simple interest, how much must you repay at the end of the 2 years?
If you lend $3500 to a friend for 15 months at 8% annual simple interest, find the future value of the loan.

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