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mathematics
calculus
Questions and Answers of
Calculus
Write the logarithmic equation in exponential form. 1. In ½ = -0.693 . . . 2. In 7 = 1.945 . . . 3. In 250 = 5.521 . . . 4. In 1 = 0
Write the exponential equation in logarithmic form. 1. e2 = 7.3890 . . . 2. e−3/4 = 0.4723 . . . 3. e−4x = ½ 4. e2x = 3
Use a calculator to evaluate the function at the given value of x. Round your result to three decimal places. Function Value 1. f (x) = ln x x = 18.42 2. f (x) = 3 ln x x = 0.74 3. g(x) = 8
Use the properties of natural logarithms to simplify the expression. 1. eln 4 2. In 1/e2 3. 2.5 ln 1 4. In e/π 5. ln eln e 6. eln(1/e)
Find the domain, x-intercept, and vertical asymptote of the logarithmic function and sketch its graph. 1. f (x) = ln(x − 4) 2. h(x) = ln(x + 5)
Write the logarithmic equation in exponential form. For example, the exponential form of log5 25 = 2 is 52 = 25. 1. log4 16 = 2 2. log9 1/81 = −2 3. log12 12 = 1 4. log32 4 = 2/5
Use a graphing utility to graph the function. Be sure to use an appropriate viewing window. 1. f (x) = ln(x − 1) 2. f (x) = ln(x + 2) 3. f (x) = − ln x + 8 4. f (x) = 3 ln x − 1
Use the One-to-One Property to solve the equation for x. 1. ln(x + 4) = ln 12 2. ln(x − 7) = ln 7 3. ln(x2 − x) = ln 6 4. ln(x2 − 2) = ln 23
The model t = 16.625 ln x/x - 750, x > 750 approximates the length of a home mortgage of $150,000 at 6% in terms of the monthly payment. In the model, t is the length of the mortgage in years and x
The percent P of households in the United States with wireless-only telephone service from 2005 through 2014 can be approximated by the model P = −3.42 + 1.297t ln t, 5 ≤ t ≤ 14 where t
1. The time t (in years) for the world population to double when it is increasing at a continuous rate r (in decimal form) is given by t = (ln 2)/r.(a) Complete the table and interpret your
Students in a mathematics class took an exam and then took a retest monthly with an equivalent exam. The average scores for the class are given by the human memory modelf (t) = 80 17
The relationship between the number of decibels and the intensity of a sound I (in watts per square meter) is β = 10 log I/10−12. (a) Determine the number of decibels of a sound with an
Determine whether the statement is true or false. Justify your answer. 1. The graph of f (x) = log6 x is a reflection of the graph of g(x) = 6x in the x-axis. 2. The graph of f (x) = ln(− x) is a
Use a graphing utility to graph f and g in the same viewing window and determine which is increasing at the greater rate as x approaches + ∞. What can you conclude about the rate of growth of the
1. (a) Complete the table for the function f (x) = (In x)/x.(b) Use the table in part (a) to determine what value f (x) approaches as x increases without bound. (c) Use a graphing utility to confirm
1. To evaluate a logarithm to any base, use the ________ formula. 2. The change-of-base formula for base e is loga x = ________. 3. When you consider loga x to be a constant multiple of logb x, the
In Exercises 1 and 2, describe the error. 1. (In e)2 = 2(In e) = 2(1) = 2 2. log2 8 = log2 (4 + 4) = log2 4 + log2 4 = log2 22 + log2 22 =2 + 2 = 4
Use a graphing utility to graph the functions y1 = ln x − ln(x − 3) and y2 = In x / x - 3 in the same viewing window. Does the graphing utility show the functions with the same domain? If not,
It For which integers between 1 and 20 can you approximate natural logarithms, given the values ln 2 ≈ 0.6931, ln 3 ≈ 1.0986, and ln 5 ≈ 1.6094? Approximate these logarithms. (Do not use a
In Exercises 1-4, use the properties of logarithms to write the logarithm in terms of log3 5 and log3 7. 1. log3 35 2. log3 5 / 7 3. log3 7 / 25 4. log3 175
In Exercises 1-4, find the exact value of the logarithmic expression without using a calculator. (If this is not possible, state the reason.) 1. log3 9 2. log5 1 / 125 3. log6 3 √1 / 6 4. log2 4
In Exercises 1-4, approximate the logarithm using the properties of logarithms, given logb 2 ≈ 0.3562, logb 3 ≈ 0.5646, and logb 5 ≈ 0.8271. 1. logb 10 2. logb 2/3 3. logb 0.04 4. logb
Name the property of logarithms illustrated by each statement. a. In(un) = In u + In v b. Loga un = n loga u c. In u/v = In u - In v
In Exercises 1-4, use the properties of logarithms to expand the expression as a sum, difference, andor constant multiple of logarithms. (Assume all variables are positive.) 1. In 7x 2. log3
In Exercises 1-4, rewrite the logarithm as a ratio of (a) Common logarithms and (b) Natural logarithms. 1. log5 16 2. log1/5 4 3. logx 3/10 4. log2.6x
In Exercises 1-4, condense the expression to the logarithm of a single quantity. 1. In 3 + In x 2. log5 8 - log5 t 3. 2 / 3 log7 (ɀ - 2) 4. - 4 In 3x
In Exercises 1 and 2, determine which (if any) of the logarithmic expressions are equal. Justify your answer. 1. log2 32 / log2 4, log2 32 / 4, log2 32 - log2 4 2. log7 √70, log7 35, 1 / 2 + log7
In Exercises 79-82, use the following information. The relationship between the number of decibels and the intensity of a sound I (in watts per square meter) isβ = 10 log 1 / 10-121. Use
In Exercises 1-4, find a logarithmic equation that relates y and x.1.2. 3. 4.
Four-legged animals run with two different types of motion: trotting and galloping. An animal that is trotting has at least one foot on the ground at all times, whereas an animal that is galloping
The approximate lengths and diameters (in inches) of bright common wire nails are shown in the table. Find a logarithmic equation that relates the diameter y of a bright common wire nail to its
A cup of water at an initial temperature of 78°C is placed in a room at a constant temperature of 21°C. The temperature of the water is measured every 5 minutes during a half-hour period. The
In Exercises 1-4, evaluate the logarithm using the change-of-base formula. Round your result to three decimal places. 1. log3 17 2. log0.4 12 3. logn 0.5 4. log2/3 0.125
Write a short paragraph explaining why the transformations of the data in following Exercise were necessary to obtain the models. Why did taking the logarithms of the temperatures lead to a linear
In Exercises 1-4, determine whether the statement is true or false given that f (x) ln x. justify your answer. 1. f (0) = 0 2. f (ax) = f (a) + f (x), a > 0, x > 0 3. f (x - 2) = f (x) - f (2), x >
In Exercises 1-4, use the change-of-base formula to rewrite the logarithm as a ratio of logarithms. Then use a graphing utility to graph the ratio. 1. f (x) = log2 x 2. f (x) = log1/2 x 3. f (x) =
1. To solve exponential and logarithmic equations, you can use the One-to-One and Inverse Properties below. (a) ax = ay if and only if ________. (b) loga x = loga y if and only if ________. (c)
In Exercises 1 and 2, approximate the point of intersection of the graphs of f and g. Then solve the equation f (x) = g (x) algebraically to verify your approximation.1. f (x) = 2x, g (x) = 82. f (x)
In Exercises 1-4, solve the exponential equation algebraically. Approximate the result to three decimal places, if necessary. 1. ex = ex2-2 2. ex2-3 = ex-2 3. 4(3x) = 20 4. 4ex = 91
In Exercises 1-2, determine whether each x-value is a solution (or an approximate solution) of the equation. 1. 42x - 7 = 64 a. x = 5 b. x = 2 c. x = 1 / 2 (log4 64 + 7) 2. 4ex-1 = 60 a. x = 1 + In
In Exercises 1-4, solve the logarithmic equation algebraically. Approximate the result to three decimal places, if necessary. 1. ln x = −3 2. ln x − 7 = 0 3. 2.1 = ln 6x 4. log 3ɀ = 2
In Exercises 1-4, use a graphing utility to graphically solve the equation. Approximate the result to three decimal places. Verify your result algebraically. 1. 5x = 212 2. 6e1-x = 25 3. 8e-2x/3 =
In Exercises 1-4, solve for x. 1. 4x = 16 2. (1/2)x = 32 3. In x - In 2 = 0 4. log x - log 10 = 0
In Exercises 71 and 72, you invest $2500 in an account at interest rate r, compounded continuously. Find the time required for the amount to (a) Double and (b) triple. 1. r = 0.025 2. r = 0.0375
In Exercises 1-4, solve the equation algebraically. Round your result to three decimal places, if necessary. Verify your answer using a graphing utility. 1. 2x2e2x + 2xe2x = 0 2. −x2e−x + 2xe−x
The percent m of American males between the ages of 20 and 29 who are under x inches tall is modeled by m (x) = 100 / 1 + e -0.5536(x - 69.51)' 64 ¤ x ¤ 78 and the percent f
The demand equation for a smart phone is p = 5000(1 - 4 / 4 + e 0.002x). Find the demand x for each price. a. p = $169 b. p = $229
The number N of beavers in a given area after x years can be approximated by N = 5.5 · 100.23x, 0 ¤ x ¤ 10. Use the model to approximate how many years it will take
The number N of trees of a given species per acre is approximated by the model N = 3500(10−0.12x), 3 ≤ x ≤ 30 where x is the average diameter of the trees (in inches) 4.5 feet above the ground.
The population P (in thousands) of Alaska in the years 2005 through 2015 can be modeled by P = 75 ln t + 540, 5 ≤ t ≤ 15 where t represents the year, with t = 5 corresponding to 2005. During
The population P (in thousands) of Montana in the years 2005 through 2015 can be modeled by P = 81 ln t + 807, 5 ≤ t ≤ 15 where t represents the year, with t = 5 corresponding to 2005. During
An object at a temperature of 80°C is placed in a room at 20°C. The temperature of the object is given by T = 20 + 60e−0.06m where m represents the number of minutes after the object is placed in
An object at a temperature of 160°C was removed from a furnace and placed in a room at 20°C. The temperature T of the object was measured each hour h and recorded in the table. A model for
In Exercises 1-2, rewrite each verbal statement as an equation. Then decide whether the statement is true or false. Justify your answer. 1. The logarithm of the product of two numbers is equal to the
In Exercises 1-2, rewrite each verbal statement as an equation. Then decide whether the statement is true or false. Justify your answer. 1. The logarithm of the difference of two numbers is equal to
You are investing P dollars at an annual interest rate of r, compounded continuously, for t years. Which change below results in the highest value of the investment? Explain. (a) Double the amount
Are the times required for the investments in following Exercises to quadruple twice as long as the times for them to double? Give a reason for your answer and verify your answer algebraically.
The effective yield of an investment plan is the percent increase in the balance after 1 year. Find the effective yield for each investment plan. Which investment plan has the greatest effective
Let f (x) = loga x and g(x) = ax, where a > 1. (a) Let a = 1.2 and use a graphing utility to graph the two functions in the same viewing window. What do you observe? Approximate any points of
1. An exponential growth model has the form ________, and an exponential decay model has the form ________. 2. A logarithmic model has the form ________ or ________. 3. In probability and statistics,
Determine the principal P that must be invested at rate r, compounded monthly, so that $500,000 will be available for retirement in t years. 1. r = 5%, t = 10 2. r = 3 1/2%, t = 15
Determine the time necessary for P dollars to double when it is invested at interest rate r compounded (a) Annually, (b) Monthly, (c) Daily, (d) Continuously. r = 10%
Complete the table for the time t (in years) necessary for P dollars to triple when it is invested at an interest rate r compounded(a) Continuously(b) Annually.
Draw scatter plots of the data in Exercise 17. Use the regression feature of a graphing utility to find models for the data.
If $1 is invested over a 10-year period, then the balance A after t years is given by either A = 1 + 0.075 [t] or A = e0.07t depending on whether the interest is simple interest at 71 2% or
If $1 is invested over a 10-year period, then the balance A after t years is given by either A = 1 + 0.06 [t] or A = [1 + (0.055/365)][365t] depending on whether the interest is simple interest at 6%
Find the missing value for the radioactive isotope.
Find the exponential model that fits the points shown in the graph or table.1.2.
The populations P (in thousands) of Horry County, South Carolina, from 1971 through 2014 can be modeled by P = 76.6e0.0313t where t represents the year, with t = 1 corresponding to 1971. (a) Use the
The table shows the mid-year populations (in millions) of five countries in 2015 and the projected populations (in millions) for the year 2025.(a) Find the exponential growth or decay model y = aebt
The number y of hits a new website receives each month can be modeled by y = 4080ekt, where t represents the number of months the website has been operating. In the website's third month, there were
The population P (in thousands) of Tallahassee, Florida, from 2000 through 2014 can be modeled by P = 150.9ekt, where t represents the year, with t = 0 corresponding to 2000. In 2005, the population
The number of bacteria in a culture is increasing according to the law of exponential growth. After 3 hours there are 100 bacteria, and after 5hours there are 400 bacteria. How many bacteria will
The number of bacteria in a culture is increasing according to the law of exponential growth. The initial population is 250 bacteria, and the population after 10 hours is double the population after
A laptop computer that costs $575 new has a book value of $275 after 2 years. (a) Find the linear model V = mt + b. (b) Find the exponential model V = aekt. (c) Use a graphing utility to graph the
The management at a plastics factory has found that the maximum number of units a worker can produce in a day is 30. The learning curve for the number N of units produced per day after a new employee
The ratio of carbon-14 to carbon-12 in a piece of wood discovered in a cave is R = 1/814. Estimate the age of the piece of wood.
The ratio of carbon-14 to carbon-12 in a piece of paper buried in a tomb is R = 1/1311. Estimate the age of the piece of paper.
The IQ scores for a sample of students at a small college roughly follow the normal distribution y = 0.0266e−(x−100)2/450, 70 ≤ x ≤ 115 where x is the IQ score. (a) Use a graphing utility to
The amount of time (in hours per week) a student utilizes a math-tutoring center roughly follows the normal distribution y = 0.7979e−(x−5.4)2/0.5, 4 ≤ x ≤ 7 where x is the number of
A cell site is a site where electronic communications equipment is placed in a cellular network for the use of mobile phones. The numbers y of cell sites from 1985 through 2014 can be modeled by y =
The population P (in thousands) of a city from 2000 through 2016 can be modeled by P = 2632/1 + 0.083e0.050t where t represents the year, with t = 0 corresponding to 2000. (a) Use the model to find
A conservation organization released 100 animals of an endangered species into a game preserve. The preserve has a carrying capacity of 1000 animals. The growth of the pack is modeled by the logistic
After discontinuing all advertising for a tool kit in 2010, the manufacturer noted that sales began to drop according to the modelS = 500,000 / 1 + 0.1ektwhere S represents the number of units sold
Find the intensity I of an earthquake measuring R on the Richter scale (let I0 = 1). (a) Peru in 2015: R = 7.6 (b) Pakistan in 2015: R = 5.6 (c) Indonesia in 2015: R = 6.6 Use the Richter scale R =
Find the magnitude R of each earthquake of intensity I (let I0 = 1). (a) I = 199,500,000 (b) I = 48,275,000 (c) I = 17,000 Use the Richter scale R = log I/I0 for measuring the magnitude R of an
(a) I = 10−10 watt per m2 (quiet room) (b) I = 10−5 watt per m2 (busy street corner) (c) I = 10−8 watt per m2 (quiet radio) (d) I = 10−3 watt per m2 (loud car horn) Use the following
(a) I = 10−11 watt per m2 (rustle of leaves) (b) I = 102 watt per m2 (jet at 30 meters) (c) I = 10−4 watt per m2 (door slamming) (d) I = 10−6 watt per m2 (normal conversation) Use the following
Due to the installation of noise suppression materials, the noise level in an auditorium decreased from 93 to 80 decibels. Find the percent decrease in the intensity of the noise as a result of the
(a) solve for P and (b) solve for t. A = Pert
Due to the installation of a muffler, the noise level of an engine decreased from 88 to 72 decibels. Find the percent decrease in the intensity of the noise as a result of the installation of the
1. Find the pH when [H+] = 2.3 × 10−5. 2. Find the pH when [H+] = 1.13 × 10−5. Use the acidity model pH log[H+], where acidity (pH) is a measure of the hydrogen ion concentration [H+] (measured
1. Compute [H+] for a solution in which pH = 5.8. 2. Compute [H+] for a solution in which pH = 3.2. Use the acidity model pH log[H+], where acidity (pH) is a measure of the hydrogen ion concentration
Apple juice has a pH of 2.9 and drinking water has a pH of 8.0. The hydrogen ion concentration of the apple juice is how many times the concentration of drinking water? Use the acidity model pH
The pH of a solution decreases by one unit. By what factor does the hydrogen ion concentration increase? Use the acidity model pH log[H+], where acidity (pH) is a measure of the hydrogen ion
At 8:30A.M, a coroner went to the home of a person who had died during the night. In order to estimate the time of death, the coroner took the person's temperature twice. At 9:00A.M. the temperature
A $120,000 home mortgage for 30 years at 7 1/2% has a monthly payment of $839.06. Part of the monthly payment covers the interest charge on the unpaid balance, and the remainder of the payment
The total interest u paid on a home mortgage of P dollars at interest rate r (in decimal form) for t years isConsider a $120,000 home mortgage at 7 1/2%. (a) Use a graphing utility to graph the total
The table shows the time t (in seconds) required for a car to attain a speed of s miles per hour from a standing start. Speed, s Time,
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