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mathematics
calculus with applications
Calculus With Applications 10th Edition Margaret L Lial, Raymond N Greenwell, Nathan P Ritchey - Solutions
LetUse the limit rules to find each limit. lim f(x) = 9 and lim g(x) = 27. x-4 x-4
The graph shows annual sales (in thousands of dollars) of a Nintendo game at a particular store. Find the average annual rate of change in sales for the following changes in years.a. 1 to 4 b. 4 to 7 c. 7 to 12d. What do your answers for parts a–c tell you about the sales of this
The graph shows the money remaining in the Medicare Trust Fund at the end of the fiscal year.Find the approximate average rate of change in the trust fund for each time period.a. From 2000 to 2008 (the peak) b. From 2008 to 2018 Medicare Trust Fund 400 300 200 214.0- 100 Dollars (in
The graph shows the population in millions of bacteria t minutes after an antibiotic is introduced into a culture. Find and interpret the average rate of change of population with respect to time for the following time intervals.a. 1 to 2 b. 2 to 3 c. 3 to 4 d. 4 to 5e. How long
The following graph shows how immigration (in thousands) to the United States has varied over the past century.a. Find the average annual rate of change in immigration for the first half of the century (from 1905 to 1955).b. Find the average annual rate of change in immigration for the second half
In 2008, the price of gasoline in the United States inexplicably spiked and then dropped. The average monthly price (in cents) per gallon of unleaded regular gasoline for 2008 is shown in the following chart. Find the average rate of change per month in the average price per gallon for each time
To send international first class mail (large envelopes) from the United States to Australia in 2010, it cost $1.24 for the first ounce, $0.84 for each additional ounce up to a total of 8 oz, and $1.72 for each additional four ounces after that up to a total of 64 oz. Let C(x) be the cost to mail x
Epidemiologists in College Station, Texas, estimate that t days after the flu begins to spread in town, the percent of the population infected by the flu is approximated by p(t) = t2 + t for 0 ≤ t ≤ 5.a. Find the average rate of change of p with respect to t over the interval from 1 to 4
The following chart shows how the percentage of eighth graders, tenth graders, and twelfth graders who have used marijuana in their lifetime has varied in recent years.a. Find the average annual rate of change in the percent of eighth graders who have used marijuana in their lifetime over the
Explain whyshould give a reasonable approximation of f'(x) when f'(x) exists and h is small. f(x +h)-f(x - h) 2h
Leta. Findb. Find the vertical asymptote of the graph of F(x)c. Compare your answers for parts a and b. What can you conclude? F(x) = 3x (x + 2)3
Use the definition of the derivative to find the derivative of the following.y = 4x2 + 3x – 2
Leta. Findb. Find the vertical asymptote of the graph of G(x)c. Compare your answers for parts a and b. Are they related? How? G(x): = -6 (x-4)²*
Use the definition of the derivative to find the derivative of the following.y = 5x2 – 6x + 7
How can you tell that the graph in Figure 10 is more representative of the function f(x) = 1/(x – 1) than the graph in Figure 11?Figure 10.Figure 11. 0 10 -10 FIGURE 10 1 2
Use a graph of f(x) = ex to answer the following questions.a. Findb. Where does the function ex have a horizontal asymptote? lim e*. 00
The annual unemployment rates of the U.S. civilian non-institutional population for 1990–2009 are shown in the graph. Sketch a graph showing the rate of change of the annual unemployment rates for this period. Use the given graph and your sketch to estimate the annual unemployment rate and rate
The graph below shows how the postage required to mail a letter in the United States has changed in recent years. Let C(t) be the cost to mail a letter in the year t. Find the following. a. lim C(t) 1-2009- b. lim C(t) 1-2009 c. lim C(t) t→2009 d. C(2009) Cost (in
Explain in your own words what Rule 4 for limits means.
Solve each equation.2|x| = 8
Evaluate each logarithm without using a calculator.ln e3
Solve each equation.5x2+x = 1
Consider the function defined bya. Graph the function on [−3.4, 3.4] by [−3, 3] From your graph, estimate how many vertical asymptotes the function has and where they are located.b. Now graph the function on [−1.5, −1.4] by [−10, 10] and on [1.4, 1.5] by [−1000, 1000] From
Solve each equation.8x2 = 2x+4
Solve each equation in Exercises 41–64. Round decimal answers to four decimal places.log9 27 = m
Since 1960, the growth in world population (in millions) closely fits the exponential function defined bywhere t is the number of years since 1960. a. World population was about 3686 million in 1970. How closely does the function approximate this value?b. Use the function to approximate world
On January 1, 2000, Jack deposited $1000 into Bank X to earn interest at the rate of j per annum compounded semiannually. On January 1, 2005, he transferred his account to Bank Y to earn interest at the rate of k per annum compounded quarterly. On January 1, 2008, the balance at Bank Y was
Salmonella bacteria, found on almost all chicken and eggs, grow rapidly in a nice warm place. If just a few hundred bacteria are left on the cutting board when a chicken is cut up, and they get into the potato salad, the population begins compounding. Suppose the number present in the potato salad
Solve each equation in Exercises 41–64. Round decimal answers to four decimal places.log5 (9x – 4) = 1
The demand for physicians is expected to increase in the future, as shown in the table on the following page.a. Plot the data, letting t = 0 correspond to 2000. Does fitting an exponential curve to the data seem reasonable?b. Use the data for 2006 and 2015 to find a function of the form f(x) =
Solve each equation in Exercises 41–64. Round decimal answers to four decimal places.log4 x – log4(x + 3) = –1
According to the U.S. Census Bureau, the United States is becoming more diverse. Based on U.S. Census population projections for 2000 to 2050, the projected Hispanic population (in millions) can be modeled by the exponential function h(t) = 37.79(1.021)t, where t = 0 corresponds to 2000 and 0 ≤ t
Solve each equation in Exercises 41–64. Round decimal answers to four decimal places.log (x + 5) + log(x + 2) = 1
Solve each equation in Exercises 41–64. Round decimal answers to four decimal places.log3 (x – 2) + log3 (x + 6) = 2
The table gives the estimated global carbon dioxide (CO2) emissions from fossil-fuel burning, cement production, and gas flaring over the last century. The CO2 estimates are expressed in millions of metric tons.a. Plot the data, letting x = 0 correspond to 1900. Do the emissions appear to grow
Suppose the quantity (in grams) of a radioactive substance present at time t iswhere t is measured in months.a. How much will be present in 6 months?b. How long will it take to reduce the substance to 8 g? Q(t) = 1000(5-0.31),
The manager of an 80-unit apartment complex is trying to decide what rent to charge. Experience has shown that at a rent of $800, all the units will be full. On the average, one additional unit will remain vacant for each $25 increase in rent.a. Let x represent the number of $25 increases. Find an
For the years 1998 to 2009, the number of applicants to U.S. medical schools can be closely approximated bywhere t is the number of years since 1998.a. Graph the number of applicants onb. Based on the graph in part a, during what years did the number of medical school applicants increase? A(t) =
The following table gives the total world wind energy capacity (in megawatts) in recent years.a. Let t be the number of years since 2000, and C the capacity (in MW). Find functions of the form C = mt + b, C = at2 + b, and C= abt that fit the data at 2001 and 2009.b. Use a graphing calculator to
The power of personal computers has increased dramatically as a result of the ability to place an increasing number of transistors on a single processor chip. The following table lists the number of transistors on some popular computer chips made by Intel.a. Let t be the year, where t = 0
Solve each equation.ex2+5x+6 = 1
Graph each of the following.y = 5ex + 2
Graph each of the following.y = –2ex – 3
Graph each of the following.y = 4e–x/2 – 1
Find the interest earned on $10,000 invested for 5 years at 4% interest compounded as follows.a. Annually b. Semiannually (twice a year)c. Quarterly d. Monthly e. Continuously
Ron Hampton needs to choose between two investments: One pays 6% compounded annually, and the other pays 5.9% compounded monthly. If he plans to invest $18,000 for 2 years, which investment should he choose? How much extra interest will he earn by making the better choice?
Find the interest rate required for an investment of $5000 to grow to $7500 in 5 years if interest is compounded as follows.a. Annually b. Quarterly
Ali Williams invests a $25,000 inheritance in a fund paying 5.5% per year compounded continuously. What will be the amount on deposit after each time period?a. 1 year b. 5 years c. 10 years
Leigh Jacks plans to invest $500 into a money market account. Find the interest rate that is needed for the money to grow to $1200 in 14 years if the interest is compounded quarterly.
Solve each equation in Exercises 41–64. Round decimal answers to four decimal places.log8 16 = z
Consider the function defined bya. Graph the function on [−6, 6] by [−6, 6] From your graph, estimate how many x-intercepts the function has and what their values are.b. Now graph the function on [−1.5, −1.4] by [−10−4, 10−4] and on [1.4, 1.5] by [−10−5. 10−5] From your graphs,
Kristi Perez puts $10,500 into an account to save money to buy a car in 12 years. She expects the car of her dreams to cost $30,000 by then. Find the interest rate that is necessary if the interest is computed using the following methods.a. Compounded quarterly b. Compounded continuously
Solve each equation.27x = 9x2+x
Solve each equation.5–|x| = 1/25
Evaluate each logarithm without using a calculator.ln e5/3
Solve each equation.2x2 – 4x = (1/16)x–4
Solve each equation.e–x = (e4)x+3
Solve each equation.4x = 8x+1
Evaluate each logarithm without using a calculator.log2 1/16
Solve each equation.25x = 125x+2
Evaluate each logarithm without using a calculator.log3 1/81
Evaluate each logarithm without using a calculator. log₂ 4
Solve each equation.16x+3 = 642x–5
Evaluate each logarithm without using a calculator. logs 8 4 2
Solve each equation.(e3)–2x = e–x+5
Solve each equation.3x = 1/81
Evaluate each logarithm without using a calculator.log4 64
Solve each equation.ex = 1/e5
Evaluate each logarithm without using a calculator.log3 27
Write each logarithmic equation in exponential form.log 100,000 = 5
Write each logarithmic equation in exponential form.log 0.001 = –3
In Exercises 13–24, graph each parabola and give its vertex, axis, x-intercepts, and y-intercept.y = x2 + 5x + 6
Solve each equation.2x = 32
Evaluate each logarithm without using a calculator.log8 64
Solve each equation.4x = 64
Evaluate each logarithm without using a calculator.log9 81
Write each exponential equation in logarithmic form.27 = 128
Write each exponential equation in logarithmic form.3–2 = 1/9
Write each exponential equation in logarithmic form.(5/4)–2 = 16/25
Write each logarithmic equation in exponential form.log2 32 = 5
Write each logarithmic equation in exponential form.ln 1/e = – 1
Write each logarithmic equation in exponential form.log2 1/8 = –3
Write each exponential equation in logarithmic form.53 = 125
Write each exponential equation in logarithmic form.34 = 81
Write each exponential equation in logarithmic form.72 = 49
The enrollment in Head Start for some recent years is included in the table.a. Plot the points from the table using 0 for 1960, and so on.b. Use the quadratic regression feature of a graphing calculator to get a quadratic function that approximates the data. Graph the function on the same window as
Solve each equation in Exercises 41–64. Round decimal answers to four decimal places.3x+1 = 5x
From 1975 to 2007, the age-adjusted incidence rate of invasive lung and bronchial cancer among women can be closely approximated bywhere t is the number of years since 1975. Based on this model, in what year did the incidence rate reach a maximum? On what years was the rate increasing? Decreasing?
Solve each equation in Exercises 41–64. Round decimal answers to four decimal places.2x+1 = 6x–1
The following table gives the median age at their first marriage of women in the United States for some selected years.a. Plot the data using x = 40 for 1940, and so on.b.Would a linear or quadratic function best model this data? Explain.c. If your graphing calculator has a regression feature, find
Solve each equation in Exercises 41–64. Round decimal answers to four decimal places.5(0.10)x = 4(0.12)x
Classify each of the functions in Exercises 63–70 as even, odd, or neither.f(x) = 1/(x2 +4)
Classify each of the functions in Exercises 63–70 as even, odd, or neither.f(x) = x/(x2 - 9)
Prove: loga xr = r loga x.
In New York state in 2010, the income tax rates for a single person were as follows:4% of the first $8000 earned,4.5% of the next $3000 earned,5.25% of the next $2000 earned,5.9% of the next $7000 earned,6.85% of the next $180,000 earned,7.85% of the next $300,000 earned, and8.97% of any amount
Leigh Jacks plans to invest $500 into a money market account. Find the interest rate that is needed for the money to grow to $1200 in 14 years if the interest is compounded continuously. (Compare with Exercise 43 in the previous section.)Data from exercises 43 section 2.4Leigh Jacks plans to invest
You are offered two jobs starting July 1, 2013. Humongous Enterprises offers you $45,000 a year to start, with a raise of 4% every July 1. At Crabapple Inc. you start at $30,000, with an annual increase of 6% every July 1. On July 1 of what year would the job at Crabapple Inc. pay more than the job
Over the last century, the world has shifted from using high-carbon sources of energy such as wood to lower carbon fuels such as oil and natural gas, as shown in the figure. The rise in carbon emissions during this time has caused concern because of its suspected contribution to global warming.a.
Mary Klingman invests $15,000 in an account paying 7% per year compounded annually.a. How many years are required for the compound amount to at least double?b. In how many years will the amount at least triple?c. Check your answer to part a using either the rule of 70 or the rule of 72, whichever
In Exercises 15–24, find an equation in slope-intercept form for each line.Through (–8, 1) with undefined slope
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