(a) The half-life of radium-226 is 1620 years. If the initial quantity of radium is Q0

In Exercises 14, decide whether the graph is concave up, concave down, or neither.

In Exercises 14, decide whether the graph is concave up, concave down, or neither.

In Exercises 14, decide whether the graph is concave up, concave down, or neither.

In Exercises 14, decide whether the graph is concave up, concave down, or neither.

The functions in Exercises 58 represent exponential growth or decay. What is the initial quantity? What is the growth rate? State if the growth rate is continuous.

The functions in Exercises 58 represent exponential growth or decay. What is the initial quantity? What is the growth rate? State if the growth rate is continuous.

The functions in Exercises 58 represent exponential growth or decay. What is the initial quantity? What is the growth rate? State if the growth rate is continuous.

The functions in Exercises 58 represent exponential growth or decay. What is the initial quantity? What is the growth rate? State if the growth rate is continuous.

Write the functions in Exercises 912 in the form P = P0at . Which represent exponential growth and which represent exponential decay?

Write the functions in Exercises 912 in the form P = P0at . Which represent exponential growth and which represent exponential decay?

Write the functions in Exercises 912 in the form P = P0at . Which represent exponential growth and which represent exponential decay?

Write the functions in Exercises 912 in the form P = P0at . Which represent exponential growth and which represent exponential decay?

In Exercises 1314, let f(t) = Q0at = Q0(1 + r) t . (a) Find the base, a. (b) Find the percentage growth rate, r.

In Exercises 1314, let f(t) = Q0at = Q0(1 + r) t . (a) Find the base, a. (b) Find the percentage growth rate, r.

A town has a population of 1000 people at time t = 0. In each of the following cases, write a formula for the population, P, of the town as a function of year t. (a) The population increases by 50 people a year. (b) The population increases by 5% a year.

An air-freshener starts with 30 grams and evaporates. In each of the following cases, write a formula for the quantity, Q grams, of air-freshener remaining t days after the start and sketch a graph of the function. The decrease is: (a) 2 grams a day (b) 12% a day

For which pairs of consecutive points in Figure 1.26 is the function graphed: (a) Increasing and concave up? (b) Increasing and concave down? (c) Decreasing and concave up? (d) Decreasing and concave down?

The table gives the average temperature in Wallingford, Connecticut, for the first 10 days in March. (a) Over which intervals was the average temperature increasing? Decreasing? (b) Find a pair of consecutive intervals over which the average temperature was increasing at a decreasing rate. Find another pair of consecutive intervals over which the average temperature was increasing at an increasing rate.

(a) Which (if any) of the functions in the following table could be linear? Find formulas for those functions. (b) Which (if any) of these functions could be exponential? Find formulas for those functions.

In Problems 2021, find all the tables that have the given characteristic.

In Problems 2021, find all the tables that have the given characteristic.

In 2010, the worlds population reached 6.91 billion and was increasing at a rate of 1.1% per year. Assume that this growth rate remains constant. (In fact, the growth rate has decreased since 1987.) (a) Write a formula for the world population (in billions) as a function of the number of years since 2010. (b) Estimate the population of the world in the year 2020. (c) Sketch world population as a function of years since 2010. Use the graph to estimate the doubling time of the population of the world.

(a) A population, P, grows at a continuous rate of 2% a year and starts at 1 million. Write P in the form P = P0ekt, with P0, k constants. (b) Plot the population in part (a) against time

A certain region has a population of 10,000,000 and an annual growth rate of 2%. Estimate the doubling time by guessing and checking.

A photocopy machine can reduce copies to 80% of their original size. By copying an already reduced copy, further reductions can be made. (a) If a page is reduced to 80%, what percent enlargement is needed to return it to its original size? (b) Estimate the number of times in succession that a page must be copied to make the final copy less than 15% of the size of the original.

When a new product is advertised, more and more people try it. However, the rate at which new people try it slows as time goes on. (a) Graph the total number of people who have tried such a product against time. (b) What do you know about the concavity of the graph?

Sketch reasonable graphs for the following. Pay particular attention to the concavity of the graphs. (a) The total revenue generated by a car rental business, plotted against the amount spent on advertising. (b) The temperature of a cup of hot coffee standing in a room, plotted as a function of time.

Each of the functions g, h, k in Table 1.10 is increasing, but each increases in a different way. Which of the graphs in Figure 1.27 best fits each function?

Each of the functions in Table 1.11 decreases, but each decreases in a different way. Which of the graphs in Figure 1.28 best fits each function?

One of the main contaminants of a nuclear accident, such as that at Chernobyl, is strontium-90, which decays exponentially at a continuous rate of approximately 2.47% per year. After the Chernobyl disaster, it was suggested that it would be about 100 years before the region would again be safe for human habitation. What percent of the original strontium-90 would still remain then?

Give a possible formula for the functions in Problems 3134

Give a possible formula for the functions in Problems 3135

Give a possible formula for the functions in Problems 3136

Give a possible formula for the functions in Problems 3137

Table 1.12 shows some values of a linear function f and an exponential function g. Find exact values (not decimal approximations) for each of the missing entries.

Match the functions h(s), f(s), and g(s), whose values are in Table 1.13, with the formulas y = a(1.1)s , y = b(1.05)s , y = c(1.03)s , assuming a, b, and c are constants. Note that the function values have been rounded to two decimal places.

(a) Estimate graphically the doubling time of the exponentially growing population shown in Figure 1.29. Check that the doubling time is independent of where you start on the graph. (b) Show algebraically that if P = P0at doubles between time t and time t+d, then d is the same number for any t.

A deposit of P0 into a bank account has a doubling time of 50 years. No other deposits or withdrawals are made. (a) How much money is in the bank account after 50 years? 100 years? 150 years? (Your answer will involve P0.) (b) How many times does the amount of money double in t years? Use this to write a formula for P, the amount of money in the account after t years.

A 325 mg aspirin has a half-life of H hours in a patients body. (a) How long does it take for the quantity of aspirin in the patients body to be reduced to 162.5 mg? To 81.25 mg? To 40.625 mg? (Note that 162.5 = 325/2, etc. Your answers will involve H.) (b) How many times does the quantity of aspirin, A mg, in the body halve in t hours? Use this to give a formula for A after t hours.

(a) The half-life of radium-226 is 1620 years. If the initial quantity of radium is Q0, explain why the quantity, Q, of radium left after t years, is given by Q = Q0 1 2 t/1620 . (b) What percentage of the original amount of radium is left after 500 years?

In the early 1960s, radioactive strontium-90 was released during atmospheric testing of nuclear weapons and got into the bones of people alive at the time. If the half-life of strontium-90 is 29 years, what fraction of the strontium-90 absorbed in 1960 remained in peoples bones in 2010? [Hint: Write the function in the form Q = Q0(1/2)t/29.]

Aircraft require longer takeoff distances, called takeoff rolls, at high altitude airports because of diminished air density. The table shows how the takeoff roll for a certain light airplane depends on the airport elevation. (Takeoff rolls are also strongly influenced by air temperature; the data shown assume a temperature of 0 C.) Determine a formula for this particular aircraft that gives the takeoff roll as an exponential function of airport elevation

(a) According to the US Department of Energy, the US consumed 91 million gallons of biodiesel in 2005. Approximately how much biodiesel (in millions of gallons) did the US consume in 2006? In 2007? (b) Graph the points showing the annual US consumption of biodiesel, in millions of gallons of biodiesel, for the years 2005 to 2009. Label the scales on the horizontal and vertical axes.

(a) True or false: The annual US consumption of biodiesel grew exponentially from 2003 to 2005. Justify your answer without doing any calculations. (b) According to this data, during what single year(s), if any, did the US consumption of biodiesel at least double? (c) According to this data, during what single year(s), if any, did the US consumption of biodiesel at least triple?

Hydroelectric power is electric power generated by the force of moving water. The table shows the annual percent change in hydroelectric power consumption by the US industrial sector.15 Year 2005 2006 2007 2008 2009 % growth over previous yr 1.9 10 45.4 5.1 11 (a) According to the US Department of Energy, the US industrial sector consumed about 29 trillion BTUs of hydroelectric power in 2006. Approximately how much hydroelectric power (in trillion BTUs) did the US consume in 2007? In 2005? (b) Graph the points showing the annual US consumption of hydroelectric power, in trillion BTUs, for the years 2004 to 2009. Label the scales on the horizontal and vertical axes.

(a) According to the US Department of Energy, the US consumption of wind power was 341 trillion BTUs in 2007. How much wind power did the US consume in 2006? In 2008? (b) Graph the points showing the annual US consumption of wind power, in trillion BTUs, for the years 2005 to 2009. Label the scales on the horizontal and vertical axes. (c) Based on this data, in what year did the largest yearly increase, in trillion BTUs, in the US consumption of wind power occur? What was this increase?

(a) According to Figure 1.30, during what single year(s), if any, did the US consumption of wind power energy increase by at least 40%? Decrease by at least 40%? (b) Did the US consumption of wind power energy double from 2006 to 2008?

The function y = e0.25x is decreasing and its graph is concave down.

The function y = 2x is increasing, and its graph is concave up.

A formula representing the statement q decreases at a constant percent rate, and q = 2.2 when t = 0.

A function that is increasing at a constant percent rate and that has the same vertical intercept as f(x)=0.3x + 2.

A function with a horizontal asymptote at y = 5 and range y > 5

The function y =2+3et has a y-intercept of y = 3.

The function y = 5 3e4t has a horizontal asymptote of y = 5.

If y = f(x) is an exponential function and if increasing x by 1 increases y by a factor of 5, then increasing x by 2 increases y by a factor of 10

If y = Abx and increasing x by 1 increases y by a factor of 3, then increasing x by 2 increases y by a factor of 9

An exponential function can be decreasing

If a and b are positive constants, b = 1, then y = a+abx has a horizontal asymptote.

The function y = 20/(1 + 2ekt) with k > 0, has a horizontal asymptote at y = 20.