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mathematics
applied calculus
Applied Calculus 6th Edition Deborah Hughes Hallett, Patti Frazer Lock, Andrew M. Gleason, Daniel E. Flath, Sheldon P. Gordon, David O. Lomen, David Lovelock, William G. McCallum, Brad G. Osgood, Andrew Pasquale - Solutions
When a deposit of $1000 is made into an account paying 2% interest, compounded annually, the balance, $B, in the account after t years is given by B = 1000(1.02)t. Find the average rate of change in the balance over the interval t = 0 to t = 5. Give units and interpret your answer in terms of the
(a) The graph of r = f(p) is in Figure 1.6. What is the value of r when p is 0? When p is 3?(b) What is f(2)? 10 8 6 4 2 r 1 2 3 4 5 6 Р
determine whether or not the function is a power function. If it is a power function, write it in the form y = kxp and give the values of k and p. y = 2x² 10
Find the doubling time of a quantity that is increasing by 7% per year.
Coober Pedy is a town in Australia that mines most of the world’s precious opals. Because of the scorching heat, much of the population lives in underground “dugouts.” The Coober Pedy monthly high temperatures are shown in Figure 1.116, with H(n) the temperature in month n, where n = 1 is
solve for t using natural logarithms.10 = 6e0.5t
A city’s population was 28,600 in the year 2015 and is growing by 750 people a year. (a) Give a formula for the city’s population, P, as a function of the number of years, t, since 2015. (b) What is the population predicted to be in 2020? (c) When is the population expected to
Give the cost, revenue, and profit functions.An online seller of T-shirts pays $500 to start up the website and $6 per T-shirt, then sells the T-shirts for $12 each.
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.
Interpret the expression in terms of Arctic Sea ice extent, the area of sea covered by ice. Let E(x) and F(t) be the Arctic Sea ice extent, both in millions of square kilometers, as a function of time, x, in years, since February 1979, and time, t, in days, since December 31, 2014.E(29) = 15
In Problem determine whether or not the function is a power function. If it is a power function, write it in the form y = kxp and give the values of k and p.y = (5x)3
Find f(5). f(x) = 2x + 3
Persistent organic pollutants (POPS) are a serious environmental hazard. Figure 1.80 shows their natural decay over time in human fat.(a) How long does it take for the concentration to decrease from 100 units to 50 units? (b) How long does it take for the concentration to decrease from 50
solve for t using natural logarithms.40 = 100e−0.03t
A person breathes in and out every three seconds. The volume of air in the person’s lungs varies between a minimum of 2 liters and a maximum of 4 liters. Which of the following is the best formula for the volume of air in the person’s lungs as a function of time?(a)(b)(c)(d) y = 2 + 2 sin (¹)
A car rental company charges a daily fee of $35 plus $0.20 per mile driven. Find a formula for the daily charge, C, in dollars, as a function of the number of miles, m, driven that day.
Give the cost, revenue, and profit functions.A car wash operator pays $35,000 for a franchise, then spends $10 per car wash, which costs the consumer $15.
A product costs $80 today. How much will the product cost in t days if the price is reduced by(a) $4 a day(b) 5% a day
Interpret the expression in terms of Arctic Sea ice extent, the area of sea covered by ice. Let E(x) and F(t) be the Arctic Sea ice extent, both in millions of square kilometers, as a function of time, x, in years, since February 1979, and time, t, in days, since December 31, 2014.E(4) = 16
determine whether or not the function is a power function. If it is a power function, write it in the form y = kxp and give the values of k and p.y = 3x2 + 4
solve for t using natural logarithms.a = bt
Find f(5).f(x) = 10x − x2
The half-life of nicotine in the blood is 2 hours. A person absorbs 0.4 mg of nicotine by smoking a cigarette. Fill in the following table with the amount of nicotine remaining in the blood after t hours. Estimate the length of time until the amount of nicotine is reduced to 0.04 mg. t
Values of a function are given in the following table. Explain why this function appears to be periodic. Approximately what are the period and amplitude of the function? Assuming that the function is periodic, estimate its value at t = 15, at t = 75, and at t = 135. 20 25 30 35 40 45 50 55 60 1.4
A company rents cars at $40 a day and 15 cents a mile. Its competitor’s cars are $50 a day and 10 cents a mile. (a) For each company, give a formula for the cost of renting a car for a day as a function of the distance traveled. (b) On the same axes, graph both functions. (c) How
Interpret the expression in terms of Arctic Sea ice extent, the area of sea covered by ice. Let E(x) and F(t) be the Arctic Sea ice extent, both in millions of square kilometers, as a function of time, x, in years, since February 1979, and time, t, in days, since December 31, 2014. E(29) - E(4) 29
Give the cost, revenue, and profit functions.A couple running a house-cleaning business invests $5000 in equipment, and they spend $15 in supplies to clean a house, for which they charge $60.
write a formula representing the function.The strength, S, of a beam is proportional to the square of its thickness, ℎ.
solve for t using natural logarithms.B = Pert
World wind energy generating 95 capacity, W , was 371 gigawatts by the end of 2014 and has been increasing at a continuous rate of approximately 16.8% per year. Assume this rate continues. (Generating capacity is the maximum amount of power generated per unit time.) (a) Give a formula for w,
Make a table of values for each of the following functions using Table 1.37:(a) f(x) +3 (b) f(x-2)(c) 5g(x)(d) -f(x) + 2(e) g(x -3)(f) f(x) + g(x) Table 1.37 x 0 1 f(x) 10 6 3 4 g(x) 2 3 4 5 7 11 15 2 3 5 8 12
Average daily high temperatures in Ottawa, the capital of Canada, range from a low of −6◦ Celsius on January 1 to a high of 26◦ Celsius on July 1 six months later. See Figure 1.117. Find a formula for H, the average daily high temperature in Ottawa in, ◦C, as a function of t, the number of
An air-freshener starts with 30 grams and evaporates over time. 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
A phone company charges $14.99 a month and 10 cents for every minute above 120 minutes. Write an expression for the monthly phone charge, P, in dollars, as a function of the number of minutes, m, used that month.
Give the cost, revenue, and profit functions.A lemonade stand operator sets up the stand for free in front of the neighbor’s house, makes 5 quarts of lemonade for $4, then sells each 8-oz cup for 25 cents.
The population of Washington DC grew from 1900 to 1950, stayed approximately constant during the 1950s, and decreased from about 1960 to 2005. Graph the population as a function of years since 1900.
Use the description of the function to sketch a possible graph. Put a label on each axis and state whether the function is increasing or decreasing.The amount of carbon dioxide in the atmosphere is a function of time, and is going up over time.
Use the description of the function to sketch a possible graph. Put a label on each axis and state whether the function is increasing or decreasing.The number of air conditioning units sold is a function of temperature, and goes up as the temperature goes up.
(a) The exponential functions in Figure 1.73 have b, d, q positive. Which of the constants a, c, and p must be positive? (b) Which of the constants a, b, c, d, p, and q must be between 0 and 1? (c) Which two of the constants a, b, c, d, p, and q must be equal? (d) What information
Table 1.12 gives values of a function ω = f(t). Does this function appear to be increasing or decreasing? Does its graph appear to be concave up or concave down? 4 8 0 100 58 16 20 20 18 12 16 32 24 20 24 17
Use the description of the function to sketch a possible graph. Put a label on each axis and state whether the function is increasing or decreasing.The noise level, in decibels, is a function of distance from the source of the noise, and the noise level goes down as the distance increases.
Determine whether or not the function is a power function. If it is a power function, write it in the form y = kxp and give the values of k and p.y = 2x
Suppose $1000 is invested in an account paying interest at a rate of 1.5% per year. How much is in the account after 8 years if the interest is compounded (a) Annually?(b) Continuously?
Solve for t using natural logarithms.10 = et
Graph the function. What is the amplitude and period?y = 5 − sin 2t
For g(x) = x2 + 2x + 3, find and simplify: (a) g(2 + ℎ) (b) g(2) (c) g(2 + ℎ) − g(2)
The cost C, in millions of dollars, of producing q items is given by C = 5.7 + 0.002q. Interpret the 5.7 and the 0.002 in terms of production. Give units.
Determine the slope and the y-intercept of the line whose equation is given.7y + 12x − 2 = 0
Give a possible formula for the functions in Problems. 500 y 12 (3, 2000) 3 4 t
Graph a function f(x) which is increasing everywhere and concave up for negative x and concave down for positive x.
Solve for t using natural logarithms.100 = 25(1.5)t
Graph the function. What is the amplitude and period? y = 4 cos $ (1)
(a) Give an example of a possible company where the fixed costs are zero (or very small). (b) Give an example of a possible company where the marginal cost is zero (or very small).
Determine the slope and the y-intercept of the line whose equation is given.3x + 2y = 8
Give a possible formula for the functions in Problems. 30 У 10 20 (25,6) L 30
For which pairs of consecutive points in Figure 1.38 is the function graphed: (a) Increasing and concave up?(b) Increasing and concave down? (c) Decreasing and concave up? (d) Decreasing and concave down? A B C D E F G H X
Let W = f(t) represent wheat production in Argentina, in millions of metric tons, where t is years since 2010. Interpret the statement f(5) = 49.2 in terms of wheat production.
Determine whether or not the function is a power function. If it is a power function, write it in the form y = kxp and give the values of k and p.y = (3x5)²
If you need $20,000 in your bank account in 6 years, how much must be deposited now? The interest rate is 10%, compounded continuously.
Solve for t using natural logarithms.50 = 10 ⋅ 3t
Figure 1.115 shows quarterly US beer production during the period 2013 to 2015. Quarter 1 reflects production during the first three months of each year, etc.(a) Explain why a periodic function should be used to model these data.(b) Approximately when does the maximum occur? The minimum? Why does
Suppose that q = f(p) is the demand curve for a product, where p is the selling price in dollars and q is the quantity sold at that price. (a) What does the statement f(12) = 60 tell you about demand for this product?(b) Do you expect this function to be increasing or decreasing? Why?
Give a possible formula for the functions in Problems. 3 y (2, 12) X
Determine the slope and the y-intercept of the line whose equation is given.12x = 6y + 4
Find the average rate of change of f(x) = 2x2 between x = 1 and x = 3.
Determine whether or not the function is a power function. If it is a power function, write it in the form y = kxp and give the values of k and p. y= 5 2√x
The chirp rate, C, in chirps per minute, of the snowy tree cricket is given by C = f(T) = 4T − 160 where T is degrees Fahrenheit. (a) Find an appropriate domain of f in the context of the model assuming a maximum temperature of 134◦F, the highest recorded at a weather station.(b) Find the
If $15,000 is deposited in an account paying 1.5% interest per year, compounded continuously, how long will it take for the balance to reach $20,000?
Solve for t using natural logarithms.5 = 2et
Sketch a possible graph of sales of sunscreen in the northeastern US over a 3-year period, as a function of months since January 1 of the first year. Explain why your graph should be periodic. What is the period?
A company has cost and revenue functions, in dollars, given by C(q) = 6000 + 10q and R(q) = 12q.(a) Find the cost and revenue if the company produces 500 units. Does the company make a profit? What about 5000 units?(b) Find the break-even point and illustrate it graphically.
Determine the slope and the y-intercept of the line whose equation is given.−4y + 2x + 8 = 0
Figure 1.22 shows four lines given by equation y = b + mx. Match the lines to the conditions on the parameters m and b.(a) m > 0, b > 0 (b) m < 0, b > 0 (c) m > 0, b < 0 (d) m < 0, b < 0Figure 1.22 y * 14 13 X
Give a possible formula for the functions in Problems. 18 y (2,8) X
The demand curve for a quantity q of a product is q = 5500 − 100p where p is price in dollars. Interpret the 5500 and the 100 in terms of demand. Give units.
Find the average rate of change of f(x) = 3x2 + 4 between x = −2 and x = 1. Illustrate your answer graphically.
The concentration of carbon dioxide, C = f(t), in the atmosphere, in parts per million (ppm), is a function of years, t, since 2000.(a) Interpret f(15) = 400 in terms of carbon dioxide.(b) What is the meaning of f(20)?
In Problem determine whether or not the function is a power function. If it is a power function, write it in the form y = kxp and give the values of k and p.y = 3 . 5x
If a bank pays 1.25% per year interest compounded continuously, how long does it take for the balance in an account to double?
For Problems solve for t using natural logarithms.e3t = 100
Values of a linear cost function are in Table 1.27. What are the fixed costs and the marginal cost? Find a formula for the cost function.Table 1.27 9 0 C(q) 5000 5 10 15 20 5020 5040 5060 5080
Find an equation for the line that passes through the given points.(4, 5) and (2,−1)
Determine whether or not the function is a power function. If it is a power function, write it in the form y = kxp and give the values of k and p. _y = 3 x2
(a) Estimate the fixed costs and the marginal cost for the cost function in Figure 1.62. (b) Estimate C(10) and interpret it in terms of cost.Figure 1.62 300 200 100 $ C(q) 5 10 15 20 25 30 9
Decide whether the graph is concave up, concave down, or neither. X
Let f(x) = x2 and g(x) = 3x − 1. Find the following: (a) f(2) + g(2) (b) f(2) ⋅ g(2) (c) f(g(2)) (d) g(f(2))
Figure 1.72 shows graphs of several cities’ populations against time. Match each of the following descriptions to a graph and write a description to match each of the remaining graphs.(a) The population increased at 5% per year.(b) The population increased at 8% per year.(c) The population
Graph the function. What is the amplitude and period?y = 3 sin 2x
Solve for t using natural logarithms.130 = 10t
The exponential function y(x) = Ceax satisfies the conditions y(0) = 2 and y(1) = 1. Find the constants C and a. What is y(2)?
Find an equation for the line that passes through the given points.(−2, 1) and (2, 3)
Find the following: (a) f(g(x)) (b) g(f(x)) (c) f(f(x))f(x) = 3x and g(x) = e2x
Graph the function. What is the amplitude and period?y = −3 sin 2θ
Solve for t using natural logarithms.2 = (1.02)t
Figure 1.79 shows the balances in two bank accounts. Both accounts pay the same interest rate, but one compounds continuously and the other compounds annually. Which curve corresponds to which compounding method? What is the initial deposit in each case? 100 20 s 5 10 15 Figure 1.79 A B 20 t (time)
Determine whether or not the function is a power function. If it is a power function, write it in the form y = kxp and give the values of k and p. y = x 100⁰
Decide whether the graph is concave up, concave down, or neither. ㄱ X
Figure 1.61 shows cost and revenue for a company. (a) Approximately what quantity does this company have to produce to make a profit? (b) Estimate the profit generated by 600 units.Figure 1.61 2500 1500 500 $ 100 R(q) 300 500 700 C(q) 900 q (quantity)
Figure 1.71 shows Q = 50(1.2)t, Q = 50(0.6)t, Q = 50(0.8)t, and Q = 50(1.4)t. Match each formula to a graph.Figure 1.71 Q (1) (II) (M)
Find an equation for the line that passes through the given points.(0, 0) and (1, 1)
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