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mathematics
applied calculus

Calculus And Its Applications 14th Edition Larry Goldstein, David Lay, David Schneider, Nakhle Asmar - Solutions

Crude oil production in the US increased approximately linearly from 1210 barrels per day in 1920 to 9637 barrels per day in 1970.17(a) Find the average daily oil production between 1920 and 1970 using an integral.(b) Find the average daily oil production by averaging the 1920 and 1970 values.
Figure 5.77 shows P'(t), the rate of change of the price of stock in a certain company at time t.(a) At what time during this five-week period was the stock at its highest value? At its lowest value?(b) If P(t) represents the price of the stock, arrange the following quantities in increasing
Use Figure 5.48 to find the values of(a) ∫ba f(x) dx(b) ∫cb f(x) dx(c) ∫ca f(x) dx (d) ∫ca |f(x)|dx a f(x) Area = 13 Figure 5.48 X Area = 2
Show the velocity, in cm/sec, of a particle moving along a number line. (Positive velocities represent movement to the right; negative velocities to the left.) Find the change in position and total distance traveled between times t = 0 and t = 5 seconds. 2 -3 v(t) 3 + 1 (sec) t 5
Show the velocity, in cm/sec, of a particle moving along a number line. (Positive velocities represent movement to the right; negative velocities to the left.) Find the change in position and total distance traveled between times t = 0 and t = 5 seconds. 10 v(t) 5 1 (sec)
The velocity of a car is f(t) = 5t meters/sec. Use a graph of f(t) to find the exact distance traveled by the car, in meters, from t = 0 to t = 10 seconds.
The graph of a derivative f'(x) is shown in Figure 5.78. Fill in the table of values for f(x) given that f(0) = 2. X 0 1 2 3 4 5 f(x) 2 2 3 4 5 5 6 f'(x) + x 6 Figure 5.78: Graph of f', not f
Show the velocity, in cm/sec, of a particle moving along a number line. (Positive velocities represent movement to the right; negative velocities to the left.) Find the change in position and total distance traveled between times t = 0 and t = 5 seconds. 8 -2 v(t) Aur 4 5 t (sec)
Match the graph with one of the following possible values for the integral ∫50 f(x) dx:I. − 10.4 II. − 2.1 III. 5.2 IV. 10.4 20 10 -10 j(x) 2 5
Annual income for ages 25 to 85 is given graphically. People sometimes spend less than their income (to save for retirement) or more than their income (taking out a loan). The process of spreading out spending over a lifetime is called consumption smoothing.(a) Find the average annual income for
Use the following table to estimate ∫2610 f(x) dx. 10 14 18 22 26 f(x) 100 88 72 50 28 x
A bicyclist accelerates at a constant rate, from 0 ft/sec to 15 ft/sec in 10 seconds.(a) Figure 5.17 shows the velocity of the bike while it is accelerating. What is the value of b in the figure?(b) How far does the bike travel while it is accelerating? velocity (ft/sec) b Figure 5.17 10 t (secs)
Use the table to estimate ∫400 f(x)dx. What values of n and Δx did you use? 0 f(x) 350 x 10 20 410 435 435 30 450 40 460
Problems refer to Figure 5.86, which shows human arterial blood pressure during the course of one heartbeat.(a) Estimate the maximum blood pressure, called the systolic pressure.(b) Estimate the minimum blood pressure, called the diastolic pressure.(c) Calculate the average of the systolic and
The derivative f'(x) is graphed in Figure 5.79. Fill in the table of values for f(x) given that f(0) = −10. X 0 1 2 3 4 5 6 f(x) -10 8 4 f'(x) 12 3 4 5. 6 Figure 5.79: Graph of f', not f X
Match the graph with one of the following possible values for the integral ∫50 f(x) dx:I. − 10.4 II. − 2.1 III. 5.2 IV. 10.4 5 -5 -10 f(x) 2 X
A forest fire covers 2000 acres at time t = 0. The fire is growing at a rate of 8√t acres per hour, where t is in hours. How many acres are covered 24 hours later?
Use the following table to estimate ∫150 f(x) dx. 0 3 48 X : j(x) 50 6 9 12 15 44 36 24 8
A car accelerates at a constant rate from 44 ft/sec to 88 ft/sec in 5 seconds.(a) Figure 5.18 shows the velocity of the car while it is accelerating. What are the values of a, b and c in the figure?(b) How far does the car travel while it is accelerating? velocity (ft/sec) b a Figure 5.18 С t
Problems refer to Figure 5.86, which shows human arterial blood pressure during the course of one heartbeat.Estimate the average arterial blood pressure over one cardiac cycle. arterial pressure (mm Hg) 100 Figure 5.86 1 one cardiac cycle
Figure 5.80 shows the rate of change in the average plasma concentration of the drug Omeprazole (in ng/ml per hour) for six hours after the first dose is administered using two different capsules: immediate-release and delayed-release.(a) Which graph corresponds to which capsule?(b) Do the two
In Problems match the graph with one of the following possible values for the integral ∫50 f(x) dx:I. − 10.4 II. − 2.1 III. 5.2 IV. 10.4 2 1 -1 f(x) 5 X
Water is pumped out of a holding tank at a rate of 5 − 5e−0.12t liters/minute, where t is in minutes since the pump is started. If the holding tank contains 1000 liters of water when the pump is started, how much water does it hold one hour later?
A village wishes to measure the quantity of water that is piped to a factory during a typical morning. A gauge on the water line gives the flow rate (in cubic meters per hour) at any instant. The flow rate is about 100 m3∕hr at 6 am and increases steadily to about 280 m3∕hr at 9 am. Using only
Use the following table to estimate ∫43 W(t) dt. What are n and Δt? t W (1) 3.0 3.2 3.4 3.6 3.8 4.0 25 23 20 15 9 2
Figure 5.87 shows the rate, f(x), in thousands of algae per hour, at which a population of algae is growing, where x is in hours.(a) Estimate the average value of the rate over the interval x = −1 to x = 3.(b) Estimate the total change in the population over the interval x = −3 to x = 3.
In Problems oil is pumped from a well at a rate of r(t) barrels per day, with t in days. Assume r'(t) < 0 and t0 > 0.What does the value of ∫0t0 r(t) dt tell us about the oil well?
(a) Graph f(x) = x(x + 2)(x − 1).(b) Find the total area between the graph and the x-axis between x = −2 and x = 1.(c) Find ∫1−2 f(x) dx and interpret it in terms of areas.
With t in seconds, the velocity of an object is v(t) = 10 + 8t − t2 m/sec.(a) Represent the distance traveled during the first 5 seconds as a definite integral and as an area.(b) Estimate the distance traveled by the object during the first 5 seconds by estimating the area.(c) Calculate the
The number of hours, H, of daylight in Madrid as a function of date is approximated by the formulaH = 12 + 2.4 sin (0.0172(t − 80)),where t is the number of days since the start of the year.Find the average number of hours of daylight in Madrid:(a) In January (b) In June (c) Over a
Filters at a water treatment plant become less effective over time. The rate at which pollution passes through the filters into a nearby lake is given in the following table.(a) Estimate the total quantity of pollution entering the lake during the 30-day period.(b) Your answer to part (a) is only
Using Figure 5.29, draw rectangles representing each of the following Riemann sums for the function f on the interval 0 ≤ t ≤ 8. Calculate the value of each sum.(a) Left-hand sum with Δt = 4(b) Right-hand sum with Δt = 4(c) Left-hand sum with Δt = 2(d) Right-hand sum with Δt = 2
Oil is pumped from a well at a rate of r(t) barrels per day, with t in days. Assume r'(t) < 0 and t0 > 0.Rank in order from least to greatest: 210 r(t) dt, 10 210 r(t) dt, -310 210 r(t) dt.
The rate of sales (in sales per month) of a company is given, for t in months since January 1, byr(t) = t4 − 20t3 + 118t2 − 180t + 200.(a) Graph the rate of sales per month during the first year (t = 0 to t = 12). Does it appear that more sales were made during the first half of the year, or
The following table gives world oil consumption, in billions of barrels per year. Estimate total oil consumption during this 25-year period. Year 1985 1990 1995 2000 2005 2010 Oil (bn barrels/yr) 20.8 23.2 25.5 27.9 30.7 32.0
(a) Using Figure 5.49, find ∫0−3 f(x) dx.(b) If the area of the shaded region is A, estimate ∫4−3 f(x) dx. -4 -2 -1 Figure 5.49 2 f(x) 3 4 5 x
A bungee jumper leaps off the starting platform at time t = 0 and rebounds once during the first 5 seconds. With velocity measured downward, for t in seconds and 0 ≤ t ≤ 5, the jumper’s velocity is approximated by v(t) = −4t2 + 16t meters/sec.(a) How many meters does the jumper travel
Use Figure 5.30 to estimate ∫200 f(x) dx. 5432 1 f(x) 4 8 12 Figure 5.30 16 20 x
A car initially going 50 ft/sec brakes at a constant rate (constant negative acceleration), coming to a stop in 5 seconds.(a) Graph the velocity from t = 0 to t = 5.(b) How far does the car travel?(c) How far does the car travel if its initial velocity is doubled, but it brakes at the same constant
A bar of metal is cooling from 1000◦C to room temperature, 20◦C. The temperature, H, of the bar t minutes after it starts cooling is given, in ◦C, byH = 20 + 980e−0.1t.(a) Find the temperature of the bar at the end of one hour.(b) Find the average value of the temperature over the first
In Problems let C(n) be a city’s cost, in millions of dollars, for plowing the roads when n inches of snow have fallen. Let c(n) = C'(n). Evaluate the expressions and interpret your answers in terms of the cost of plowing snow, given c'(n) < 0, c(24) = 0.4, 15 Jo c(n) dn = 7.5, C(15) = 8, c(15) =
Use the following table to estimate the area between f(x) and the x-axis on the interval 0 ≤ x ≤ 20. x 0 5 10 15 20 f(x) 15 18 20 16 12
Use Figure 5.31 to estimate ∫15−10 f(x)dx. -10 -30- -20- -10- 0 10 Figure 5.31 f(x) X
Let C(n) be a city’s cost, in millions of dollars, for plowing the roads when n inches of snow have fallen. Let c(n) = C'(n). Evaluate the expressions and interpret your answers in terms of the cost of plowing snow, givenC(0) c'(n) < 0, c(24) = 0.4, 15 Jo c(n) dn = 7.5, C(15) = 8, c(15) =
Figure 5.19 shows the rate of change of a fish population. Estimate the total change in the population during this 12-month period. rate (fish per month) 25 20 15 10 5 4 8 Figure 5.19 12 time (months)
Use the graphs in Problems to estimate ∫30 f(x) dx. 8 6 4 2 0 -f(x)- 12 3 4 X
Figure 5.56 shows the length growth rate of a human fetus.(a) What feature of a graph of length as a function of age corresponds to the maximum in Figure 5.56?(b) Estimate the length of a baby born in week 40. length growth rate (cm/week) 2 1 0 8 16 24 age of fetus 32 40 (weeks after last
Use an integral to find the specified area. Under y = 5 ln(2x) and above y = 3 for 3 ≤ x ≤ 5.
Let C(n) be a city’s cost, in millions of dollars, for plowing the roads when n inches of snow have fallen. Let c(n) = C'(n). Evaluate the expressions and interpret your answers in terms of the cost of plowing snow, given c'(n) < 0, c(24) = 0.4, 15 Jo c(n) dn = 7.5, C(15) = 8, c(15) = 0.7, C(24)
Throughout much of the 20th century, the yearly consumption of electricity in the US increased exponentially at a continuous rate of 7% per year. Assume this trend continues and that the electrical energy consumed in 1900 was 1.4 million megawatt-hours.(a) Write an expression for yearly electricity
The following table gives the annual natural gas production, in trillions of cubic feet per year, in the US between 2002 and 2014.(a) Estimate the total natural gas produced in the US between 2002 and 2014 using a (i) Left sum, n = 6 (ii) Right sum, n = 6(b) Can you determine whether the right sum
Use the graphs in Problems to estimate ∫30 f(x) dx. 16 12 8 4 J (x) 12 3 x
Use an integral to find the specified area. Under y = 6x3 − 2 for 5 ≤ x ≤ 10.
One or more initial conditions are given for each differential equation in the following exercises. Use the qualitative theory of autonomous differential equations to sketch the graphs of the corresponding solutions. Include a yz-graph if one is not already provided. Always indicate the constant
Solve the differential equations in Exercises.yy′ + t = 6t2, y(0) = 7
At one point in his study of a falling body starting from rest, Galileo conjectured that its velocity at any time is proportional to the distance it has dropped. Using this hypothesis, set up the differential equation whose solution is y = f (t), the distance fallen by time t. By making use of the
One or more initial conditions are given for each differential equation in the following exercises. Use the qualitative theory of autonomous differential equations to sketch the graphs of the corresponding solutions. Include a yz-graph if one is not already provided. Always indicate the constant
The National Automobile Dealers Association reported that the average retail selling price of a new vehicle was $30,303 in 2012. A person purchased a new car at the average price and financed the entire amount. Suppose that the person can only afford to pay $500 per month. Assume that the payments
Let f (t) be the solution of y′ = y(2t - 1), y(0) = 8. Use Euler’s method with n = 4 to estimate f (1).
Is the constant function f (t) = -4 a solution of the differential equation y′ = t2 (y + 4)?
What is an integrating factor and how does it help you solve a first-order linear differential equation?
One or more initial conditions are given for each differential equation in the following exercises. Use the qualitative theory of autonomous differential equations to sketch the graphs of the corresponding solutions. Include a yz-graph if one is not already provided. Always indicate the constant
Solve the given equation using an integrating factor. Take t > 0.y′ + 2ty = 0
Solve the differential equations in Exercises. 1 2(1 + 1) = 1+t, t≥ 0
Solve the differential equations in Exercises.y′ = 5 - 8y, y(0) = 1
In an autocatalytic reaction, one substance is converted into a second substance in such a way that the second substance catalyzes its own formation. This is the process by which trypsinogen is converted into the enzyme trypsin. The reaction starts only in the presence of some trypsin, and each
The Federal Housing Finance Board reported that the national average price of a new onefamily house in 2012 was $278,900. At the same time, the average interest rate on a conventional 30-year fixed-rate mortgage was 3.1%. A person purchased a home at the average price, paid a down payment equal to
Solve the differential equations in Exercises. 2 1-t - y = (1 - 1)4
Let f (t) be the solution of y′ = -(t + 1)y2, y(0) = 1. Use Euler’s method with n = 5 to estimate f (1). Then, solve the differential equation, find an explicit formula for f (t), and compute f (1). How accurate is the estimated value of f (1)?
Find a constant solution of y′ = t2y - 5t2.
What is an autonomous differential equation?
Solve the given equation using an integrating factor. Take t > 0.y′ - 2ty = -4t
One or more initial conditions are given for each differential equation in the following exercises. Use the qualitative theory of autonomous differential equations to sketch the graphs of the corresponding solutions. Include a yz-graph if one is not already provided. Always indicate the constant
Let q = f (p) be the demand function for a certain commodity, where q is the demand quantity and p the price of 1 unit. We defined the elasticity of demand as(a) Find a differential equation satisfied by the demand function if the elasticity of demand is a linear function of price given by E( p) =
A porous material dries outdoors at a rate that is proportional to the moisture content. Set up the differential equation whose solution is y = f (t), the amount of water at time t in a towel on a clothesline. Sketch the solution.
Answer parts (a), (b), and (c) of Exercise 9 if the person takes a 15-year fixed-rate mortgage with a 6% interest rate and intends to pay off the entire loan in 15 years.Exercise 9The Federal Housing Finance Board reported that the national average price of a new onefamily house in 2012 was
Let f (t) be the solution of y′ = 10 - y, y(0) = 1. Use Euler’s method with n = 5 to estimate f (1). Then, solve the differential equation and find the exact value of f (1).
Find two constant solutions of y′ = 4y ( y - 7).
How do you recognize an autonomous differential equation from its slope field?
Solve the given equation using an integrating factor. Take t > 0.y′ = 2(20 - y)
One or more initial conditions are given for each differential equation in the following exercises. Use the qualitative theory of autonomous differential equations to sketch the graphs of the corresponding solutions. Include a yz-graph if one is not already provided. Always indicate the constant
Find a curve in the xy-plane passing through the origin and whose slope at the point (x, y) is x + y.
Suppose that the Consumer Products Safety Commission issues new regulations that affect the toy-manufacturing industry. Every toy manufacturer will have to make certain changes in its manufacturing process. Let f (t) be the fraction of manufacturers that have complied with the regulations within t
If the function f (t) is a solution of the initial-value problem y′ = 2y - 3, y(0) = 4, find f (0) and f′(0).
One or more initial conditions are given for each differential equation in the following exercises. Use the qualitative theory of autonomous differential equations to sketch the graphs of the corresponding solutions. Include a yz-graph if one is not already provided. Always indicate the constant
Solve the given equation using an integrating factor. Take t > 0.y′ = .5(35 - y)
Let P(t) denote the price in dollars of a certain commodity at time t in days. Suppose that the rate of change of P is proportional to the difference D - S of the demand D and supply S at any time t. Suppose further that the demand and supply are related to the price by D = 10 - .3P and S = -2 +
An experimenter reports that a certain strain of bacteria grows at a rate proportional to the square of the size of the population. Set up a differential equation that describes the growth of the population. Sketch a solution.
Find the demand function if the elasticity of demand is a linear function of price given by E( p) = ap + b, where a and b are constants.
The Los Angeles Zoo plans to transport a California sea lion to the San Diego Zoo. The animal will be wrapped in a wet blanket during the trip. At any time t, the blanket will lose water (due to evaporation) at a rate proportional to the amount f (t) of water in the blanket, with constant of
If the function f (t) is a solution of the initial-value problem y′ = et + y, y(0) = 0, find f (0) and f′(0).
Solve the given equation using an integrating factor. Take t > 0. y' + y 10+ t = 0
One or more initial conditions are given for each differential equation in the following exercises. Use the qualitative theory of autonomous differential equations to sketch the graphs of the corresponding solutions. Include a yz-graph if one is not already provided. Always indicate the constant
What is the logistic differential equation?
Solve the given equation using an integrating factor. Take t > 0.y′ + y = e-t + 1
If f (t) is a solution of y′ = (2 - y)e-y, is f (t) increasing or decreasing at some value of t where f (t) = 3?
Suppose that substance A is converted into substance B at a rate that, at any time t, is proportional to the square of the amount of A. This situation occurs, for instance, when it is necessary for two molecules of A to collide to create one molecule of B. Set up the differential equation that is
When a red-hot steel rod is plunged in a bath of water that is kept at a constant temperature 10C, the temperature of the rod at time t, f (t), satisfies the differential equation y′ = k[10 - y], where k > 0 is a constant of proportionality. Determine f (t) if the initial temperature of the
The differential equation y′ = .5(1 - y)(4 - y) has five types of solutions labeled A–E. For each of the following initial values, graph the solution of the differential equation and identify the type of solution. Use a small value of h, let t range from 0 to 4, and let y range from -1 to
Solve the initial-value problem y' = e² (cos y)(1-e³-¹), y(0) = 1.

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