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mathematics
applied calculus
Calculus And Its Applications 14th Edition Larry Goldstein, David Lay, David Schneider, Nakhle Asmar - Solutions
If y = 5(1 - e-2x), compute y′ and show that y′ = 10 - 2y.
One thousand dollars is deposited in a savings account at 10% interest compounded continuously. How many years are required for the balance in the account to reach $3000?
Ten thousand dollars is deposited in a savings account at 4.6% yearly interest compounded continuously.(a) What differential equation is satisfied by A(t), the balance after t years?(b) What is the formula for A(t)?(c) How much money will be in the account after 3 years?(d) When will the balance
Find the logarithmic derivative and then determine the percentage rate of change of the functions at the points indicated.f (x) = e-0.05x at x = 1 and x = 10
State the formula for each of the following quantities:(a) The compound amount of P dollars in t years at interest rate r, compounded continuously(b) The present value of A dollars in n years at interest rate r, compounded continuously
If f (x) = 3(1 - e-10x), show that y = f (x) satisfies the differential equation y = 10(3 - y), f(0) = 0.
The half-life of the radioactive element tritium is 12 years. Find its decay constant.
An investment earns 4.2% yearly interest compounded continuously. How fast is the investment growing when its value is $9000?
Determine the growth constant k, then find all solutions of the given differential equation. 2y'. ע 2 = 0
Find the logarithmic derivative and then determine the percentage rate of change of the functions at the points indicated.f(t) = e0.3t2 at t = 1 and t = 5
What is the difference between a relative rate of change and a percentage rate of change?
An investment earns 5.1% yearly interest compounded continuously and is currently growing at the rate of $765 per year. What is the current value of the investment?
Find the logarithmic derivative and then determine the percentage rate of change of the functions at the points indicated.G(s) = e-0.05s2 at s = 1 and s = 10
Determine the growth constant k, then find all solutions of the given differential equation. y = 1.6y'
Define the elasticity of demand, E( p), for a demand function. How is E( p) used?
When a grand jury indicted the mayor of a certain town for accepting bribes, the newspaper, online news outlets, radio, and television immediately began to publicize the news. Within an hour, one-quarter of the citizens heard about the indictment. Estimate when three-quarters of the town heard the
From January 1, 2010, to January 1, 2017, the population of a state grew from 17 million to 19.3 million.(a) Give the formula for the population t years after 2010.(b) If this growth continues, how large will the population be in 2020?(c) In what year will the population reach 25 million?
One thousand dollars is deposited in a savings account at 6% yearly interest compounded continuously. How many years are required for the balance in the account to reach $2500?
Find the logarithmic derivative and then determine the percentage rate of change of the functions at the points indicated.f(p) = 1/( p + 2) at p = 2 and p = 8
Describe an application of the differential equation y′ = k(M - y).
Examine formula (8) for the amount A(t) of excess glucose in the bloodstream of a patient at time t. Describe what would happen if the rate r of infusion of glucose were doubled.
A stock portfolio increased in value from $100,000 to $117,000 in 2 years. What rate of interest, compounded continuously, did this investment earn?
Ten thousand dollars is invested at 6.5% interest compounded continuously. When will the investment be worth $41,787?
Find the logarithmic derivative and then determine the percentage rate of change of the functions at the points indicated.g( p) = 5/(2p + 3) at p = 1 and p = 11
Describe an application of the differential equation y′ = ky(M - y).
A news item is spread by word of mouth to a potential audience of 10,000 people. After t days,people will have heard the news. The graph of this function is shown in Fig. 7.(a) Approximately how many people will have heard the news after 7 days?(b) At approximately what rate will the news spread
The annual sales S (in dollars) of a company may be approximated by the formulawhere t is the number of years beyond some fixed reference date. Use a logarithmic derivative to determine the percentage rate of growth of sales at t = 4. S = 50,000 √evi,
Determine the growth constant k, then find all solutions of the given differential equation. y 3 = 4y'
A news item is broadcast by mass media to a potential audience of 50,000 people. After t days,people will have heard the news. The graph of this function is shown in Fig. 8.(a) How many people will have heard the news after 10 days?(b) At what rate is the news spreading initially?(c) When will
An investor initially invests $10,000 in a risky venture. Suppose that the investment earns 20% interest, compounded continuously, for 5 years and then 6% interest, compounded continuously, for 5 years thereafter.(a) How much does the $10,000 grow to after 10 years?(b) The investor has the
One hundred shares of a technology stock were purchased on January 2, 1990, for $1200 and sold on January 2, 1998, for $12,500. What rate of interest compounded continuously did this investment earn?
Physiologists usually describe the continuous intravenous infusion of glucose in terms of the excess concentration of glucose, C(t) = A(t)/V, where V is the total volume of blood in the patient. In this case, the rate of increase in the concentration of glucose due to the continuous injection is
Two different bacteria colonies are growing near a pool of stagnant water. The first colony initially has 1000 bacteria and doubles every 21 minutes. The second colony has 710,000 bacteria and doubles every 33 minutes. How much time will elapse before the first colony becomes as large as the second?
Determine the growth constant k, then find all solutions of the given differential equation.5y′ - 6y = 0
Pablo Picasso’s Angel Fernandez de Soto was acquired in 1946 for a postwar splurge of $22,220. The painting was sold in 1995 for $29.1 million. What yearly rate of interest compounded continuously did this investment earn?
Describe an experiment that a doctor could perform to determine the velocity constant of elimination of glucose for a particular patient. y 50000 45000 40000 35000 30000 25000 20000 15000 10000 5000 Figure 8 [(y = f(t) y = f'( 5 10 15
The price of wheat per bushel at time t (in months) is approximated by f (t) = 4 + .001t + .01e-t. What is the percentage rate of change of f (t) at t = 0? t = 1? t = 2?
The population of a city t years after 1990 satisfies the differential equation y′ = .02y. What is the growth constant? How fast will the population be growing when the population reaches 3 million people? At what level of population will the population be growing at the rate of 100,000 people
Solve the given differential equation with initial condition.y′ = 3y, y(0) = 1
How many years are required for an investment to double in value if it is appreciating at the yearly rate of 4% compounded continuously?
The wholesale price in dollars of one pound of ground beef is modeled by the function f (t) = 3.08 + .57t - .1t2 + .01t3, where t is measured in years from January 1, 2010.(a) Estimate the price in 2011 and find the rate in dollars per year at which the price was rising in 2011.(b) What is the
A colony of bacteria is growing exponentially with growth constant .4, with time measured in hours. Determine the size of the colony when the colony is growing at the rate of 200,000 bacteria per hour. Determine the rate at which the colony will be growing when its size is 1 million.
Solve the given differential equation with initial condition.y′ = 4y, y(0) = 0
What yearly interest rate (compounded continuously) is earned by an investment that doubles in 10 years?
The wholesale price in dollars of one pound of pork is modeled by the function f (t) = 1.4 + .26t - .1t2 + .01t3, where t is measured in years from January 1, 2010.(a) Estimate the price in 2012 and find the percentage rate of increase of the price in 2012?(b) Answer part (a) for the year 2017.
After a drug is taken orally, the amount of the drug in the bloodstream after t hours is f (t) = 122(e-0.2t - e-t) units.(a) Graph f (t), f′(t), and f″(t) in the window [0, 12] by [-20, 75](b) How many units of the drug are in the bloodstream after 7 hours?(c) At what rate is the level of drug
Solve the given differential equation with initial condition.y′ = 2y, y(0) = 2
If an investment triples in 15 years, what yearly interest rate (compounded continuously) does the investment earn?
For each demand function, find E( p) and determine if demand is elastic or inelastic (or neither) at the indicated price.q = 700 - 5p, p = 80
Solve the given differential equation with initial condition.y′ = y, y(0) = 4
You have 80 grams of a certain radioactive material, and the amount remaining after t years is given by the function f (t) shown in Fig. 1.(a) How much will remain after 5 years?(b) When will 10 grams remain?(c) What is the half-life of this radioactive material?(d) At what rate will the
A model incorporating growth restrictions for the number of bacteria in a culture after t days is given by f (t) = 5000(20 + te-0.04t).(a) Graph f′(t) and f′′(t) in the window [0, 100] by [-700, 300].(b) How fast is the culture changing after 100 days?(c) Approximately when is the culture
Solve the given differential equation with initial condition. y' y 7 0, y(0) = 6
If real estate in a certain city appreciates at the yearly rate of 15% compounded continuously, when will a building purchased in 2010 triple in value?
For each demand function, find E( p) and determine if demand is elastic or inelastic (or neither) at the indicated price.q = 600e-0.2p, p = 10
Solve the given differential equation with initial condition.y′ - .6y = 0, y(0) = 5
A few years after money is deposited into a bank, the compound amount is $1000, and it is growing at the rate of $60 per year. What interest rate (compounded continuously) is the money earning?
Suppose that the bank in Example 3 increased its fees by charging a negative annual interest rate of -.9%. Find the balance after two years in a savings account if P0 = 10, 000 SFr.
For each demand function, find E( p) and determine if demand is elastic or inelastic (or neither) at the indicated price.q = 400(116 - p2), p = 6
The current balance in a savings account is $1230, and the interest rate is 4.5%. At what rate is the compound amount currently growing?
How is the account in Exercise 15 changing when the balance is 9,500 SFr?
For each demand function, find E( p) and determine if demand is elastic or inelastic (or neither) at the indicated price.q = (77/p2) + 3, p = 1
Solve the given differential equation with initial condition.6y′ = y, y(0) = 12
Find the percentage rate of change of the function f (t) = 50e0.2t2 at t = 10.
A farm purchased in 2000 for $1 million was valued at $3 million in 2010. If the farm continues to appreciate at the same rate (with continuous compounding), when will it be worth $10 million?
For each demand function, find E( p) and determine if demand is elastic or inelastic (or neither) at the indicated price.q = p2e-(p+3), p = 4
In Exercises, solve the given differential equation with initial condition.5y = 3y′, y(0) = 7
Find E(p) for the demand function q = 4000 - 40p2, and determine if demand is elastic or inelastic at p = 5.
A parcel of land bought in 1990 for $10,000 was worth $16,000 in 1995. If the land continues to appreciate at this rate, in what year will it be worth $45,000?
For each demand function, find E( p) and determine if demand is elastic or inelastic (or neither) at the indicated price.q = 700/( p + 5), p = 15
For a certain demand function, E(8) = 1.5. If the price is increased to $8.16, estimate the percentage decrease in the quantity demanded. Will the revenue increase or decrease?
Let P(t) be the population (in millions) of a certain city t years after 2015, and suppose that P(t) satisfies the differential equation P′(t) = .01P(t), P(0) = 2.(a) Find a formula for P(t).(b) What was the initial population, that is, the population in 2015?(c) Estimate the population in 2019.
Find the present value of $1000 payable at the end of 3 years, if money may be invested at 8% with interest compounded continuously.
A colony of fruit flies exhibits exponential growth. Suppose that 500 fruit flies are present. Let P(t) denote the number of fruit flies t days later, and let k = .08 denote the growth constant.(a) Write a differential equation and initial condition that model the growth of this colony.(b) Find a
Find the present value of $2000 to be received in 10 years, if money may be invested at 8% with interest compounded continuously.
An electronic store can sell q = 10,000/( p + 50) - 30 cellular phones at a price p dollars per phone. The current price is $150.(a) Is demand elastic or inelastic at p = 150?(b) If the price is lowered slightly, will revenue increase or decrease?
A company can sell q = 1000p2e-0.02(p+5) calculators at a price of p dollars per calculator. The current price is $200. If the price is decreased, will the revenue increase or decrease?
A bacteria culture that exhibits exponential growth quadruples in size in 2 days.(a) Find the growth constant if time is measured in days.(b) If the initial size of the bacteria culture was 20,000, what is its size after just 12 hours?
How much money must you invest now at 4.5% interest compounded continuously to have $10,000 at the end of 5 years?
A movie theater has a seating capacity of 3000 people. The number of people attending a show at price p dollars per ticket is q = (18,000/p) - 1500. Currently, the price is $6 per ticket.(a) Is demand elastic or inelastic at p = 6?(b) If the price is lowered, will revenue increase or decrease?
Consider a demand function of the form q = ae-bp, where a and b are positive numbers. Find E(p), and show that the elasticity equals 1 when p = 1>b.
The highest price ever paid for an artwork at auction was for Pablo Picasso’s 1955 painting Les femmes d’Alger, which fetched $179.4 million in a Christie’s auction in 2015. The painting was last sold in 1997 for $31.9 million. If the painting keeps on appreciating at its current rate, then a
The graphs of y = x + ln x and y = ln 2x are shown in Fig. 6.(a) Show that both functions are increasing for x > 0.(b) Find the point of intersection of the graphs. 2 1 -2. -3 Figure 6 0.5 y = x + ln x 1.0 y = ln 2x + 1.5 + 2.0
Repeat the previous exercise with y = √x ln x.Exercises 29Find the coordinates of the relative extreme point of y = x2 ln x, x > 0. Then, use the second derivative test to decide if the point is a relative maximum point or a relative minimum point.
Solve the given equation for x.ln(x + 1) - ln(x - 2) = 1
Find the values of x at which the function has a possible relative maximum or minimum point. (Recall that ex is positive for all x.) Use the second derivative to determine the nature of the function at these points.f (x) = (5x - 2)e1-2x
Repeat Exercise 31 with the functions y = x + ln x and y = ln 5x.Exercise 31The graphs of y = x + ln x and y = ln 2x are shown in Fig. 6.(a) Show that both functions are increasing for x > 0.(b) Find the point of intersection of the graphs. 3 2+ 1 -1 -2 -3 Figure 7 0.5 y = x + ln x 1.0 y = In
Solve the equation 4 · 2x = ex.
Solve the given equation for x.ln[(x - 3)(x + 2)] - ln(x + 2)2 - ln 7 = 0
The graph of the function y = x2 - ln x is shown in Fig. 8. Find the coordinates of its minimum point. Figure 8 5 4- Co 3- y 2 0 f(x) = x² - Inx + 0.5 + 1.0 + 1.5 Xx
Find the values of x at which the function has a possible relative maximum or minimum point. (Recall that ex is positive for all x.) Use the second derivative to determine the nature of the function at these points.f (x) = (2x - 5)e3x-1
Solve the equation 3x = 2ex.
Differentiate.y = ln[(x + 5)(2x - 1)(4 - x)]
The value of an investment portfolio consisting of two stocks is given by f (t) = 3e0.06t + 2e0.02t, where t is the number of years since the inception of the portfolio, and f (t) is in thousands of dollars.(a) What is the initial dollar amount invested?(b) What is the value of the portfolio after
Find the points on the graph of y = ex where the tangent line has slope 4.
Differentiate.y = ln[(x + 1)(2x + 1)(3x + 1)]
Find the points on the graph of y = ex + e-2x where the tangent line is horizontal.
The function y = 2x2 - ln 4x (x > 0) has one minimum point. Find its first coordinate.
The graph of y = x - ex has one extreme point. Find its coordinates and decide whether it is a maximum or a minimum.
Differentiate.y = ln[(1 + x)2(2 + x)3(3 + x)4]
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