AP Calculus AB Unit 7 review

Express the following as a differential equation: The population of deer in a certain area is changing at a rate that is proportional to the current population.

dy/dx = kx

Express the following as a differential equation: The temperature of after a certain amount of hours is changing at a rate that is three times the square of the hours.

dy/dx = 3x^2

Express the following as a differential equation: The number of gallons of juice left after a certain amount of days is changing at a rate that is proportional to the square root of the days.

dy/dx = ksqrt(x)

Express the following as a differential equation: The amount of money you have saved is changing at a rate that is inversely proportional to the amount of money you currently have saved.

dy/dx = k/y

Express the following as a differential equation: The temperature of a solution is changing at a rate that is inversely proportional to cube of the current temperature.

dy/dx = k/(y^3)

dy/dx = xe^y. Find the general solution.

y = -ln[-(x^2)/2 + C]

dy/dx = e^(y-x). Find the general solution.

y = ln(e^x + C)

dy/dx = 2yx + yx^2. Find the general solution.

y = Ce^[(x^2) + (x^3)/3]

dy/dx = 2y - 1. Find the general solution.

y = [Ce^(2x) + 1]/2

dy/dx = (2x)/[e^(2y)]. Find the general solution.

y = [ln(2x^2 + C)]/2

Solve the differential equation: y' = ycosx

y = e^(sinx)

Solve the differential equation: (y^3)y' - 6x^2 = 0

y = (8x^3 + C)^(1/4)

(sinx)(dy/dx) = cosx. Find the general solution

y = ln|sinx| + C

dy/dx - ax = b. Find the general solution

y = [ke^(ax) - b]/a

dy/dx = (y^2 + 6)/(2y). Find the general solution.

y = sqrt(Ce^x - 6)

2y(dy/dx) = x + sinx. Find the general solution.

y = sqrt[(x^2)/2 - cosx + C]

dy/dx = x/(y^2 + 1). Find the general solution.

(y^3)/3 + y = (x^2)/2 + C

(e^y)(dy/dx) = 2x. Find the general solution.

y = ln(x^2 + C)

dy/dx = (-x^2)/2y. Find the general solution.

(x^2) + (2y^2) = C

dy/dx = k(10 - y). Find the general solution.

y = 10 - Ce^(-kx)

dy/dx = 3x^2. Find the particular solution that passes through the point (2, 2)

y = x^3 - 6

(e^y)(dy/dx) = 2x. Find the particular solution that passes through the point (3, 0)

y = ln(x^2 - 8)

dy/dx = (-x^2)/2y. Find the particular solution that passes through the point (5, 6sqrt(2))

x^2 + 2y^2 = 169

dy/dx = (y^2 + 6)/(2y). Find the particular solution that passes through the point (0, 5)

y = sqrt(31e^x - 6)

(y^3)y' - 6x^2 = 0. Find the particular solution that passes through the point (2, 3)

y = (8x^3 + 17)^(1/4)

dy/dx = y. Find the particular solution that passes through the point (0, 500)

y = 500e^x

dy/dx = y + 8. Find the particular solution that passes through the point (0, 4)

y = 12e^x - 8

e^x - y(dy/dx) = 0. Find the particular solution that passes through the point (0, 3)

y = sqrt(2e^x + 7)

2y(dy/dx) = x + sinx. Find the particular solution that passes through the point (0, 1)

y = sqrt[(x^2)/2 - cosx + 2]

dy + k(y - 25)dx = 0. Find the particular solution that passes through the point (0, 130).

y = 65e^(-kx) + 65

dy/dx = 1/x. Find the particular solution that passes through the point (e, -2)

y = lnx - 3

dy/dx = (3x^2)/(y^2). Find the particular solution that passes through the point (0, 2)

y = (3x^3 + 8)^(1/3)

dy/dx = y + 1. Find the particular solution that passes through the point (0, 3)

ln(y + 1) = x + ln4

dy/dx = y. Find the particular solution that passes through the point (0, 10)

y = 10e^x

dy/dx = x/(y^2 + 1). Find the particular solution that passes through the point (3, 2)

(y^3)/3 + y = (x^2)/2 + 10

The half life of an element is 10 years. Find k.

k = 0.0693

You have 62 grams of an element that has a half life of 625 years. How long would it take for the 62 grams to decay to 22 grams?

About 934 years

The rate of change of y is proportional to y. Find the equation of y if y(0) = 2 and y(2) = 4

y = 2e^(0.347x)

dy/dt = -2y and y(0) = 1. Find y(3).

y = e^(-6)

The growth rate of a certain type of bacteria is proportional to the current number. If the bacteria doubles in three hours, how long will it take for it to triple?

(3ln3)/2

The rate of change of a deer population follows the equation dy/dx = ky. If a deer population doubles every ten years, what is k?

0.069

The spread rate of a virus follows the equation dy/dx = ky. If 1000 people are originally infected with it and 1200 people are infected seven days later, how many people will be infected in 12 days?

1367

A piece of metal is taken out of a furnace and has a temperature of 1500. It is put into a room with a constant temperature of 90. An hour later, its temperature is 1120. Write an equation for the temperature of the metal. Use Newton's Law of Cooling: dy/dx = k(T - Ta).

y = 90 + 1410e^(-0.314x)

You have a pot of water that has a temperature of 95 and you place it into a freezer with a temperature of 20. After a minute, the temperature has decreased to 90. How long will it take for the temperature to be 65? Use Newton's Law of Cooling: dy/dx = k(T - Ta).

7.403

You're trying to determine exactly what time you made your food. You check its temperature and find that it's 80 degrees. You check again two hours later and it is 75 degrees. You know the temperature of the room is 60 degrees. If it's midnight now, when did you make the food? Use Newton's Law of Cooling: dy/dx = k(T - Ta).

7: 26

For a period of time, an island's population grows at a rate proportional to its population. If the growth rate is 3.8% per year and the current population is 1543, what will the population be 5.2 years from now?

1880

A savings account balance is compounded continuously. If the interest rate is 3.1% per year and the current balance is $1077.00, in how many years will the balance reach $1486.73?

10.4

A cup of coffee cools at rate proportional to the difference between the constant room temperature of 20.0°C and the temperature of the coffee. If the temperature of the coffee was 86.1°C 3.0 minutes ago and the current temperature of the coffee is 79.9°C, what will the temperature of the coffee be 29.0 minutes from now? Use Newton's Law of Cooling: dy/dx = k(T - Ta)

43.1

Radioactive isotope Carbon-14 decays at a rate proportional to the amount present. If the decay rate is 12.10% per thousand years and the current mass is 135.2 mg, what will the mass be 2.2 thousand years from now?

103.6

During the exponential phase, E. coli bacteria in a culture increase in number at a rate proportional to the current population. If the population doubles in 20.4 minutes, in how many minutes will the population triple?

32.3