AP Classroom Unit 7 Calc

Let y=f(x)y=f(x) be the particular solution to the differential equation dydx=2x−1y2dydx=2x−1y2 with the initial condition y(0)=3y(0)=3. Which of the following is an expression for f(x)f(x) ?

(3x2−3x+27)^1/3

Let y=f(x)y=f(x) be the particular solution to the differential equation dydx=ex−1eydydx=ex−1ey with the initial condition f(1)=0f(1)=0. What is the value of f(−2)f(−2) ?

0.349

The amount of bacteria in a petri dish increases at a rate proportional to the amount present. At time t=0t=0, the amount of bacteria in the dish is 10 grams. At time t=2t=2, the amount of bacteria in the dish is 30 grams. What is the amount of bacteria in the dish at time t=6t=6 ?

270 grams

What is the general solution to the differential equation dy/dx=cos(8x)/cos(4y) ?

2sin(4y)−sin(8x)=C

Which of the following could be a slope field for the differential equation dydx=y3−xdydx=y3−x ?

B

The value of a car t years after it is purchased is given by the decreasing function V, where VV(t) is measured in dollars. The rate of change of the car's value in dollars per year is proportional to the car's value. Which of the following differential equations could be used to model the value of the car, where kk is a constant?

B. dVdt=kV

Of the following, which are solutions to the differential equation 4y+y′′=04y+y″=0 ? y=3sin(2x)y=3sin⁡(2x) y=5cos(2x)y=5cos⁡(2x) y=e2xy=e2x

I and II only

Which of the following is the solution to the differential equation dPdt+P=10 with the initial condition P(0)=4

P=10−6e^−t

As a glacier melts, the volume V of the ice, measured in cubic kilometers, decreases at a rate modeled by the differential equation dVdt=kVd, where t is measured in years. The volume of the glacier is 400km3 at time t=0. At the moment when the volume of the glacier is 300km3, the volume is decreasing at the rate of 15km33 per year. What is the volume VV in terms of time t

V=400e^−0.05t

Out of a total of NN students at a school, the number of students who have seen a new television program increases at a rate proportional to the product of the number of students who have seen the program and the number of students who have not seen the program. If SS denotes the number of students who have seen the program at time tt, which of the following differential equations could be used to model this situation, where kk is a positive constant?

dSdt=kS(N−S)

A hard-boiled egg is removed from a pot of hot water and set on the table to cool. The rate of change of the egg's temperature TT with respect to time tt is proportional to the difference between the egg's temperature in degrees Fahrenheit (°F°F) and the room temperature of 75°F75°F. Which of the following is a differential equation that describes this relationship, where kk is a constant?

dTdt=k(T−75)

The rate of change of the volume V with respect to time t of water leaking from a tank is proportional to the cube of the volume. Which of the following is a differential equation that could describe this relationship?

dVdt=−0.2V3

Shown above is a slope field for which of the following differential equations?

dy/dx=x−y+1

Shown above is the slope field for which of the following differential equations?

dy/dx=−xy

Which of the following is the particular solution to the differential equation dydx=sin(x2)dydx=sin(x2) with the initial condition y(π√)=4y(π)=4 ?

y=4+∫xπ√sin(t2) dt

Of the following, which is not a solution to the differential equation y′′′+4y′=0y‴+4y′=0 ?

y=4e−2x

Which of the following is a solution to the differential equation xy′−3y=6xy′−3y=6 ?

y=5x3−2

Let y=f(x)y=f(x) be the particular solution to the differential equation dydx=3x2−12ydydx=3x2−12y with the initial condition f(1)=4f(1)=4. Which of the following is an expression for f(x)f(x) ?

y=√x3−x+16