AP Calculus AB: Chapter 11 Practice Test
This content covers solving various types of differential equations, including separable and second-order forms, applying initial conditions for particular solutions, using Euler’s method for numerical approximations, and modeling exponential population growth to predict future values.
Separate the variables of the equation: x^2ydydx=e^y
yeydy=1x2dx
Key Terms
Separate the variables of the equation: x^2ydydx=e^y
yeydy=1x2dx
Solve the following differential equation for the general solution: dy/dx=3x+2y2−1
32x2+2x=13y3−y+C
Evaluate the following as true of false.The equation xy+xxy−y=dydx is a separable differential equation.
true
Find the particular solution to dy/dx=x if y(2)=5.
y=12x2+3
Which of the following is not a solution of d2ydx2=6x?
y = x 3 + x 2
Use Euler’s method with step size 0.5 to compute the approximate y-value y (2) of the solution of the initial-value problem y′ = xy, y (0) = 2.
y(2)≈6.5625
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| Term | Definition |
|---|---|
Separate the variables of the equation: x^2ydydx=e^y | yeydy=1x2dx |
Solve the following differential equation for the general solution: dy/dx=3x+2y2−1 | 32x2+2x=13y3−y+C |
Evaluate the following as true of false.The equation xy+xxy−y=dydx is a separable differential equation. | true |
Find the particular solution to dy/dx=x if y(2)=5. | y=12x2+3 |
Which of the following is not a solution of d2ydx2=6x? | y = x 3 + x 2 |
Use Euler’s method with step size 0.5 to compute the approximate y-value y (2) of the solution of the initial-value problem y′ = xy, y (0) = 2. | y(2)≈6.5625 |
Use Euler’s method with step size 0.5 to compute the approximate y-value y (2) of the solution of the initial-value problem y′ = xy, y (0) = 2. | 480 |
The population of a colony of 300 bacteria grows exponentially. After 2 hours, the population reaches 500. How much time will it take for the population to reach 9,600? Give the answer to the nearest tenth of an hour. | 13.6 hours |
One thousand dollars is invested at 5% continuous annual interest. This means the value of the investment will grow exponentially, with k equaling the decimal rate of interest. What will the value of the investment be after 7 1/2 years? | $1,454.99 |
The half-life of iodine-126 (I 126 ) is 13 days. Of an original sample of 1,000.0 grams, how may grams of I 126 will remain after 98 days? Give the answer to the nearest tenth of a gram. | 5.4 g |
Solve the following differential equation for the general solution: dy/dx=e2x. | y=12e2x+C |
Suppose the fish population in a local lake increases at a yearly rate of 0.3 times the population at each moment. Which of the following differential equations describes the rate of change of the fish population in the lake? | dPdt=0.3P |