AP Calc AB Modeling with Differential Equations

Write a differential equation that describes each relationship. If necessary, use k as the constant of proportionality. The rate of change of Y with respect to w is directly proportional to the square of x.

• dY/dw = kx^2

• dY/dw = kx

• dY/dw = k/x^2

• dY/dw = k/x

Write a differential equation that describes each relationship. If necessary, use k as the constant of proportionality. The rate of change of S with respect to y is proportional to the square root of u and inversely proportional to v.

• dS/dy = k * sqrt(u) / v

• dS/dy = k * v / sqrt(u)

• dS/dy = k * u^2 / v

• dS/dy = k * v^2 / sqrt(u)

Write a differential equation that describes each relationship. If necessary, use k as the constant of proportionality. L is increasing with respect to x at a rate that is proportional to the cube root of m. The rate of change of L is 12 when m = 5.

• dL/dx = k * m^(1/3)

• dL/dx = k * m^(1/2)

• dL/dx = k * m^3

• dL/dx = k * m

Write a differential equation that describes each relationship. If necessary, use k as the constant of proportionality. The rate of change of U with respect to a is inversely proportional to the cube of v. The rate of change of U is –5 when v = 1/2.

• dU/da = k/v^3

• dU/da = -5v^3

• dU/da = -5/v^3

• dU/da = k*v^3

Write a differential equation that describes each relationship. If necessary, use k as the constant of proportionality. The height of a rocket is given by the function h(t), where t is measured in seconds since the launch and h is measured in meters. The acceleration is proportional to the cube root of the time since the start of the launch. At 12 seconds, the acceleration is 3 meters per second per second.

• d²h/dt² = k * t^(1/3)

• d²h/dt² = k * t^(3)

• d²h/dt² = k * t^(1/2)

• d²h/dt² = k * t^(2)

Write a differential equation that describes each relationship. If necessary, use k as the constant of proportionality. A scientist is studying the relationship of two quantities A and B in an experiment. The scientist finds that the quantity of A decreases and the quantity of B increases. The scientist determines that the rate of change of the quantity of A with respect to the quantity of B is inversely proportional to the square of the quantity of B.

• dA/dB = kB^2

• dA/dB = k/B^2

• dA/dB = kB

• dA/dB = k/B

Write a differential equation that describes each relationship. If necessary, use k as the constant of proportionality. The number of packets, p, Mr. Sullivan completes for Pre-Calculus is increasing as he nears the end of the school year. The rate of change of p with respect to time t is inversely proportional to the natural log of t.

• dp/dt = k * ln(t)

• dp/dt = k / ln(t)

• dp/dt = k * t

• dp/dt = k / t

Write a differential equation that describes each relationship. If necessary, use k as the constant of proportionality. Mr. Brust is running down his street. His position is given by the function p(t), where t is measured in minutes since the start of his run. His acceleration is inversely proportional to the cube of the time since the start of his run.

• a(t) = k/t^3

• a(t) = kt^3

• a(t) = k/t^2

• a(t) = kt^2