Differential Equations

The rate of change of a quantity A is directly proportional to the square of A and inversely proportional to the time t. Which of the following differential equations represents this situation?

A scientist observes that, when the temperature $T$ of a substance is doubled, the rate of change in its mass $m$ with respect to the time $t$ is also doubled. Which of the following differential equations could represent the situation?

Given that a particle moves along a line so that its velocity at time t is given by $v(t)=3t^{2}-6t+5$, for what intervals of time t does it accumulate distance?

t seconds after a Calculus textbook is dropped from a building, the rate of change of the textbook’s position is proportional to the amount of time passed. If y(t) is the position of the textbook at time t, which of the following differential equations represents this situation?

The rate of change of a parachuter’s speed is the sum of two terms: a constant gravity that speeds up the parachuter, and air resistance that slows down the parachuter, which is proportional to the parachuter’s speed. Which of the following differential equations represents this situation?

Which technique should be used to solve the logistic differential equation $\frac{dP}{dt} = P(100-P)$ for population P(t), considering that Euler's method is not applicable here?

Given a logistic growth model $\frac{dP}{dt} = kP \left(1-\frac{P}{L}\right)$, where $k$ is a positive constant, $P$ is the population at time $t$, and $L$ is the carrying capacity, what would be the effect of doubling $k$ on the rate of growth when $P=\frac{L}{2}$?

If a function $f(x)$ is differentiable at $x = a$, which of the following must be true about $f$ at $x = a$?

The rate of change of the temperature of a cylindrical rod in Kelvin per second, $T$, is proportional to the radius of the cylinder in inches, $r$, and the height of the cylinder in inches, $h$. If the rate of change of the temperature is 60 Kelvin per second for a rod with radius 3 inches and height 2 inches, wh...

Which type of test gives information on whether a critical point is a local maximum or minimum for a differentiable function?