Rates of Change & Related Rates (College Board AP® Calculus AB): Exam Questions

Léon swims back and forth along a straight path in a 50-meter-long pool for 126 seconds. Léon's velocity is modeled by $v\left(t\right)=1.957e^{-0.015t}\sin \left(\frac{\pi }{63}t\right)$, where $t$ is measured in seconds and $v\left(t\right)$ is measured in meters per second.

Find all times $t$ in the interval $0<t<126$ at which Léon changes direction. Give a reason for your answer.

Find Léon's acceleration at time $t=10$ seconds and indicate units of measure. Is Léon speeding up or slowing down at time $t=10$ seconds? Give a reason for your answer.

The height of a cone increases at a rate of 3 centimeters per hour whilst the radius increases at a rate of 1 centimeter per hour. At time $t=4$ hours, the radius is 300 centimeters and the height is 100 centimeters. Find the rate of change of the volume of the cone with respect to time in cubic centimeters per hour, at time $t=4$ hours. (The volume $V$ of a cone with radius $r$ and height $h$ is $V=\frac{1}{3}\pi r^{2}h$.)

A young animal's weight, in kilograms, can be modeled by the function $W\left(L\right)=\frac{8L}{10+L}$, where $L$ is the animal's length in centimeters. When the animal weighs 4 kilograms, its length is increasing at a rate of 3 centimeters per month.

According to this model, what is the rate of change of the animal's weight with respect to time, in kilograms per month, at the time when the animal is 4 kilograms?

For $0\le t\le 12$, a particle moves along the $x$-axis. The velocity of the particle at time $t$ is given by $v\left(t\right)=\sin \left(\frac{\pi }{5}t\right)$.

For $0\le t\le 12$, when is the particle moving to the left?

Find the acceleration of the particle at time $t=\frac{5}{6}$. Is the speed of the particle increasing, decreasing, or neither at time​ $t=\frac{5}{6}$? Explain your reasoning.

The radius $r$ of a sphere is increasing at a constant rate of 0.05 centimeters per second.

At the time when the radius of the sphere is 3 centimeters, what is the rate of increase of its volume?

(The volume of a sphere with radius $r$ is $V=\frac{4}{3}\pi r^3$.)

At the time when the volume of the sphere is $\frac{500\pi }{3}$ cubic centimeters, what is the rate of increase of the area of a cross section through the center of the sphere?

At the time when the volume and the radius of the sphere are increasing at the same numerical rate, what is the radius?

The density of a bacteria population in a circular petri dish at a distance $r$ centimeters from the center of the dish is given by an increasing, differentiable function $f$, where $f\left(r\right)$ is measured in milligrams per square centimeter. Values of $f\left(r\right)$ for selected values of $r$ are given in the table below.

Use the data in the table to estimate $f\text{'}(2.25)$. Using correct units, interpret the meaning of your answer in the context of this problem.

A particle moves along the $x$-axis so that its velocity at time $t\ge 0$ is given by

$v\left(t\right)=\ln (t^2-4t+5)-0.2t$

There is one time, $t=t_R$, in the interval $0<t<2$ when the particle is at rest (not moving). Find $t_R$. For $0<t<t_R$, is the particle moving to the right or to the left? Give a reason for your answer.

Find the acceleration of the particle at time $t=1.5$. Show the setup for your calculations. Is the speed of the particle increasing or decreasing at time $t=1.5$? Explain your reasoning.

For time $t\ge 0$, a particle is moving along another curve defined by the equation $y^3+2xy=24$. At the instant the particle is at the point $(4,2)$, the $y$-coordinate of the particle's position is decreasing at a rate of 2 units per second. At that instant, what is the rate of change of the $x$-coordinate of the particle's position with respect to time?

Particle $P$ moves along the $x$-axis such that, for time $t>0$, its position is given by $x_P\left(t\right)=6-4e^{-t}$. Particle $Q$ moves along the $y$-axis such that, for time $t>0$, its velocity is given by $v_Q\left(t\right)=\frac{1}{t^2}$. At time $t=1$, the position of particle $Q$ is $y_Q\left(1\right)=2$.

Find $v_P\left(t\right)$, the velocity of particle $P$ at time $t$.

Find $a_Q\left(t\right)$, the acceleration of particle $Q$ at time $t$. Find all times $t$, for $t>0$, when the speed of particle $Q$ is decreasing. Justify your answer.

The height of the tree, in meters, can also be modeled by the function $G$, given by $G\left(x\right)=\frac{100x}{1+x}$, where $x$ is the diameter of the base of the tree, in meters. When the tree is 50 meters tall, the diameter of the base of the tree is increasing at a rate of 0.03 meter per year. According to this model, what is the rate of change of the height of the tree with respect to time, in meters per year, at the time when the tree is 50 meters tall?

For $t\ge 0$, a particle moves along the $x$-axis. The velocity of the particle at time $t$ is given by

$v\left(t\right)=1+2\text{sin}\left(\frac{t^2}{2}\right)$

The particle is at position $x=2$ at time $t=4$.

At time $t=4$, is the particle speeding up or slowing down? Give a reason for your answer.

Find all times $t$ in the interval $0<t<3$ when the particle changes direction. Justify your answer.

A cylindrical barrel with a diameter of 2 feet contains collected rainwater, as shown in the figure below. The water drains out through a valve (not shown) at the bottom of the barrel. The rate of change of the height $h$ of the water in the barrel with respect to time $t$ is modelled by

$\frac{dh}{dt}=-\frac{1}{10}\sqrt{h}$

where $h$ is measured in feet and $t$ is measured in seconds. The volume $V$ of a cylinder with radius $r$ and height $h$ is $V=\pi r^{2}h$.

Find the rate of change of the volume of water in the barrel with respect to time when the height of the water is 4 feet. Indicate units of measure.

Two particles move along the $x$-axis. For $0\le t\le 8$, the position of particle $P$ at time $t$ is given by $x_P\left(t\right)=\text{ln}(t^2-2t+10)$, while the velocity of particle $Q$ at time $t$ is given by $v_Q\left(t\right)=t^2-8t+15$. Particle $Q$ is at position $x=5$ at time $t=0$.

For $0\le t\le 8$, when is particle $P$ moving to the left?

For $0\le t\le 8$, find all times $t$ during which the two particles travel in the same direction.

Find the acceleration of particle $Q$ at time $t=2$. Is the speed of particle $Q$ increasing, decreasing, or neither at time $t=2$? Explain your reasoning.

Particle $Q$ moves along the $y$-axis such that, for time $0<t<2\pi$, its velocity is given by $v_Q\left(t\right)=3\sin \left(2t\right)$.

Find all times $t$, when the speed of the particle $Q$ is decreasing. Justify your answer.

Planes A and B are flying at the same, constant altitude.

Plane A is flying due east toward a control tower at a speed of 400 kilometers per hour (km/hr). Plane B is flying due south away from the same control tower at a speed of 300 km/hr.

Let $x$ be the distance between Plane A and the control tower at time $t$, and let $y$ be the distance between Plane B and the control tower at time $t$, as shown in the figure below.

Find the rate of change, in km/hr, of the distance between the two planes when $x=12$ km and $y=5$ km.

Let $\theta$ be the angle shown in the figure. Find the rate of change of $\theta$, in radians per hour, when $x=12$ km and $y=5$ km.

A container has the shape of an open right circular cone, as shown in the figure below. The height of the container is 20 cm and the diameter of the opening is 10 cm. Water in the container is evaporating so that its depth $h$ is changing at the constant rate of $-\frac{4}{5}$ cm/hr.

(The volume of a cone of height $h$ and radius $r$ is given by $V=\frac{1}{3}\pi r^{2}h$.)

Find the volume $V$ of water in the container when $h=10$ cm. Indicate units of measure.

Find the rate of change of the volume of water in the container, with respect to time, when $h=10$ cm. Indicate units of measure.

Show that the rate of change of the volume of water in the container due to evaporation is directly proportional to the exposed surface area of the water. What is the constant of proportionality?

A circle is inscribed in a square as shown in the figure below. The circumference of the circle is increasing at a constant rate of 6 centimeters per second. As the circle expands, the square expands so that the sides of the square are always tangents to the circle.

At the instant when the area of the circle is $36\pi$ square centimeters, find the rate of increase of the area enclosed between the circle and the square.

(A circle with radius $r$ has circumference $C=2\pi r$ and area $A=\pi r^2$.)

A right-circular cone has a radius and height which can vary, whilst its volume remains constant. For the point in time where the radius is the same length as the height, find the rate at which the radius is changing with respect to the height.

(A cone with radius $r$ and height $h$ has volume $V=\frac{1}{3}\pi r^{2}h$.)

Two particles, $H$ and $J$, are moving along the $x$-axis. For $0\le t\le 5$, the position of particle $H$ at time $t$ is given by $x_H\left(t\right)={\text{e}}^{t^2-4t}$ and the velocity of particle $J$ at time $t$ is given by $v_J\left(t\right)=2t{\left(t^2-1\right)}^3$.

Find the velocity of particle $H$ at time $t=1$. Show the work that leads to your answer.

During what open intervals of time $t$, for $0<t<5$, are particles $H$ and $J$ moving in opposite directions? Give a reason for your answer.

It can be shown that $v_J\text{'}\left(2\right)>0$. Is the speed of particle $J$ increasing, decreasing, or neither at time $t=2$? Give a reason for your answer.

A particle moves along the curve $H$ defined by the equation $2xy+\text{ln}y=8$. At the instant when the particle is at the point $\left(4,1\right)$, $\frac{\text{d}x}{\text{d}t}=3$. Find $\frac{\text{d}y}{\text{d}t}$ at that instant. Show the work that leads to your answer.