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

Léon swims back and forth along a straight path in a 50-meter-long pool for 126 seconds. Léon's velocity is modeled by , where is measured in seconds and is measured in meters per second.

Find all times in the interval at which Léon changes direction. Give a reason for your answer.

Find Léon's acceleration at time seconds and indicate units of measure. Is Léon speeding up or slowing down at time 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 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 hours. (The volume of a cone with radius and height is .)

A young animal's weight, in kilograms, can be modeled by the function , where 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 , a particle moves along the -axis. The velocity of the particle at time is given by .

For , when is the particle moving to the left?

Find the acceleration of the particle at time . Is the speed of the particle increasing, decreasing, or neither at time​ ? Explain your reasoning.

The radius 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 is .)

At the time when the volume of the sphere is 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?

Particle moves along the -axis such that, for time , its velocity is given by .

Find all times , when the speed of the particle 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 be the distance between Plane A and the control tower at time , and let be the distance between Plane B and the control tower at time , as shown in the figure below.

Find the rate of change, in km/hr, of the distance between the two planes when km and km.

Let be the angle shown in the figure. Find the rate of change of , in radians per hour, when km and 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 is changing at the constant rate of cm/hr.

(The volume of a cone of height and radius is given by .)

Find the volume of water in the container when cm. Indicate units of measure.

Find the rate of change of the volume of water in the container, with respect to time, when 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 square centimeters, find the rate of increase of the area enclosed between the circle and the square.

(A circle with radius has circumference and area .)

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 and height has volume .)