Rate of Change

AP Chemistry

AP Calculus AB/BC - 2.3 Estimating Derivatives of a Function at a Point

AP Calculus AB/BC - 4.5 Solving Related Rates Problems

AP Calculus AB/BC - 5.12 Exploring Behaviors of Implicit Relations

AP Calculus AB/BC - 7.1 Modeling Situations with Differential Equations

AP Calculus AB/BC - Unit 6 Overview: Integration and Accumulation of Change

You are using calculus to model a population of pandas. In your model the first derivative represents the rate of change of the population with respect to time. What are the units of the first derivative?

The volume of a sphere is given by V = (4/3)πr^3. If the radius is increasing at a rate of 4 cm/s, what is the rate of change of the volume with respect to time when the radius is 3 cm?

A marathon runner is jogging around a circular track with a radius of 60 meters at a constant speed of 4 m/s, while her running partner stands still at the starting point. Exactly 10π seconds after she leaves the starting point, what is the rate of change in the distance between the runner and her partner?

The length of a rectangular prism is increasing at a rate of 6 in/s, the width is increasing at a rate of 4 in/s, and the height is decreasing at a rate of 2 in/s. If the length is 8 in, the width is 6 in, and the height is 4 in, what is the rate of change of the volume with respect to time?

A policeman is 60 feet away from an intersection and runs toward it at a speed of 10 ft/s. A robber is running away from the intersection on a perpendicular street; he is 80 feet away and runs at 5 ft/s. What is the rate of change of the straight-line distance between the policeman and the robber?

An ant is moving on the coordinate plane. When the ant is at point (x, y), the angle of the ant is defined as arctan(y/x). If the ant is currently located at the point (5, 5), its x-coordinate is changing at a rate of -2 units/second, and its y-coordinate is changing at the rate of 4 units/second, what is the rate of change of the angle of the ant?

What does the concavity of a function indicate about its rate of change?

As a snowball melts, the rate of change of its mass is inversely proportional to the square of its mass at a given time. If m(t) is the mass of the snowball at time t, which of the following differential equations can represent this situation?

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 the height h of a tree with respect to time t is directly proportional to the natural logarithm of h. Which of the following differential equations represents this situation, where k is a proportionality constant?

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?

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?

The amount of money in a savings account is given by the function A(t), where t is measured in years and A is measured in dollars. The rate of change of money is proportional to the square root of the current amount of money. When the account contains $10,000, the amount of money is increasing at $1,000/year. What is the differential equation that represents this situation?

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, what is the differential equation that represents this situation?

The rate of change of the surface area of a sphere with respect to time is proportional to the square of the radius. At t = 0, the radius is 2 cm, and the rate of change of the surface area is 16 cm^2/s. If A(t) is the surface area of the sphere, what is the differential equation that represents this situation?

Given a population that grows at a continuous rate of 3% per year, which of the following represents the rate of change of the population at any given time t?