This video from MIT Open CourseWare on the CK-12 website will explore the meaning of exponential growth or decay by solving a differential equation that models such growth or decay.
This self-checking online assessment has 10 questions that will help students practice solving differential equations that represent exponential change. There are hints available on the screen, and there is an online scratchpad that students can use to work on the problems.
This self-checking online assessment has 10 questions that will help students practice calculating and interpreting the average rate of change of a function. There are hints available on the screen, and there is an online scratchpad that students can use to work the problems.
When the rate of change of the amount of a substance, or a population, is proportional to the amount present at any time, we say that this substance or population is going through either a decay or a growth, depending on the sign of the constant of proportionality. Do you know how to write a differential equation that expresses this condition? This kind of growth or decay, common in nature and in the business world, is called exponential growth or exponential decay and is characterized by rapid change.
This informational material will explain how to find solutions to differential equations that represent rapid change. It will explain real-life applications of these equations, such as radioactive decay and compound interest. There are corresponding videos available. Practice questions with a PDF answer key are provided.
This interactive will model exponential growth using the following scenario:
An ancient legend tells of a wise man who advises a king during a time of famine. As a reward for his help, the man asks the miserly king for grains of rice every day. He asks the king to put a single grain on the first square of a chessboard on the first day, two grains on the second square on the second day, double that amount on the third square on the third day, and so on. The amount of rice grows exponentially.
In this interactive, students will:
After engaging with this interactive, students should be able to explain the key features of exponential functions.
This informational material will explain average and instantaneous rates of change by using the average velocity and velocity at a point using the slope of tangents. The article includes many examples of graphs related to this concept. There are corresponding videos available. Practice questions with a PDF answer key are provided.
Students will test their knowledge of average and instantaneous rates of change represented graphically with the following scenario:
Here you are given two points, C and D, which are located along the function f(x) = x2. If you draw a line crossing those two points, you will create a secant line. Now suppose you take point D and move it closer and closer to point C. What happens to the average rate of change as the two points get closer to each other?
In the interactive, students will:
In 1979 China introduced its one-child policy. Communist leaders hoped to raise the average annual income to $1000 a person. They felt that the rising population was holding back China's economy. Today, China's rate of population growth has slowed and its economy has soared. Did the one-child policy cause the change?
This informational material will apply a precalculus concept--the rate of change of a function--to a current issue in sociology--patterns of population change. There are links to additional information included.
