Exponential Growth and Decay

Learning Resource Type

Classroom Resource

Subject Area

Mathematics

Grade(s)

9, 10, 11, 12

Overview

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.   

Mathematics (2019) Grade(s): 09-12 - Mathematical Modeling

MA19.MM.4

Organize and display financial information using geometric sequences to represent compound interest and proportional depreciation, including periodic (yearly, monthly, weekly) and continuous compounding.

UP:MA19.MM.4

Vocabulary

  • Compound Interest
  • Geometric Sequence
  • Proportional Depreciation
  • Periodic
  • Annual Percentage Rate
  • Annual Percentage Yield

Knowledge

Students know:
  • how to select information from a real-world financial problem, such as the initial amount of the investment and its periodic rate of change, and use it along with a geometric sequence to model compound interest and proportional depreciation.

Skills

Students are able to:
  • Identify the first term and common ratio in a geometric sequence.
  • Recognize that the first term of a geometric sequence is the initial value of the loan or investment.
  • Recognize that the common ratio is either (1+rate of growth) or (1-Rate of decay).
  • Use a geometric sequence to model compound interest or proportional depreciation.
  • Display data found using a geometric sequence to model compound interest or proportional depreciation. Relate APR to APY using compound interest formulas.

Understanding

Students understand that:
  • the initial amount of an investment or a loan and its periodic rate of change correlate to the first term and the common difference in a geometric sequence.
  • Geometric sequences can be used to model compound interest and proportional depreciation.
  • The annual percentage rate is the yearly rate of interest while the annual percentage yield is the rate you actually pay when compound interest is included.
Mathematics (2019) Grade(s): 09-12 - Precalculus

MA19.PRE.25

Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Extend from polynomial, exponential, logarithmic, and radical to rational and all trigonometric functions.

UP:MA19.PRE.25

Vocabulary

  • Average rate of change
  • Specified interval
  • Difference Quotient

Knowledge

Students know:
  • Techniques for graphing.
  • Techniques for finding a rate of change over an interval on a table or graph.
  • Techniques for estimating a rate of change over an interval on a graph.

Skills

Students are able to:
  • Calculate average rate of change on a specified interval when given an equation or table ofa polynomial, exponential, logarithmic, and radical to rational and all trigonometric functions.
  • Interpret the average rate of change of a polynomial, exponential, logarithmic, and radical to rational and all trigonometric functions in the context of a problem when given symbolic representations, tables, graphs, or contextual situations.
  • Estimate the average rate of change for a specific interval of a polynomial, exponential, logarithmic, and radical to rational and all trigonometric functions functions when given agraph.

Understanding

Students understand that:
  • The rate of change is the ratio of the change between the dependent and independent variable.

CR Resource Type

Informational Material

Resource Provider

CK-12
https://www.ck12.org/c/calculus/differential-equations-representing-growth-and-…

Custom License Type (URL)

https://www.ck12info.org/terms-of-use/

Accessibility

Text Resources: Content is organized under headings and subheadings
Video resources: includes closed captioning or subtitles

Approved Date

Owner (Author)

habradley
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