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Exponential Growth and Decay

Classroom Resource Information

Title:

Exponential Growth and Decay

URL:

https://www.ck12.org/c/calculus/differential-equations-representing-growth-and-decay/lesson/Exponential-Growth-and-Decay-CALC/?referrer=concept_details

Content Source:

Other
CK-12

Type:

Informational Material

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.

Content Standard(s):

Mathematics
MA2019 (2019)
Grade: 9-12
Mathematical Modeling

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

a. Explain the relationship between annual percentage yield (APY) and annual percentage rate (APR) as values for r in the formulas A=P(1+r)^t and A=Pe^rt.

Unpacked Content

Evidence Of Student Attainment:

Students:

given information about an investment or loan involving compound interest or proportional depreciation, can identify the initial value and periodic rate of change.
Can apply a geometric sequence to generate data for the problem.
Can display the data in an organized manner and answer questions about the trends and results found.
Can explain the relationship between APR (annual percentage rate) and APY (annual percentage yield) using compound interest formulas.

Teacher 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.

Diverse Learning Needs:

Essential Skills:

Learning Objectives:
MMOD.4.1: Define geometric sequence, compound interest, proportional depreciation, frequent compounding, continuous compounding, annual percentage yield and annual percentage rate.
MMOD.4.2: Calculate proportional depreciation.
MMOD.4.3: Identify the formula for proportional depreciation.
MMOD.4.4: Calculate compound interest.
MMOD.4.5: Calculate simple interest.
MMOD.4.6: Compare compound and simple interest.
MMOD.4.7: Identify the formula to compute compound interest.
MMOD.4.8: Identify the formula to compute simple interest.

Prior Knowledge Skills:

Evaluate a function rule given the independent variable.
Define arithmetic and geometric sequence and input-output pairs.
Define sequences and recursively-defined sequences.
Recognize that sequences are functions whose domain is the set of all positive integers and zero
Calculate the common ratio of a geometric sequence.

Mathematics
MA2019 (2019)
Grade: 9-12
Precalculus

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.

a. Find the difference quotient ^{f(x+�"x)-f(x)}/_�"x of a function and use it to evaluate the average rate of change at a point.

b. Explore how the average rate of change of a function over an interval (presented symbolically or as a table) can be used to approximate the instantaneous rate of change at a point as the interval decreases.

Unpacked Content

Tags:

continuous compounding, decay, exponential, function, growth, mathematical modeling, precalculus, radioactive, rate of change

License Type:

Custom Permission Type
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Accessibility

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

Comments

This resource provided by:

Author:

Hannah Bradley

ALEX Classroom Resource

Exponential Growth and Decay