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Understanding Linear and Exponential Growth Patterns: Key Differences and Insights

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mikkel
7 diapositives · 280 Views · Sep 01, 2014

In this lesson, we explore two fundamental growth patterns: linear and exponential growth. Linear growth occurs when a quantity increases by a consistent absolute amount over time, exemplified by a steady rise in students over three years. In contrast, exponential growth involves a consistent relative increase, such as the price of milk rising due to inflation. We also discuss the concepts of linear and exponential decrease. Finally, we highlight key facts about exponential growth, including repeated doublings and the limits of indefinite growth.

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Understanding Linear and Exponential Growth Patterns: Key Differences and Insights
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  1. Unit 8A Growth: Linear versus Exponential

  2. TWO BASIC GROWTH PATTERNS • We are going to consider two basic growth patterns: • Linear growth • Exponential growth

  3. LINEAR VERSUS EXPONENTIAL GROWTH

  4. LINEAR GROWTH Linear growth occurs when a quantity grows by the same absolute amount to each unit of time. EXAMPLE: The number of students taking Math 1001 has increased by 30 in each of the past three years. If the number of students taking Math 1001 was 150 three years ago, what is it this year?

  5. EXPONENTIAL GROWTH Exponential growth occurs when a quantity grows by the same relative amount—that is, by the same percentage—in each unit of time. EXAMPLE: The price of milk has been rising with inflation at 3% per year. If the price of a gallon of milk is now $2.00, and the inflation rate of 3% continues, what will be the price of a gallon of milk (a) two years from now and (b) ten years from now? How long will it take for the price of a gallon of milk to (a) double, (b) triple, and (c) quadruple?

  6. LINEAR AND EXPONENTIAL DECREASE The terms linear and exponential can also be applied to quantities that decrease with time. Other terms for decrease are decline and decay. • Linear decrease (or linear decay) occurs when a quantity decreases by the same absolute amount in each unit of time. • Exponential decay or (exponential decline) occurs when a quantity decreases by the same relative amount (same percentage) in each unit of time.

  7. KEY FACTS ABOUT EXPONENTIAL GROWTH • Exponential growth leads to repeated doublings. With each doubling, the amount of increase is approximately equal to the sum of all the preceding doublings. • Exponential growth cannot continue indefinitely. After only a relative small number of doublings, exponential growing quantities reach impossible proportions.