Exponential Growth and Decay

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What is exponential growth?

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Exponential growth occurs when the increase of a quantity is proportional to its current value, leading to growth at an increasing rate over time.

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What is exponential decay?

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Exponential decay is the process by which a quantity decreases at a rate proportional to its current value, resulting in a rapid decrease that slows over time.

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What is the formula for exponential growth?

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The formula for exponential growth is y = a(1 + r)^t, where 'a' is the initial amount, 'r' is the growth rate, and 't' is time.

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What is the formula for exponential decay?

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The formula for exponential decay is y = a(1 - r)^t, where 'a' is the initial amount, 'r' is the decay rate, and 't' is time.

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How do you calculate compound interest?

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Compound interest can be calculated using the formula A = P(1 + r/n)^(nt), where A is the amount of money accumulated after n years, P is the principal amount, r is the annual interest rate, n is the number of times that interest is compounded per year, and t is the number of years.

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What is the significance of the base in exponential functions?

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In exponential functions, the base determines the rate of growth or decay; a base greater than 1 indicates growth, while a base between 0 and 1 indicates decay.

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What does a 30% annual decay rate mean in terms of a population?

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A 30% annual decay rate means that each year, the population will retain 70% of its previous year's population.

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How do you interpret the value of 'x' in the equation y = a(0.7)^x?

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In the equation y = a(0.7)^x, 'x' represents the number of time periods (e.g., years) that have passed since the initial measurement.

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What is the impact of a 5% annual increase on an investment?

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A 5% annual increase means that the investment grows by 5% of its current value each year, leading to compound growth over time.

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How do you determine the future value of an investment using exponential growth?

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To determine the future value of an investment using exponential growth, apply the formula A = P(1 + r)^t, substituting the principal, growth rate, and time.