Exam Review Alg.2 Part 5

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What is continuous compounding in finance?

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Continuous compounding refers to the process of earning interest on an investment where the interest is calculated and added to the principal balance continuously, rather than at discrete intervals. The formula used is A = Pe^(rt), where A is the amount of money accumulated after n years, P is the principal amount, r is the annual interest rate, and t is the time in years.

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How do you calculate the future value of an investment with continuous compounding?

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To calculate the future value of an investment with continuous compounding, use the formula A = Pe^(rt), where A is the future value, P is the principal amount, r is the annual interest rate (as a decimal), and t is the time in years.

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What is exponential growth?

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Exponential growth occurs when the growth rate of a value is proportional to its current value, leading to growth that accelerates over time. It can be modeled with the function y = a(1 + r)^t, where y is the final amount, 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 y is the final amount, a is the initial amount, r is the decay rate, and t is time.

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How do you determine the population after a certain time with exponential growth?

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To determine the population after a certain time with exponential growth, use the formula P(t) = P0 * e^(rt), where P(t) is the population at time t, P0 is the initial population, r is the growth rate, and t is the time.

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What does a percentage decrease mean in terms of area or population?

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A percentage decrease indicates how much a quantity has reduced relative to its original amount. For example, a 6.5% decrease means that the new amount is 93.5% of the original amount.

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How do you calculate the area after a percentage decrease?

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To calculate the area after a percentage decrease, use the formula: New Area = Original Area * (1 - Percentage Decrease). For example, if the original area is 5000 km² and it decreases by 6.5%, the new area is 5000 * (1 - 0.065).

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What is the significance of the base 'e' in continuous growth models?

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The base 'e' (approximately 2.718) is a mathematical constant that is used in continuous growth models because it represents the limit of (1 + 1/n)^n as n approaches infinity, making it ideal for modeling growth processes that occur continuously.

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What is the difference between linear and exponential growth?

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Linear growth increases by a constant amount over time, while exponential growth increases by a percentage of the current value, leading to faster growth as time progresses.

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How do you express a growth rate as a decimal?

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To express a growth rate as a decimal, divide the percentage by 100. For example, a growth rate of 5% is expressed as 0.05.