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 occurs when a quantity decreases at a rate proportional to its current value, leading to a decrease that slows over time.

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How do you calculate the future value of an investment with exponential growth?

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The future value can be calculated using the formula: FV = P(1 + r)^t, where P is the principal amount, r is the growth rate, and t is the time in years.

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What is the formula for exponential decay?

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The formula for exponential decay is: N(t) = N0 * e^(-kt), where N0 is the initial quantity, k is the decay constant, and t is time.

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What does the variable 'b' represent in the exponential function f(x) = a(b^x)?

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In the function f(x) = a(b^x), 'b' represents the base of the exponential function, which indicates the growth (b > 1) or decay (0 < b < 1) rate.

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How do you determine if a function represents exponential growth or decay?

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If the base 'b' in the function f(x) = a(b^x) is greater than 1, it represents exponential growth; if 'b' is between 0 and 1, it represents exponential decay.

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What is the continuous growth formula?

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The continuous growth formula is A = Pe^(rt), where A is the amount after time t, P is the initial amount, r is the growth rate, and e is Euler's number (approximately 2.71828).

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What is the continuous decay formula?

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The continuous decay formula is A = Pe^(-rt), where A is the amount after time t, P is the initial amount, r is the decay rate, and e is Euler's number.

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How can you model a situation with a percentage increase?

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You can model a situation with a percentage increase using the formula: y = a(1 + r)^t, where a is the initial amount, r is the percentage increase (as a decimal), and t is time.

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How can you model a situation with a percentage decrease?

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You can model a situation with a percentage decrease using the formula: y = a(1 - r)^t, where a is the initial amount, r is the percentage decrease (as a decimal), and t is time.