Exponential Growth and Decay

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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 the value increasing rapidly over time. It can be represented by the formula A = a(1 + r)^t, where A is the amount after time t, a is the initial amount, r is the growth rate, and t is time.

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What is exponential decay?

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Exponential decay is the process of reducing an amount by a consistent percentage rate over a period of time. It can be represented by the formula A = a(1 - r)^t, where A is the amount after time t, a is the initial amount, r is the decay rate, and t is time.

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What is the general formula for exponential functions?

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The general formula for exponential functions is y = a(b)^x, where a is the initial value, b is the base (growth factor if b > 1, decay factor if 0 < b < 1), and x is the exponent.

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How can you identify exponential growth from a function?

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Exponential growth can be identified if the base of the exponent is greater than 1 (b > 1) in the function y = a(b)^x.

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How can you identify exponential decay from a function?

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Exponential decay can be identified if the base of the exponent is between 0 and 1 (0 < b < 1) in the function y = a(b)^x.

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What does the initial value represent in an exponential function?

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The initial value in an exponential function represents the starting amount before any growth or decay occurs, denoted by 'a' in the formula y = a(b)^x.

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What is the significance of the growth rate in exponential functions?

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The growth rate (r) in exponential functions determines how quickly the value increases over time. A higher growth rate results in a steeper increase.

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What is the difference between linear and exponential functions?

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Linear functions increase by a constant amount over equal intervals, while exponential functions increase by a percentage of the current value, leading to rapid growth.

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What is the effect of compounding in exponential growth?

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Compounding in exponential growth means that the growth is applied to the accumulated amount, leading to faster increases as time progresses.

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What is the formula for continuous exponential growth?

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The formula for continuous exponential growth 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).