Exponential Functions (Word Problems)

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What is an exponential function?

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An exponential function is a mathematical function of the form f(x) = a(b^x), where 'a' is a constant, 'b' is the base (a positive real number), and 'x' is the exponent. It represents growth or decay processes.

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What does it mean for a quantity to grow exponentially?

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Exponential growth occurs when a quantity increases by a fixed percentage over equal time intervals, leading to rapid increases as time progresses.

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What does it mean for a quantity to decay exponentially?

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Exponential decay occurs when a quantity decreases by a fixed percentage over equal time intervals, leading to rapid decreases as time progresses.

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How do you write an exponential growth equation?

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An exponential growth equation can be written as y = a(1 + r)^t, where 'a' is the initial amount, 'r' is the growth rate, and 't' is time.

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How do you write an exponential decay equation?

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An exponential decay equation can be written as y = a(1 - r)^t, where 'a' is the initial amount, 'r' is the decay rate, and 't' is time.

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What is the formula for calculating 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)^n, where FV is the future value, P is the principal amount, r is the rate of return, and n is the number of periods.

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What is the formula for calculating the remaining value of an asset with exponential decay?

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The remaining value can be calculated using the formula RV = P(1 - r)^n, where RV is the remaining value, P is the initial value, r is the depreciation rate, and n is the number of periods.

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If a population of bacteria doubles every hour, how would you express this mathematically?

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This can be expressed as N(t) = N0(2^t), where N0 is the initial population and t is the time in hours.

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What is the significance of the base in an exponential function?

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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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How do you determine the growth rate from an exponential function?

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The growth rate can be determined from the equation y = a(b^x) by calculating (b - 1) * 100%.