Exponential Functions

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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 describes 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 initially, slowing down over time.

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

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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 decay rate, and t is the time in years.

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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 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 2.718).

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What is the formula for continuous exponential decay?

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The formula for continuous exponential decay 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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If a population is decreasing at a rate of 1% per year, what is the decay factor?

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The decay factor is 0.99, which means the population retains 99% of its value each year.

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If a population is increasing at a rate of 50% per year, what is the growth factor?

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The growth factor is 1.5, which means the population increases to 150% of its value each year.

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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.