Exponential Function Real World Application

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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 shows rapid growth or decay.

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What does it mean for a function to exhibit growth?

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Growth in a function means that as the input (x) increases, the output (f(x)) also increases. In exponential functions, this is characterized by a base greater than 1.

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What does it mean for a function to exhibit decay?

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Decay in a function means that as the input (x) increases, the output (f(x)) decreases. In exponential functions, this is characterized by a base between 0 and 1.

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What is the formula for compound interest?

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The formula for compound interest is A = P(1 + r/n)^(nt), where A is the amount of money accumulated after n years, including interest, P is the principal amount, r is the annual interest rate (decimal), n is the number of times that interest is compounded per year, and t is the time in years.

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In the compound interest formula A = P(1 + r/n)^(nt), what does 'P' stand for?

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'P' stands for the principal amount, which is the initial sum of money invested or loaned.

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In the compound interest formula A = P(1 + r/n)^(nt), what does 'r' represent?

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'r' represents the annual interest rate expressed as a decimal.

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In the compound interest formula A = P(1 + r/n)^(nt), what does 'n' indicate?

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'n' indicates the number of times that interest is compounded per year.

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In the compound interest formula A = P(1 + r/n)^(nt), what does 't' represent?

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't' represents the time the money is invested or borrowed for, in years.

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How do you calculate the total amount after 15 years with a principal of $1,000 at 6.75% interest compounded semi-annually?

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Using the formula A = P(1 + r/n)^(nt), we find A = 1000(1 + 0.0675/2)^(2*15) = $2,706.86.

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If a population of mosquitoes grows at a rate of 25% per month, how do you calculate the population after 4 months starting with 4,000 mosquitoes?

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Use the formula for exponential growth: P = P0 * (1 + r)^t, where P0 = 4000, r = 0.25, and t = 4. Thus, P = 4000 * (1 + 0.25)^4 = 9765.