Log/Exp Test Review (2025)

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What is continuous compounding?

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Continuous compounding is the process of earning interest on an investment where the interest is calculated and added to the principal continuously, rather than at discrete intervals. The formula used is A = Pe^(rt), where A is the amount of money accumulated after time t, P is the principal amount, r is the annual interest rate, and t is the time in years.

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How do you calculate the average rate of return needed for continuous compounding?

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To calculate the average rate of return needed for continuous compounding, use the formula: r = (ln(A/P))/t, where A is the future value, P is the present value, and t is the time in years.

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What is the formula for continuous growth?

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The formula for continuous growth is N(t) = N0 * e^(rt), where N(t) is the amount at time t, N0 is the initial amount, r is the growth rate, and t is time.

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What does the variable 'r' represent in the context of continuous compounding?

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'r' represents the annual interest rate or growth rate expressed as a decimal in the context of continuous compounding.

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If you want to double your investment with continuous compounding, what is the approximate rate of return needed?

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To double your investment, you need an approximate rate of return of 69.3% per year, based on the rule of 70 (70/r = doubling time in years).

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What is the natural logarithm and how is it used in continuous compounding?

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The natural logarithm (ln) is the logarithm to the base e (approximately 2.718). It is used in continuous compounding to solve for time or rate in the formula A = Pe^(rt).

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How do you interpret a growth rate of 21% in a population model?

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A growth rate of 21% means that the population is increasing by 21% of its current size each year, leading to exponential growth over time.

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What is the significance of the time variable 't' in continuous growth equations?

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The time variable 't' represents the duration over which growth occurs, affecting the total amount of growth experienced in a continuous growth model.

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How can you determine the time it takes for a population to reach a certain size using continuous growth?

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To determine the time it takes for a population to reach a certain size, rearrange the continuous growth formula to solve for t: t = (ln(N(t)/N0))/r.

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What is the difference between discrete and continuous compounding?

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Discrete compounding calculates interest at specific intervals (e.g., annually, semi-annually), while continuous compounding calculates interest at every possible moment, leading to slightly higher returns.