Exponential Modeling

Front

What is exponential growth?

Back

Exponential growth occurs when a quantity increases by a constant percentage over a period of time, resulting in a rapid increase. For example, if a population grows by 5% each year, it will grow faster each subsequent year.

Tags

CCSS.HSF-LE.A.1A

Front

What is exponential decay?

Back

Exponential decay occurs when a quantity decreases by a constant percentage over a period of time, leading to a rapid decrease. For example, if a radioactive substance decays at a rate of 10% per year, it will lose 10% of its remaining amount each year.

Tags

CCSS.HSF-IF.C.8B

Front

How do you calculate future value with compound interest?

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The formula for future value with compound interest is: \( A = P(1 + \frac{r}{n})^{nt} \) where A is the amount of money accumulated after n years, including interest, P is the principal amount, r is the annual interest rate, n is the number of times that interest is compounded per year, and t is the number of years.

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What is the formula for continuous compounding?

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The formula for continuous compounding 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 e is the base of the natural logarithm.

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If a population of 1000 increases by 20% each year, what will be the population after 3 years?

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Using the formula for exponential growth: \( A = P(1 + r)^t \), we have \( A = 1000(1 + 0.20)^3 = 1000(1.728) \approx 1728 \).

Tags

CCSS.HSF-LE.A.1C

Front

What is the difference between simple interest and compound interest?

Back

Simple interest is calculated only on the principal amount, while compound interest is calculated on the principal plus any interest that has already been added. This means compound interest grows faster than simple interest.

Tags

CCSS.HSF.BF.A.2

Front

How do you determine if a function represents exponential growth or decay?

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If the base of the exponential function is greater than 1 (e.g., \( f(x) = a(1 + r)^x \)), it represents exponential growth. If the base is between 0 and 1 (e.g., \( f(x) = a(1 - r)^x \)), it represents exponential decay.

Tags

CCSS.HSF-IF.C.8B

Front

What is the formula for calculating the population after a certain number of months with a growth rate?

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The formula is: \( P = P_0(1 + r)^t \) where \( P_0 \) is the initial population, r is the growth rate per time period, and t is the number of time periods.

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If a bank offers 5% interest compounded monthly, how do you calculate the effective annual rate (EAR)?

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The formula for EAR is: \( EAR = (1 + \frac{r}{n})^{nt} - 1 \) where r is the nominal interest rate, n is the number of compounding periods per year, and t is the number of years.

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What is the significance of the number e in continuous compounding?

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The number e (approximately 2.718) is the base of the natural logarithm and is used in continuous compounding to represent the limit of (1 + 1/n)^n as n approaches infinity.

Tags

CCSS.HSF-IF.C.8B