Exponential Functions and Growth

Front

What is an exponential function?

Back

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.

Front

Define exponential growth.

Back

Exponential growth occurs when the growth rate of a value is proportional to its current value, leading to a rapid increase over time. It can be modeled by the function f(t) = a * e^(kt), where 'k' is a positive constant.

Front

What is the formula for calculating future value in exponential growth?

Back

The formula is: Future Value = Present Value * (1 + r)^t, where 'r' is the growth rate and 't' is the time period.

Front

What is exponential decay?

Back

Exponential decay is the process where a quantity decreases at a rate proportional to its current value, often modeled by the function f(t) = a * e^(-kt), where 'k' is a positive constant.

Front

How do you calculate the remaining amount in exponential decay?

Back

The remaining amount can be calculated using the formula: Remaining Amount = Initial Amount * (1 - r)^t, where 'r' is the decay rate and 't' is the time period.

Front

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

Back

The base 'b' in an exponential function determines the rate of growth or decay. If b > 1, the function represents growth; if 0 < b < 1, it represents decay.

Front

Evaluate the function f(x) = 3(2)^x when x = 3.

Back

f(3) = 3(2)^3 = 3 * 8 = 24.

Front

If a population of 500 grows at a rate of 10% per year, what will be the population after 5 years?

Back

Population = 500 * (1 + 0.10)^5 = 500 * 1.61051 ≈ 805.

Front

What does it mean if a function is described as exponential growth?

Back

It means that the function increases rapidly as the input increases, typically doubling at regular intervals.

Front

What is the half-life in the context of exponential decay?

Back

Half-life is the time required for a quantity to reduce to half its initial value, commonly used in radioactive decay.