Exponential Functions - Average ROC

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 a positive real number, and x is the exponent. It describes growth or decay processes.

Front

What does ROC stand for in mathematics?

Back

ROC stands for Rate of Change, which measures how a quantity changes in relation to another quantity.

Front

How do you calculate the average rate of change of a function over an interval [a, b]?

Back

The average rate of change is calculated using the formula: \( \frac{f(b) - f(a)}{b - a} \).

Front

What is the average rate of change between x = -1 and x = 0 for the function f(x) = 2^x?

Back

The average rate of change is \( \frac{f(0) - f(-1)}{0 - (-1)} = \frac{1 - 0.5}{1} = 0.5 \).

Front

If a function f represents the number of trading cards, what does a negative average rate of change indicate?

Back

A negative average rate of change indicates that the number of trading cards is decreasing over the specified interval.

Front

What is the average rate of change from day 5 to day 15 if f(5) = 528 and f(15) = 1283?

Back

The average rate of change is \( \frac{1283 - 528}{15 - 5} = \frac{755}{10} = 75.5 \).

Front

What does it mean if the average rate of change is -143.3 cards/day?

Back

It means that, on average, the store is losing 143.3 popular trading cards each day.

Front

How do you interpret the average rate of change in a real-world context?

Back

The average rate of change can be interpreted as the change in a quantity per unit of time or another variable, providing insight into trends over that interval.

Front

What is the average rate of change of the function f(x) = 22000(0.9)^x between x = 0 and x = 4?

Back

The average rate of change is approximately -$1,891.45/year.

Front

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

Back

The base determines the growth or decay rate of the function. A base greater than 1 indicates growth, while a base between 0 and 1 indicates decay.