What is the rate of increase for the function [tex]f(x)=\frac{1}{3}(\sqrt[3]{24})^{2 x}[/tex]?
Answer (1)
Rewrite the function f ( x ) = 3 1 ( 3 24 ) 2 x in the form a ⋅ b x .
Simplify the expression ( 3 24 ) 2 x as ( 2 4 3 1 ) 2 x = 2 4 3 2 x = ( 2 4 3 2 ) x .
Calculate 2 4 3 2 = ( 2 3 ⋅ 3 ) 3 2 = 2 2 ⋅ 3 3 2 = 4 ⋅ 3 3 2 = 4 ⋅ 3 9 .
The rate of increase is the base of the exponential function, which is 4 3 9 .
Explanation
Problem Analysis We are given the function f ( x ) = 3 1 ( 3 24 ) 2 x and asked to find its rate of increase. The rate of increase of an exponential function f ( x ) = a ⋅ b x is given by the base b . Therefore, we need to rewrite the given function in this form.
Simplifying the Expression First, let's simplify the expression ( 3 24 ) 2 x . We can rewrite this as ( 2 4 3 1 ) 2 x = 2 4 3 2 x = ( 2 4 3 2 ) x .
Calculating the Base Now, let's calculate 2 4 3 2 . We can rewrite 24 as 2 3 ⋅ 3 . Therefore, 2 4 3 2 = ( 2 3 ⋅ 3 ) 3 2 = ( 2 3 ) 3 2 ⋅ 3 3 2 = 2 3 ⋅ 3 2 ⋅ 3 3 2 = 2 2 ⋅ 3 3 2 = 4 ⋅ 3 3 2 = 4 ⋅ 3 3 2 = 4 ⋅ 3 9 .
Finding the Rate of Increase So, we have f ( x ) = 3 1 ( 4 ⋅ 3 9 ) x . The rate of increase is the base of the exponential function, which is 4 3 9 .
Final Answer Therefore, the rate of increase for the function f ( x ) = 3 1 ( 3 24 ) 2 x is 4 3 9 .
Examples
Understanding the rate of increase of a function is crucial in various real-world applications. For instance, in finance, if you invest money in an account with a growth function similar to the one in this problem, knowing the rate of increase helps you predict how quickly your investment will grow. If the function represents the growth of a bacterial colony, the rate of increase can help estimate how rapidly the colony expands, which is vital in medical and environmental studies. In general, exponential growth models are used to describe phenomena that increase at a constant percentage rate over time, making the rate of increase a key parameter for analysis and prediction.
