What is the growth rate of the function y=3*2^x?
Is the answer 1?
y(x) = 3*2^x
y(x+1) = 3*2^(x+1) = 3*2*2^x = 2y(x)
So, each year y doubles.
To find the growth rate of the function y = 3 * 2^x, we need to determine the rate at which y changes as x increases.
One way to do this is to calculate the derivative of the function with respect to x. The derivative represents the instantaneous rate of change of the function.
For the given function y = 3 * 2^x, we can take the derivative as follows:
dy/dx = (d/dx) (3 * 2^x)
To find the derivative of 2^x, we can use the rule that says the derivative of a function of the form a^x, where a is a constant, is equal to (ln(a)) * (a^x).
Therefore, the derivative of 2^x with respect to x is (ln(2)) * (2^x).
Now, applying the derivative rule for a constant times a function, we have:
dy/dx = (d/dx) (3 * 2^x)
= (3) * (d/dx) (2^x)
= (3) * [(ln(2)) * (2^x)]
Simplifying further, we have:
dy/dx = (3 * ln(2)) * (2^x)
The derivative of the function y = 3 * 2^x is (3 * ln(2)) * (2^x).
This derivative represents the growth rate of the function. In this case, it shows that as x increases, y grows at a rate of (3 * ln(2)) times the current value of y.
Therefore, the growth rate of the function y = 3 * 2^x is not 1.