Question

The number of bacteria in a culture after x days is represented by the function f(x)=x4−3x2
. What function represents the average number of bacteria in the culture after x days?

Enter the missing values.

(1 point)
(fg)(x)=
$$

Answer 1

To find the function that represents the average number of bacteria in the culture after \( x \) days, we need to divide the total number of bacteria \( f(x) \) over the time period. The average number of bacteria over the interval from day 0 to day \( x \) can be represented as follows:

\[
\text{Average Number of Bacteria} = \frac{1}{x} \int_0^x f(t) \, dt
\]

where \( f(t) = t^4 - 3t^2 \).

So, we first need to compute the integral \( \int_0^x (t^4 - 3t^2) \, dt \):

1. Compute the integral:

\[
\int (t^4 - 3t^2) \, dt = \frac{t^5}{5} - t^3 + C
\]

2. Now evaluate the definite integral from 0 to \( x \):

\[
\int_0^x (t^4 - 3t^2) \, dt = \left[ \frac{t^5}{5} - t^3 \right]_0^x = \left( \frac{x^5}{5} - x^3 \right) - \left( 0 - 0 \right) = \frac{x^5}{5} - x^3
\]

3. Now plug this into the average formula:

\[
\text{Average Number of Bacteria} = \frac{1}{x} \left( \frac{x^5}{5} - x^3 \right) = \frac{x^5}{5x} - \frac{x^3}{x} = \frac{x^4}{5} - x^2
\]

Therefore, the average number of bacteria after \( x \) days is represented by the function:

\[
A(x) = \frac{x^4}{5} - x^2
\]

Thus, the final answer for \( (fg)(x) \) which typically denotes the average in this context is:

\[
(fg)(x) = \frac{x^4}{5} - x^2
\]