Question

2. The functions ???? and ???? represent the population of two different kinds of bacteria, where x is the time (in hours) and ???? and ???? are the number of bacteria (in thousands).

????(x) = 2x^2 + 7 and ????(x) = 2x.
a. Between the third and sixth hour, which bacteria had a faster rate of growth?
b. Will the population of ???? ever exceed the population of ????? If so, at what hour?

Answer 1

(a) Bacterial 1 had a faster rate of growth.

(b) The population of f(x) always exceed the population of g(x). In other words, population of g(x) cannot exceed the population of f(x).

Explanation:

Consider the given functions are

`f(x)=2x^2+7`

`g(x)=2x`

where, x is the time (in hours) and f(x) and g(x) are the number of bacteria (in thousands).

(a)

The rate of change of a function f(x) on [a,b] is

`m=(f(b)-f(a))/(b-a)`

Rate of change between third and sixth hour of first function is

`m_1=(f(6)-f(3))/(6-3)`

`m_1=((2(6)^2+7)-(2(3)^2+7))/(6-3)`

`m_1=(79-25)/(3)`

`m_1=(54)/(3)`

`m_1=18`

Rate of change between third and sixth hour of second function is

`m_2=(g(6)-g(3))/(6-3)`

`m_2=(2(6)-2(3))/(6-3)`

`m_2=(12-6)/(3)`

`m_2=(6)/(3)`

`m_2=2`

Since
`m_1>m_2`, therefore bacterial 1 had a faster rate of growth.

(b)

The initial population of f(x) is 7 and it increases exponentially.

The initial population of g(x) is 0 and it increases linearly.

It means population of f(x) always exceed the population of g(x).

In other words, population of g(x) cannot exceed the population of f(x).