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

Answer:

(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).

Step-by-step 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

$m=\frac{f(b)-f(a)}{b-a}$

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

$m_1=\frac{f(6)-f(3)}{6-3}$

$m_1=\frac{(2(6)^2+7)-(2(3)^2+7)}{6-3}$

$m_1=\frac{79-25}{3}$

$m_1=\frac{54}{3}$

$m_1=18$

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

$m_2=\frac{g(6)-g(3)}{6-3}$

$m_2=\frac{2(6)-2(3)}{6-3}$

$m_2=\frac{12-6}{3}$

$m_2=\frac{6}{3}$

$m_2=2$

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).