Bacteria Population The total number of bacteria (in...

Bacteria Population The total number of bacteria (in millions) present in a culture is given by $$ N(t)=2 t(5 t+9)^{1 / 2}+12 $$ where $t$ represents time (in hours) after the beginning of an experiment. Find the rate of change of the population of bacteria with respect to time for the following numbers of hours. a. 0 b. 7$/ 5$ c. 8

Bacteria Population The total number of bacteria (in millions) present in a culture is given by
$$
N(t)=2 t(5 t+9)^{1 / 2}+12
$$
where $t$ represents time (in hours) after the beginning of an experiment. Find the rate of change of the population of bacteria with respect to time for the following numbers of hours.
a. 0
b. 7$/ 5$
c. 8

Key Concepts

Evaluation of Derivatives

This concept involves substituting specific values into the derivative to determine the instantaneous rate of change at those points. Evaluating the derivative at given time values helps to quantify how briskly the bacteria population is growing at different stages of the experiment.

Chain Rule

The chain rule is applied when differentiating a composite function—a function made up of an inner and an outer function. In this case, the square root function, which has a linear function inside it, requires the use of the chain rule to find its derivative accurately.

Rate of Change

This concept measures how one quantity changes in response to a change in another quantity. In the context of a bacteria population, it indicates how quickly the population size varies with respect to time, representing the derivative of the population function.

Derivative

A derivative is a fundamental concept in calculus that captures the instantaneous rate of change of a function with respect to one of its variables. In this problem, the derivative of the bacteria population function with respect to time reveals how fast the bacteria count is increasing at any given moment.

Product Rule

The product rule is a technique used to differentiate expressions where two functions are multiplied together. Since the bacteria population function is given as a product of time and a square root function, applying the product rule is essential to correctly compute its derivative.

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