Bacteria Population Assume that the total number (in...

Bacteria Population Assume that the total number (in millions) of bacteria present in a culture at a certain time $t$ (in hours) is given by $$N(t)=3 t(t-10)^{2}+40 \text { . }$$ a. Find $N^{\prime}(t)$ . Find the rate at which the population of bacteria is changing at the following times. b.8 hours $\qquad$ c. 11 hours d. The answer in part b is negative, and the answer in part c is positive. What does this mean in terms of the population of bacteria?

Bacteria Population Assume that the total number (in millions) of bacteria present in a culture at a certain time $t$ (in hours) is given by
$$N(t)=3 t(t-10)^{2}+40 \text { . }$$
a. Find $N^{\prime}(t)$ .
Find the rate at which the population of bacteria is changing at the following times.
b.8 hours $\qquad$ c. 11 hours
d. The answer in part b is negative, and the answer in part c is positive. What does this mean in terms of the population of bacteria?

Key Concepts

Interpretation of the Derivative's Sign

The sign of the derivative conveys important information about the behavior of the function: a positive derivative indicates that the function is increasing, meaning the bacteria population is growing, while a negative derivative indicates a decreasing population. This interpretation helps in understanding the overall dynamics described by the model.

Chain Rule

The chain rule is a method for computing the derivative of a composite function. It is essential when one function is applied inside another, such as when a term like (t - 10)^2 appears in the model, requiring differentiation of an inner function and an outer function in succession.

Rate of Change Interpretation

Evaluating the derivative at specific points in time gives the instantaneous rate of change of the population at those times. This provides practical information on whether the number of bacteria is increasing or decreasing and how rapidly this change occurs.

Derivative

The derivative of a function represents the rate at which the function’s output is changing with respect to a change in the input. In applied contexts, such as a bacteria population model, the derivative provides insight into how quickly the population is growing or declining at any given moment.

Product Rule

The product rule is a fundamental technique in differentiation used when dealing with functions that are products of two or more sub-functions. It states that the derivative of a product is the derivative of the first function multiplied by the second function plus the first function multiplied by the derivative of the second function.

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