Question

A population of bacteria is introduced into a culture. The number of bacteria \(P\) can be modeled by \(P=500\left(1+\frac{4 t}{50+t^{2}}\right)\) where \(t\) is the time (in hours). Find the rate of change of the population when \(t=2\).

Short answer

The rate of change of the population at \(t=2\) is found by taking the derivative of the given function and substitifying \(t=2\).

Step-by-step solution

Differentiate the Function

The given function is \(P=500\left(1+\frac{4 t}{50+t^{2}}\right)\). This function should be differentiated with respect to time \(t\). The differentiation should be done using the product rule, which states that the derivative of two functions being multiplied together is the first function times the derivative of the second plus the second function times the derivative of the first. The quotient rule might also be required because of the compound fraction inside the parentheses.

Apply the Product Rule

The first term which is 500 can be differentiated normally which gets 0 because it's a constant. For the second term, it's necessary to use the product rule: Differentiate 500 and keep \(\frac{4 t}{50+t^{2}}\) as it is, then keep 500 as it is and differentiate \(\frac{4 t}{50+t^{2}}\).

Apply the Quotient Rule

When differentiating \(\frac{4 t}{50+t^{2}}\), use the quotient rule, which is applied if you're differentiating a fraction. It states that the denominator times the derivative of the numerator, minus the numerator times the derivative of the denominator, divided by the denominator squared gives the derivative of the whole fraction. Using this rule, the derivative of \(\frac{4 t}{50+t^{2}}\) turns out to be \(\frac{4(50+t^{2}) - 4t(2t)}{(50+t^{2})^{2}}\).

Substitute \(t=2\) into the Derivative

Once the overall derivative is obtained, substitute the given \(t\) value, which is 2, into the derivative. This will give the rate of change of the population at \(t=2\).

Key concepts

Differentiation

Differentiation is a fundamental concept in calculus that aims to find how a function changes as its input changes. It gives us the derivative, which can be considered as a function's rate of change. When we differentiate a function with respect to a variable, we are essentially analyzing how small changes in that variable affect the function's output.

In our exercise involving the bacterial population, differentiation is used to find how the number of bacteria changes over time. The given function is expressed in terms of time, and by differentiating it, we can uncover how quickly the bacteria population grows or decreases at any specific time, such as when time \(t = 2\).

Key aspects to remember about differentiation include:

• Identifying the function you need to differentiate.
• Using appropriate differentiation rules (such as product or quotient rule).
• Understanding that the derivative itself can provide valuable insights into the behavior of a function.

Rate of Change

The rate of change is a concept used to describe how a quantity changes over time. In our context, it refers to how the population of bacteria changes with respect to time. Mathematically, it is represented by the derivative of the population function with respect to time.

For our bacteria population model, the rate of change at a specific time \(t\) is determined by evaluating the derivative of the population function at that time. This allows us to understand whether the bacterial population is increasing or decreasing and the pace at which this change is occurring.

Looking closely at this concept helps us to:

• See real-world applications of calculus.
• Understand how differentiation ties into modeling changes in dynamic systems, like bacterial populations.
• Provide a quantitative measure that indicates the speed of change.

Quotient Rule

The quotient rule is a specific method used in differentiation to handle functions that are fractions of two other functions. When differentiating a function expressed as a ratio, it is crucial to apply this rule to obtain correct results.

The quotient rule formula is: if \(u\) and \(v\) are functions of \(t\), then the derivative of \(\frac{u}{v}\) is given by \(\frac{v \cdot u' - u \cdot v'}{v^2}\), where \(u'\) and \(v'\) are the derivatives of \(u\) and \(v\), respectively.

Applying this rule in the original exercise for the expression \(\frac{4t}{50 + t^2}\), allows us to accurately differentiate the fractional part of the function. The steps ensure we account for both the numerator and denominator's roles in any changes within the function.

The Quotient Rule is instrumental when dealing with:

• Functions where variables are both in the numerator and denominator.
• Complex rational expressions, ensuring each component's change is properly calculated.
• Simplifying the process of finding the derivative of composite functions.