Population Growth A population of bacteria is introduced...

Population Growth 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 $t=2$.

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Key Concepts

Population Growth Models

This concept involves representing changes in the size of a population over time using mathematical functions. In this context, an equation is used to describe how the population of bacteria evolves, which is typical of many biological systems where environmental factors and growth limits play roles.

Rate of Change

Rate of change refers to how a quantity changes over time and is fundamental in understanding dynamic systems. It is quantified as the derivative of the function expressing the system's state, providing insight into how quickly the population increases or decreases at any given moment.

Differentiation

Differentiation is a core concept in calculus that involves finding the derivative of a function. It is used to calculate the instantaneous rate of change of a quantity with respect to time, which is essential for analyzing real-world phenomena like population dynamics.

Quotient Rule

The quotient rule is a specific technique used in differentiation to find the derivative of a function that is the ratio of two differentiable functions. This rule is critical when handling rational expressions that appear in models of population growth and other applied problems.

Transcript

00:01 Okay, so here what i see is i see i want to find the rate of change at a certain point.

00:06 I specifically want to find the rate of change.

00:09 So that means i'm really trying to find the slope at a point.

00:12 Okay, finding the slope at a point.

00:14 What helps you find the slope at a point, specifically the first derivative.

00:18 So what i'm going to end up doing is i'm going to take the first derivative.

00:22 I see that i have two functions here.

00:24 I have 500, and then i have 1 plus 4t over 50 plus t squared.

00:31 To make this a little simpler, what i'm going to do is i'm going to distribute 500 to both terms.

00:36 So i'm going to say p is 500 plus 500 is going to multiply it to the top, so it'll be 2 ,000 t over 50 plus t squared.

00:51 Now i'm going to take the first derivative here.

00:55 Derivative of 500 is just zero, so this cancels out, zero plus i'm dividing two terms.

01:00 So i'm going to do the quotient rule.

01:04 Derotive of the top term is just 2000 because of the product rule, 2000 times the second term, 50 plus t squared, minus the first term, 2000 t, times the derivative of the denominator derivative of 50 cancels.

01:21 It's just zero plus derivative of t squared is just 2t.

01:25 All over 50 plus t squared squared...

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