Population growth Consider the following population functions.
a. Find the instantaneous growth rate of the population, for t≥0.
p(t) = 600 (t²+3/t²+9)

Verified step by step guidance

1. Step 1: Identify the function for which you need to find the instantaneous growth rate. The function given is \( p(t) = 600 \left( \frac{t^2 + 3}{t^2 + 9} \right) \).
2. Step 2: To find the instantaneous growth rate, you need to compute the derivative of the function \( p(t) \) with respect to \( t \). This involves using the quotient rule for differentiation.
3. Step 3: Recall the quotient rule: if you have a function \( \frac{u(t)}{v(t)} \), its derivative is \( \frac{u'(t)v(t) - u(t)v'(t)}{(v(t))^2} \). Here, \( u(t) = t^2 + 3 \) and \( v(t) = t^2 + 9 \).
4. Step 4: Differentiate \( u(t) \) and \( v(t) \) separately. \( u'(t) = 2t \) and \( v'(t) = 2t \).
5. Step 5: Apply the quotient rule: \( p'(t) = 600 \left( \frac{(2t)(t^2 + 9) - (t^2 + 3)(2t)}{(t^2 + 9)^2} \right) \). Simplify the expression to find the derivative.

Key Concepts

Instantaneous Growth Rate

The instantaneous growth rate of a function at a given point is defined as the derivative of that function at that point. In the context of population growth, it represents how fast the population is changing at a specific time, providing insight into trends and future projections.

Derivatives

A derivative measures how a function changes as its input changes. In calculus, it is a fundamental tool used to find rates of change, slopes of tangent lines, and to analyze the behavior of functions. For population functions, derivatives help determine growth rates and can indicate whether a population is increasing or decreasing.

Population Functions

Population functions are mathematical models that describe how a population changes over time. They can take various forms, such as linear, exponential, or polynomial functions. Understanding the specific form of the population function is crucial for applying calculus techniques to analyze growth rates and predict future population sizes.