Population growth Consider the following population functions.
b. What is the instantaneous growth rate at t=5?
p(t) = 600 (t²+3/t²+9)

Verified step by step guidance

1. Step 1: Understand that the instantaneous growth rate of a population function p(t) at a specific time t is given by the derivative of the function, p'(t), evaluated at that time.
2. Step 2: Identify the given population function: p(t) = 600 \left(\frac{t^2 + 3}{t^2 + 9}\right).
3. Step 3: Use the quotient rule to differentiate p(t). The quotient rule states that if you have a function in the form of \(\frac{u(t)}{v(t)}\), 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: Compute the derivatives u'(t) and v'(t). For u(t) = t^2 + 3, u'(t) = 2t. For v(t) = t^2 + 9, v'(t) = 2t.
5. Step 5: Substitute u(t), v(t), u'(t), and v'(t) into the quotient rule formula to find p'(t). Then, evaluate p'(t) at t = 5 to find the instantaneous growth rate.

Key Concepts

Instantaneous Growth Rate

The instantaneous growth rate of a function at a specific point is determined by the derivative of that function at that point. It represents how fast the population is changing at that exact moment in time, providing a snapshot of growth rather than an average over an interval.

Derivatives

A derivative is a fundamental concept in calculus that measures how a function changes as its input changes. It is defined as the limit of the average rate of change of the function as the interval approaches zero. In the context of population functions, the derivative gives the rate of change of the population with respect to time.

Population Functions

Population functions model the size of a population over time, often using mathematical expressions that incorporate variables like time. In this case, the function p(t) = 600(t² + 3)/(t² + 9) describes how the population changes as time t progresses, allowing for analysis of growth patterns and rates.