Additional Applications


Bacterium population
When a bactericide was added to a nutrient broth in which bacteria were growing, the bacterium population continued to grow for a while, but then stopped growing and began to decline. The size of the population at time t (hours) was b = 10⁶ + 10⁴t − 10³t². Find the growth rates at


a. t = 0 hours.
b. t = 5 hours.
c. t = 10 hours.

Verified step by step guidance

1. To find the growth rate of the bacterium population at a given time, we need to calculate the derivative of the population function b with respect to time t. The function given is b(t) = 10⁶ + 10⁴t − 10³t².
2. The derivative, b'(t), represents the rate of change of the population with respect to time. To find b'(t), differentiate each term of b(t) with respect to t. The derivative of a constant is 0, the derivative of 10⁴t is 10⁴, and the derivative of −10³t² is −2 * 10³ * t.
3. Combine these derivatives to get b'(t) = 0 + 10⁴ − 2 * 10³ * t. Simplify this expression to obtain b'(t) = 10⁴ − 2 * 10³ * t.
4. Now, substitute t = 0 into b'(t) to find the growth rate at t = 0 hours. This will give you the initial growth rate of the population.
5. Similarly, substitute t = 5 and t = 10 into b'(t) to find the growth rates at t = 5 hours and t = 10 hours, respectively. This will show how the growth rate changes over time.

Key Concepts

Derivative

The derivative of a function represents the rate at which the function's value changes with respect to a change in its input. In this context, the derivative of the bacterium population function b(t) with respect to time t gives the growth rate of the population at any given time. Calculating the derivative allows us to determine how the population size is changing at specific time points.

Polynomial Functions

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. The given bacterium population function b(t) = 10⁶ + 10⁴t − 10³t² is a quadratic polynomial in t. Understanding how to differentiate polynomial functions is crucial for finding the growth rate, as it involves applying basic rules of differentiation to each term.

Evaluating Derivatives at Specific Points

Once the derivative of a function is found, it can be evaluated at specific points to determine the rate of change at those points. In this problem, after finding the derivative of b(t), we substitute t = 0, t = 5, and t = 10 into the derivative to find the growth rates at these specific times. This process helps in understanding how the population's growth rate changes over time.