An initial population, p, of 1000 bacteria grows in number...

An initial population, $p,$ of 1000 bacteria grows in number according to the equation $p(t)=1000\left(1+\frac{4 t}{t^{2}+50}\right),$ where $t$ is in hours. Find the rate at which the population is growing after $1 \mathrm{h}$ and after $2 \mathrm{h}$

Key Concepts

Instantaneous Rate of Change

This concept refers to the speed at which a quantity is changing at a precise moment in time. In calculus, it is determined using the derivative of a function with respect to time, which gives the slope of the tangent line at any given point. It is particularly useful in real-world applications such as measuring population growth rates at specific moments.

Differentiation

Differentiation is a fundamental concept in calculus that involves computing the derivative of a function. The derivative represents the rate of change of the function with respect to its independent variable. Understanding differentiation is key to solving problems that require finding how a variable changes over time.

Quotient Rule

The quotient rule is a technique used in calculus for differentiating functions that are expressed as the ratio of two differentiable functions. It provides a systematic method to find the derivative of a quotient by using the derivatives of the numerator and the denominator. This rule is essential when dealing with rational functions.

Evaluation of Derivatives

Once the derivative of a function is obtained, evaluating it at specific points allows one to find the instantaneous rate of change at those points. This process is critical in practical applications where the value of the derivative at particular instances gives important insights into the behavior of the modeled system.

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