At a time t hours after it was administered, the...

At a time $t$ hours after it was administered, the concentration of a drug in the body is $f(t)=27 e^{-0.14 t} \mathrm{ng} / \mathrm{ml}$. What is the concentration 4 hours after it was administered? At what rate is the concentration changing at that time?

Key Concepts

Exponential Decay

Exponential decay refers to a process that decreases at a rate proportional to its current value. In mathematics, this is often modeled by a function of the form f(t) = A e^(kt) with k being a negative constant. This concept is widely used in various fields such as chemistry, physics, and pharmacology to model processes like radioactive decay, cooling, and drug elimination from the body.

Function Evaluation

Evaluating a function at a specific point involves substituting a given value into the function to find the corresponding output. This process is essential for determining the state of a model at certain conditions or time points, such as finding the concentration of a substance or the amount of decay after a set time period.

Differentiation of Exponential Functions

Differentiation is a fundamental operation in calculus that computes the rate at which a function changes. For exponential functions, the derivative maintains the exponential form and is typically proportional to the original function, indicating that the rate of change is also proportional to the current value. This principle is especially useful when analyzing how quickly a decaying substance decreases over time.

Chain Rule

The chain rule is a method in calculus used to differentiate composite functions. When a function involves a composition of two or more functions, the chain rule provides a systematic way to compute its derivative. It is particularly important for exponential functions with a multiplied constant in the exponent because it allows for the correct calculation of the derivative by considering the rate of change of the inner function.

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