The drug concentration in the bloodstream t hours after...

The drug concentration in the bloodstream $t$ hours after injection is given approximately by $C(t)=4.35 e^{-t} \quad 0 \leq t \leq 5$ where $C(t)$ is concentration in milligrams per milliliter.
(A) What is the rate of change of concentration after 1 hour? After 4 hours?
(B) Graph C.

Key Concepts

Exponential Decay

Exponential decay refers to a process where the quantity decreases at a rate proportional to its current value. This type of behavior is common in many natural and engineered systems and is mathematically represented by functions that include an exponential term with a negative exponent. It is essential when modeling physical processes such as radioactive decay, cooling, and in this case, the drug concentration in a bloodstream.

Differentiation of Exponential Functions

Differentiation of exponential functions utilizes the chain rule to find the rate at which the function changes with respect to its variable. For an exponential decay function, the derivative also represents an exponential function scaled by the parameters of the original function, revealing how rapidly the quantity decreases over time.

Rate of Change

The concept of the rate of change in calculus quantifies how a function’s output changes as its input changes. In physical contexts, like the concentration of a drug in the bloodstream, it indicates how rapidly the drug level is falling, which is important for understanding pharmacokinetics and determining appropriate dosing intervals.

Graphing Exponential Functions

Graphing exponential functions involves plotting the function over a domain to visualize its behavior, including features such as the initial value, decay or growth rate, and asymptotic behavior. This helps in understanding the dynamics of the system modeled by the function, such as how the concentration diminishes over time in response to the decay process.

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