Suppose a certain drug is administered to a patient, with...

Suppose a certain drug is administered to a patient, with the percent of concentration of the drug in the bloodstream t hours later given by
$$K(t)=\frac{5 t}{t^{2}+1}$$
a. On what time intervals is the concentration of the drug increasing?
b. On what intervals is it decreasing?

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Key Concepts

Critical Points

Critical points of a function occur where its derivative is zero or undefined. These are key locations that potentially indicate local maxima or minima, and they serve as boundary points when performing a test to determine intervals where the function increases or decreases.

Rational Functions

Rational functions, which are ratios of polynomial functions, often exhibit unique characteristics such as asymptotes and discontinuities. Analyzing their behavior with calculus, especially through derivatives, helps in understanding trends and inflection points which are pivotal in many applied contexts.

Derivative

The derivative of a function represents the instantaneous rate of change of the function with respect to its independent variable. It is a fundamental tool in calculus that allows one to determine the slope of the tangent line at any given point on a function, which is critical for analyzing the behavior of functions over intervals.

First Derivative Test

The first derivative test uses the sign of the derivative to determine where a function is increasing or decreasing. If the derivative is positive over an interval, the function is increasing; if it is negative, the function is decreasing. This test is essential in understanding and characterizing the monotonic behavior of functions.

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