Drug Concentration The percent of concentration of a certain...

Drug Concentration The percent of concentration of a certain drug in the bloodstream $x$ hours after the drug is administered is given by $$K(x)=\frac{3 x}{x^{2}+4}$$ For example, after 1 hour the concentration is given by $$K(1)=\frac{3(1)}{1^{2}+4}=\frac{3}{5} \%=0.6 \%=0.006$$ a. Find the time at which concentration is a maximum. b. Find the maximum concentration.

Drug Concentration The percent of concentration of a certain drug in the bloodstream $x$ hours after the drug is administered is given by
$$K(x)=\frac{3 x}{x^{2}+4}$$
For example, after 1 hour the concentration is given by
$$K(1)=\frac{3(1)}{1^{2}+4}=\frac{3}{5} \%=0.6 \%=0.006$$
a. Find the time at which concentration is a maximum.
b. Find the maximum concentration.

Key Concepts

Rational Functions

Rational functions are expressions formed by the ratio of two polynomials. Analyzing rational functions often involves techniques such as the quotient rule and critical point analysis to understand their behavior, including finding their maximum or minimum values.

Derivative

The derivative represents the slope or rate of change of a function at any given point. In optimization problems, taking the derivative of a function helps determine where the function’s rate of increase or decrease is zero, which are the potential candidates for local maxima or minima.

Quotient Rule

The quotient rule is a method used to differentiate functions that are expressed as one function divided by another. It is particularly useful when dealing with rational functions, where both the numerator and denominator are functions of the variable.

Critical Points

Critical points are values in the domain of a function where the derivative is zero or undefined. These points are essential in optimization problems because they indicate potential locations for local maxima or minima, which can then be tested or analyzed further.

Optimization

Optimization involves finding the maximum or minimum values of a function, which is a fundamental concept in calculus. This process typically requires finding the derivative, setting it equal to zero to find critical points, and then determining which of these points correspond to the desired extrema.

Transcript

00:01 The blood concentration after x hours is given above.

00:04 If we want to find the maximum, we could go ahead and take the derivative and calculate x prime of t.

00:11 And to do so, we could use the quotient rule.

00:14 So we'll start off with low, d high.

00:19 So we take the bottom, multiply by the derivative up top.

00:24 It should just be three.

00:27 Minus, let's get rid of b for now, minus high dol, derivative of the bottom is 2x and then all divided by the derivative of the bottom square.

00:43 We could really just ignore that because we're only interested when the slope is zero to find a max.

00:51 So completely ignoring this and just looking at the top, setting this equal to zero.

00:58 This will simplify eventually, you know, just distributing and combining some terms here, we'll get negative 3 x squared and then plus 12 x sorry plus 12 and then factoring out the 0 sorry factoring out the negative 3 we'll get negative 3 and then x squared minus 4 so we're going to end up with two critical points here plus or minus 2 positive negative 2 and so we'll go with the positive one since we're interested in time.

01:40 And then from here, we could just go ahead and, well, we really only have one critical point.

01:47 So this is, in fact, our max after two hours, we'll end up with a max concentration in the bloodstream.

01:53 But we can prove this a little bit further here.

01:57 If we looked at the graph, let's say, then it is going to come to a peak...

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