Drug Concentration The percent of concentration of a drug in...

Drug Concentration The percent of concentration of a drug in the bloodstream $t$ hours after the drug is administered is given by
$K(t)=\frac{7 t}{4 t^{2}+16}$
for $t \geq 0$
(a) On what time intervals is the concentration of the drug increasing?
(b) On what intervals is it decreasing?

Key Concepts

Sign Analysis

Sign analysis involves determining the sign (positive or negative) of the derivative over different intervals. By analyzing these signs, one can accurately describe where a function's rate of change leads to increasing or decreasing values, thus outlining the monotonicity of the function.

Critical Points

Critical points occur where the derivative of a function is zero or undefined. These points often signal potential changes in the increasing or decreasing behavior of the function, making them essential in studying function behavior.

Quotient Rule

The quotient rule is a method for differentiating functions that are ratios of two differentiable functions. It provides a systematic way to compute the derivative, which is necessary for further analysis of the function's behavior.

Derivative and Monotonicity

The first derivative of a function represents its rate of change. Determining where the derivative is positive or negative helps identify intervals where the function is increasing or decreasing, respectively.

Transcript

00:01 Hello, we are looking at chapter 5, section 1, problem number 52.

00:06 So in this problem, we are looking at a formula for the percentage of concentration of a drug in the bloodstream's t hours after the drug has been administered.

00:19 So that concentration is given by the equation k of t is equal to 7t over 4t squared plus 16 for t greater than or equal to.

00:31 To zero.

00:32 So then for part a, we want to find the time intervals, the concentration of the drug is increasing.

00:38 And for b, the time interval for which the concentration of the drug is decreasing.

00:43 So the first thing we want to do when we're looking at increasing and decreasing of a function is we want to find the first derivative.

00:51 So we're going to say k prime of t is equal to.

00:56 And it looks like we have a quotient rule problem.

01:00 So i like to do the quotient.

01:01 As low -de -high minus high -de -low over the square of what's below, which means low -de -high would mean i'm taking the denominator, so 4t squared plus 16, so low -de -high, so times the derivative of the numerator, minus high -de -low, so that's high is the numerator, 7t, d -low would be the derivative of the denominator, so that's going to be 18.

01:44 And we square the bottom, so we square the denominator.

01:50 So 4t squared plus 16 quantity squared.

01:56 So again, that was low -d -high, minus, high, d, low, square the bottom, and away we go.

02:04 So we have our first derivative.

02:06 We can clean it up a bit.

02:08 So that's going to be distributing the 7.

02:11 We're going to get 28 t squared plus 16 times 7 is 112 minus 56 t squared all over our quantity 4t squared plus 16.

02:33 All squared.

02:36 So finally that means we're going to get what is that negative, 28 t squared plus 112 all over 4 t squared plus 16 quantity squared and so there's our first derivative so we know to find our critical numbers we set the first derivative equal to zero or undefined so setting it equals 0, setting the numerator equal to 0, we'll do that in a second.

03:10 Setting the denominator equal to zero to find undefined, it's not necessary because this is always positive, right? so we have 4t squared, t squared is positive.

03:20 So positive plus a positive is positive.

03:23 Squared is still positive...

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