[T] The concentration of antibiotic in the bloodstream t...

[T] The concentration of antibiotic in the bloodstream $t$ hours after being injected is given by the function $C(t)=\frac{2 t^{2}+t}{t^{3}+50},$ where $C$ is measured in milligrams per liter of blood. a. Find the rate of change of $C(t)$ . b. Determine the rate of change for $t=8,12,24,$ and $36 .$ c. Briefly describe what seems to be occurring as the number of hours increases.

[T] The concentration of antibiotic in the bloodstream $t$ hours after being injected is given by the function $C(t)=\frac{2 t^{2}+t}{t^{3}+50},$ where $C$ is measured in milligrams per liter of blood.
a. Find the rate of change of $C(t)$ .
b. Determine the rate of change for $t=8,12,24,$ and $36 .$
c. Briefly describe what seems to be occurring as the number of hours increases.

Key Concepts

Long-term Behavior Analysis

Long-term behavior analysis involves studying how a function behaves as the input variable becomes very large or very small. In practical applications, this concept helps in predicting trends and understanding the overall progression or decay of a system over time, which is particularly important in fields like pharmacokinetics or any area where changes evolve over time.

Derivative

The derivative is a core concept in calculus, representing the instantaneous rate of change of a function with respect to a variable. It allows us to determine how a function changes at any given point, which is crucial for understanding dynamic processes and modeling real-world phenomena.

Quotient Rule

The quotient rule is a method used in calculus for differentiating functions that are expressed as the ratio of two differentiable functions. This rule is essential when dealing with rational functions, and it states that the derivative of a quotient is obtained by taking the derivative of the numerator multiplied by the denominator minus the numerator multiplied by the derivative of the denominator, all divided by the square of the denominator.

Evaluation at Specific Points

Evaluating the derivative at specific points involves substituting given values into the derivative function to obtain numerical measures of the instantaneous rate of change at those points. This process helps in understanding the behavior of the function at different moments or conditions, providing localized insights into the dynamics of the system being modeled.

Transcript

00:01 We're told the concentration of an antibiotic in someone's blood at time t is given by 2 t squared plus 2 over t cube plus 50.

00:11 We're asked to find the rate of change of c and then evaluate c at these given numbers, then describe what happens as the number of hours is increasing.

00:22 So, okay, let's go ahead and compete the first one.

00:27 So we're given a quotient.

00:28 So c of t is equal to f over g like this.

00:38 So f of t is 2t squared plus t and i'm just going to write g over here.

00:48 G of t is t cubed plus 50.

00:53 Now we can differentiate both of these and then we can go ahead and use the quotient rule.

00:57 So the derivative of f is 4t plus 1.

01:02 And the derivative of g is 3t squared.

01:07 So the quotient rule says that c prime of t is going to be f prime g minus g prime f all over g squared.

01:20 Let's go ahead and calculate this.

01:22 So we're going to have 4t plus 1 times t cubed plus 50 minus 3t squared.

01:37 Times 2t squared plus t all over t cubed plus 50 squared okay and if we expand this we can simplify it down to it will end up being minus 2 t to the 4 minus 2 t cubed plus 200 t plus 50 plus 50 all over t plus 50 squared.

02:23 Okay, so that answers the first bit.

02:26 Now let's move on to b.

02:29 Let's scroll down.

02:30 There you go.

02:32 Okay, so let's just input these values.

02:38 So we want to find c prime of 8.

02:41 This is going to be minus 2 times 8 to the 4, minus 2 times 8 cubed plus 200 times 8 plus 8 plus 50 all over 50 cubed plus 50 not 50 cubed and cubed.

03:11 Okay, now this is where a calculator comes in handy and we can show that this is, if you evaluate this, we see that this is approximately equal to minus 0 .02...

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