Question

Medicine A body assimilates a 12 -hour cold tablet at a rate $$ modeled $$ by $dC/dt=8-\ln (t^2-2t+4)$, $0\le t\le 12$, where $dC/dt$ is measured in milligrams per hour and $t$ is the time in hours. Use Simpson's Rule with $n=8$ to estimate the total amount of the drug absorbed into the body during the 12 hours.

Short answer

The total amount of the drug absorbed into the body during the 12 hours can be estimated by calculating the set-up Simpson's Rule equation. The exact numerical answer will be obtained according to the given function and intervals.

Step-by-step solution

Defining the function

First, define the function $f(t)$ as the rate of assimilation given by the differential equation $f(t)=dC/dt=8-ln(t^2-2t+4)$.

Setting up Simpson's Rule

Next, set up Simpson's rule function for approximation of the integral over the interval from 0 to 12 with n=8. The formula for Simpson's rule is ${\int }_a^{b}f(x)dx\approx \frac{h}{3}[f(x_0)+4f(x_1)+2f(x_2)+4f(x_3)+\dots +2f(x_{n-2})+4f(x_{n-1})+f(x_n)]$. Here $h=(b-a)/n$, $x_i=a+i\ast h$, $i$ ranging from 0 to $n$.

Applying Simpson's Rule

Substitute the values into the Simpsons rule equation, where $\mathrm{\Delta }t=(12-0)/8=1.5$. Compute the values $f(t_i)$ by substituting $t_i$ into the function one by one, adding them up by following the pattern (ignore multipliers): start with $f(t_0)$, then 4 times $f(t_1)$, 2 times $f(t_2)$, 4 times $f(t_3)$, 2 times $f(t_4)$, and so on until $f(t_8)$. Final equation will look as follows: ${\int }_0^{12}f(t)dt\approx 1.5/3\ast [f(0)+4f(1.5)+2f(3)+4f(4.5)+2f(6)+4f(7.5)+2f(9)+4f(10.5)+f(12)]$

Calculations and Finding the Answer

Carry out the calculations, add all these terms together and multiply by $1.5/3$ to get the final estimate of the total assimilation of the drug into the system over the given 12-hour period.

Key concepts

Integral Approximation

Estimating the value of an integral is a fundamental task in calculus, often required when an exact analytical solution is elusive or impossible. Integral approximation involves using mathematical techniques to approximate the area under a curve, which represents the integral of a function over a given interval. One common method is Simpson's Rule, a technique that uses parabolic arcs instead of straight lines to estimate the area. The accuracy of Simpson's Rule increases with the number of intervals (or partitions) used, as smaller intervals generally capture the curvature of the function better.To apply Simpson’s Rule, the interval $$a,b$$ is divided into an even number of subintervals, creating a series of points along the function’s curve. The formula involves a weighted sum of the function’s values at these points, with alternating coefficients of 4 and 2, except for the first and last points which have a coefficient of 1. This weighting scheme is instrumental because it gives more importance to certain points for a more precise approximation. As the exercise illustrates, selecting an appropriate number of subintervals, such as $n=8$ in the example, is crucial to gaining an accurate estimate of the integral.

Calculus in Pharmacokinetics

Pharmacokinetics is the branch of pharmacology concerned with the movement of drugs within the body, specifically absorption, distribution, metabolism, and excretion. Calculus, especially differential equations and integral approximation methods like Simpson's Rule, plays a vital role in modeling these processes.In our exercise, the differential equation $dC/dt=8-\ln (t^2-2t+4)$ represents the rate of drug assimilation over time, where $dC/dt$ indicates how the concentration of the medicine in the bloodstream changes per hour. By integrating this rate over a particular time span, you can calculate the total amount of the drug absorbed into the body. Simpson's Rule aids in this integration by approximating the definite integral, providing an estimate for the total drug absorption over the 12-hour period. Understanding this concept is crucial for dosing regimens and ensuring therapeutic efficacy while minimizing toxicity.

Differential Equations

A differential equation is an equation that involves an unknown function and its derivatives. It's a powerful tool in expressing how a particular quantity changes over time or space. In our context, the rate of drug assimilation, $dC/dt$, is expressed as a differential equation—the core of our exercise.Differential equations are not just prevalent in pharmacokinetics; they are widely used in engineering, physics, economics, and other fields to model complex systems and phenomena. In the given problem, solving the differential equation directly may be challenging, but by integrating both sides with respect to time, one can find the total change in concentration over a given period. In cases where direct integration is not feasible, numerical methods like Simpson's Rule are invaluable. They provide a practical means to estimate integrals and solve applied problems where continuous processes must be quantified.