Concentration of a Drug in the Bloodstream The concentration (in milligrams per milliliter) of a

Concentration of a Drug in the Bloodstream The concentration...

Key Concepts

Average Value of a Function

The average value of a continuous function over an interval is determined by integrating the function over that interval and then dividing by the length of the interval. This concept is fundamental in calculus for summarizing the overall behavior of a function across a specified domain.

Definite Integral

A definite integral represents the accumulation of a function’s values over an interval and is used to calculate total quantities, areas, or averages. It serves as the mathematical tool to sum infinitesimal contributions continuously across a given range.

Exponential Functions

Exponential functions describe situations where rates of change are proportional to the current value, often modeling decay or growth processes. Their unique properties require specific integration techniques and are broadly used in various fields of science and engineering.

Integration Techniques

Integration techniques, such as substitution and integration by parts, are strategies to simplify and evaluate integrals, especially when dealing with products of functions like polynomials and exponentials. Mastery of these techniques is essential for solving complex integration problems.

Transcript

00:01 In the question, it is saying that the concentration of certain drug in term of t as a function is given that is c t is equal to 3 t times of e to the power minus t over 3 and unit is mill per ml where t is a time and unit is that is in hour.

00:29 So now in the question it is asking to find the average value of concentration of that drug over 12 hour.

00:49 So it means i have to find the value of average value of this function.

00:58 So as we know the formula for average value of function will be written as 1 divided by b minus a integral a to b fx dx will be here a to b's value at which we have to find the average value of functions a and we are limits of function so in the question it is saying that we have to find the average value of concentration over 12 hour means i can say that if that is lower limit is equal to zero over and b is equal to that is upper limit is equal to 12 hour and in the question the value of fx will be equal to function that is given c t so substitute all this data in this formula of which is used to find the average value of function so after substituting it will be written as 1 divided by b minus a b minus a means 12 minus 0 times of integral integral a to b means 0 to 12 and fx will be that is c t c d t because this function is in term of t so so i can say that 1 divided by 2 12 12 will be 2 1 2 2m minus 0 will be equal to 12 times of integral 0 to 12 and we know the value of ct that is 3 t times of e to the power minus t over 3 d t so so we have to find this value for finding concentration of average value of concentration of that drug so first for finding this i will assume that you will be equal to t so d u will become that is equal to d t and let assume dv will be equal to that is e to the power minus t divided by three d t then we will be equal to after integration value is that is minus three times of e to the power minus t over three so this is a value of v now next step is using the formula of integration by part which is written as that is integral utv will be equal to uv minus integral vdu.

04:06 Now put the value of about data in this formula so i can say that integral udv means integral udv means t and times of dv will be e to the power minus t over 3 dt will be equal to uv.

04:26 U will be t times of minus 3 times of e to the power minus t over 3 minus integral v.

04:36 V means minus 3 times of e to the power minus t over 3 times of d u will be that is dt...

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