Question

Numerical, Graphical, and Analytic Analysis The concentration \(C\) of a chemical in the bloodstream \(t\) hours after injection into muscle tissue is \(C(t)=\frac{3 t}{27+t^{3}}, \quad t \geq 0\) (a) Complete the table and use it to approximate the time when the concentration is greatest. $$ \begin{array}{|l|l|l|l|l|l|l|l|} \hline t & 0 & 0.5 & 1 & 1.5 & 2 & 2.5 & 3 \\ \hline \boldsymbol{C}(\boldsymbol{t}) & & & & & & & \\ \hline \end{array} $$ (b) Use a graphing utility to graph the concentration function and use the graph to approximate the time when the concentration is greatest. (c) Use calculus to determine analytically the time when the concentration is greatest.

Short answer

Through numerical analysis, graphical analysis and calculus, it will be determined when the concentration \(C(t)\) is greatest.

Step-by-step solution

Tabulate Values

For each time point \(t\) in the given table which are \([0, 0.5, 1, 1.5, 2, 2.5, 3]\), substitute these values respectively into the equation \(C(t)=\frac{3 t}{27+t^{3}}\) to obtain the corresponding concentration \(C(t)\).

Graph the Function

Plot the function \(C(t)=\frac{3 t}{27+t^{3}}\) for \(t \geq 0\) using a graphing utility. Then, determine the time \(t\) value at the peak of the graph. This time value is an approximation of when the concentration is greatest.

Calculus Analysis

To find the exact time when the concentration is at its maximum, differentiate the function \(C(t)\) with respect to \(t\), set this derivative equal to zero and solve for \(t\). This gives the time when the concentration reaches a maximum. A second derivative check will confirm if it's truly a maximum point.

Key concepts

Numerical Analysis

In numerical analysis, you tackle mathematical problems using numerical approximation. For calculus optimization problems that involve functions like our concentration function, numerical analysis starts with creating a table of values. By evaluating the function at specific intervals, in this case at each half-hour within a certain range, it allows you to gain a practical understanding of how the function behaves over time.

The function for the concentration of a chemical in the bloodstream over time, described as \(C(t)\), is such that it changes with varying values of \(t\). To numerically analyze it, you'd fill the table with the corresponding \(C(t)\) values for each time point and look for the maximum value. This gives you a rough estimate of the greatest concentration and its approximate time occurrence. While this method does not give an exact maximum, it is a straightforward way to begin the optimization problem.

Graphical Analysis

Graphical analysis involves visualizing mathematical concepts, and in the context of optimization problems in calculus, it's about understanding how a function's value changes over time.

You can use a graphing utility to plot the concentration function \(C(t)\) against time \(t\). This visual representation makes it easier to identify trends and patterns, such as peaks and troughs in the graph that represent maximum and minimum concentrations respectively. By observing the graph, you can approximate when the concentration is greatest. Although graphical analysis provides a good visual aid, the accuracy of your approximation can vary based on the scale and resolution of your graph.

Analytic Analysis

Analytic analysis, unlike numerical or graphical techniques, involves working directly with the mathematical function and its derivatives to find exact answers. It's a process that includes differentiation, setting derivatives to zero, and solving equations.

For our concentration function, analytic analysis means finding the first derivative of \(C(t)\), setting it equal to zero to find critical points, and then using the second derivative test to verify if these points are maxima, minima, or points of inflection. The analytic approach leads you to the precise time when the concentration is at its peak, providing a more accurate result than numerical or graphical analysis.

Concentration Function

In our exercise, the concentration function \(C(t)=\frac{3t}{27+t^3}\) represents the concentration of a chemical in the bloodstream over time. The function tells us how the concentration changes as time progresses after the chemical is injected into muscle tissue.

This specific type of function is essential in pharmacokinetics, the study of how drugs move through the body. Understanding the behavior of the concentration function helps in determining the most effective dosing schedules and in predicting times of peak concentration, which can be vital for effective treatment.

Differentiation

Differentiation is a fundamental concept in calculus that deals with how a function changes at any given point. It's the process of finding the derivative of a function.

In the given exercise, differentiation plays a crucial role in the analytic analysis. The derivative of the concentration function \( C'(t) \) with respect to time \( t \) provides information about the rate of change of concentration. By setting \( C'(t) \) equal to zero, we find the critical points that help us identify when the concentration is at a maximum or minimum. For optimization problems, differentiation is the tool that precisely pinpoints the peak concentration time.

Graphing Utility

A graphing utility is a digital tool that helps plot functions and visualize complex equations, which is especially helpful for students learning calculus. It allows turning abstract formulas into tangible graphs that can be studied to understand a function's behavior.

In our context, a graphing utility would let students plot the function \(C(t)=\frac{3t}{27+t^3}\) and clearly see the curvature of the graph. This direct visual approach is particularly useful for identifying the greatest concentration's approximate time as the highest point on the graph. Incorporating a graphing utility into the learning process can significantly enhance comprehension and engagement with mathematical analysis.