Problem 26 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Sketch the graphs of the functions \(f\) and \(g\) and find the area of the region enclosed by these graphs and the vertical lines \(x=a\) and \(x=b\). $$f(x)=x, g(x)=e^{2 x} ; a=1, b=3$$

Step-by-Step Solution

Answer

The approximate area enclosed by the graphs of the functions \(f(x)=x\) and \(g(x)=e^{2x}\) and the vertical lines \(x=1\) and \(x=3\) is \(\frac{1}{2}(e^6 - 9 - e^2 + 1)\) square units.

Step 1: Calculate Points of Intersections

To find the points where the graphs of both functions intersect, we need to solve the equation \(f(x) = g(x)\) for \(x\): \(x = e^{2x}\) Unfortunately, this equation doesn't have an elementary solution or algebraic solutions, so we will find an approximate numerical solution using a calculator or a computer program, keeping in mind our bounds \(a=1\) and \(b=3\).

Step 2: Sketch the Graphs

Now that we have the rough idea of the points of intersection, we can sketch the graphs of the functions \(f(x) = x\) and \(g(x) = e^{2x}\), as well as the vertical lines \(x = 1\) and \(x = 3\). Be sure to mark the points of intersection between the graphs on your sketch.

Step 3: Find the Area Enclosed

To find the area enclosed between the two functions and the vertical lines, we need to calculate the difference between the graphs of the functions, and integrate this difference over the interval \([a, b]\). The difference between the two functions is: \(g(x) - f(x) = e^{2x} - x\) Now, we need to calculate the integral of this difference over the given interval: \(\int_{a}^{b} (g(x) - f(x)) dx = \int_{1}^{3} (e^{2x} - x) dx\) Calculating the integral: \(\int_{1}^{3} (e^{2x} - x) dx = \frac{1}{2}e^{2x} - \frac{1}{2}x^2 \bigg |_{1}^{3}\) Now, find the definite integral: \(\frac{1}{2}e^{6} - \frac{1}{2}(3^2) - \left(\frac{1}{2}e^{2} - \frac{1}{2}(1^2)\right) \) Simplify and calculate the resulting expression: \(= \frac{1}{2}(e^6 - 9 - e^2 + 1)\) So, the area enclosed by the graphs of the functions and the vertical lines is approximately equal to \(\frac{1}{2}(e^6 - 9 - e^2 + 1)\) square units.

Key Concepts

Definite Integral

In Integral Calculus, a definite integral helps in finding the net area under a curve between two specified points, often referred to as bounds. This area can represent various quantities, like distance or probability, depending on the context. For the function given, the \(f(x)\) and \(g(x)\) curves, we need to calculate the difference between the two functions over an interval \([a, b]\).
\[ \int_{a}^{b} (g(x) - f(x)) dx = \int_{1}^{3} (e^{2x} - x) dx \]
This expression signifies the accumulation of the function \(g(x) - f(x)\) from \(x = 1\) to \(x = 3\), which represents the area between the two curves over the interval.
Definite integrals have unique properties:

• The area under the curve can be negative if the graph lies below the x-axis.
• The integral provides a precise quantity, contrasting with an indefinite integral that includes a constant.
• When calculating definite integrals, evaluate the antiderivative at both ends \(a\) and \(b\) and subtract.

Understanding how definite integrals work gives us a precise method to quantify enclosed areas or other physical interpretations.

Area Between Curves

Finding the area between curves entails calculating the integral of the difference between two functions. When calculating such an area between \(f(x)\) and \(g(x)\):
\[ \int_{a}^{b} (g(x) - f(x)) dx \]
It involves:

• Identifying points where the curves intersect, which can complicate when dealing with exponential or non-linear equations.
• Sketching both graphs for visual assistance, marking intersections and vertical lines that serve as boundaries.
• Analysing which function lies above the other within the interval. Here, \(g(x)\) is typically above \(f(x)\) given its exponential nature.

This geometric representation helps students grasp the concept by showing how calculus can determine the actual physical or hypothetical space that lies between two curves.
Crucially, this process gives a practical way to compare different phenomena, like comparing growth models or understanding differences in speed over time.

Exponential Functions

Exponential functions are vital in mathematics due to their rapid growth and natural association with phenomena like population growth, radioactive decay, or interest accumulations. The function \(g(x) = e^{2x}\) is particularly interesting because of its nature.
Key characteristics of exponential functions include:

• They grow rapidly, rapidly outpacing polynomial functions. Consequently, there's significant difference between \(g(x)\) and linear functions like \(f(x) = x\).
• The base \(e\), approximately 2.718, is crucial in natural processes since it arises naturally in calculus involving growth and decay.
• Exponential functions maintain positive values, never cross the x-axis, leading to consistently positive areas under their curves over any interval.

Understanding exponential functions assists in visualizing growth trends and logarithmic relationships. When comparing \(f(x)\) and \(g(x)\), the starkly different growth rates distinctly showcase their respective contributions to the total area between the curves.