Problem 16 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find an approximation of the area of the region \(R\) under the graph of the function \(f\) on the interval \([a, b] .\) In each case, use \(n\) subintervals and choose the representative points as indicated. \(f(x)=e^{x} ;[0,3] ; n=5 ;\) midpoints

Step-by-Step Solution

Answer

The approximation of the area of the region R under the graph of the function \(f(x) = e^x\) on the interval \([0,3]\) using \(n = 5\) subintervals with midpoint rectangles is approximately \(11.938\).

Step 1: Determine the interval width and width of each subinterval

The interval \([0,3]\) has a width of \(3\). Since we have \(n = 5\) subintervals, each subinterval will have a width of: \[ \Delta x = \frac{3-0}{5} = \frac{3}{5} \]

Step 2: Find the midpoints of each subinterval

To find the midpoints, we need to add half of the width, \(\frac{\Delta x}{2} = \frac{3}{10}\), to the start of each subinterval: \[ x_1 = 0 + \frac{3}{10} = \frac{3}{10} \] \[ x_2 = \frac{3}{5} + \frac{3}{10} = \frac{6}{10} = \frac{3}{5} \] \[ x_3 = \frac{6}{5} + \frac{3}{10} = \frac{9}{10} \] \[ x_4 = \frac{9}{5} + \frac{3}{10} = \frac{12}{10} = \frac{6}{5} \] \[ x_5 = \frac{12}{5} + \frac{3}{10} = \frac{15}{10} = \frac{3}{2} \]

Step 3: Calculate the height of each rectangle using the given function

Now we need to find the height of each rectangle by plugging in the midpoint (x-values) we found in the previous step into the given function: \[ f(x_1) = e^{\frac{3}{10}} \] \[ f(x_2) = e^{\frac{3}{5}} \] \[ f(x_3) = e^{\frac{9}{10}} \] \[ f(x_4) = e^{\frac{6}{5}} \] \[ f(x_5) = e^{\frac{3}{2}} \]

Step 4: Find the area of each rectangle and sum their areas

To find the area of each rectangle, we simply multiply their height by their width, \(\Delta x\). Then, add them up to get the total area: \[ A \approx \Delta x \sum_{i=1}^{5} f(x_i) \] \[ A \approx \frac{3}{5} \left( e^{\frac{3}{10}} + e^{\frac{3}{5}} + e^{\frac{9}{10}} + e^{\frac{6}{5}} + e^{\frac{3}{2}} \right) \]

Step 5: Write the final answer as an approximation of the area under the curve

Plugging the numbers into a calculator to get the final result gives: \[ A \approx \frac{3}{5} \left( 1.34985880758 + 1.82211880039 + 2.45960311116 + 3.32011692212 + 4.48168907034 \right) \] \[ A \approx 11.9384442268 \] Hence, the approximation of the area of the region R under the graph of the function \(f(x) = e^x\) on the interval \([0,3]\) using \(n = 5\) subintervals with midpoint rectangles is approximately \(11.938\).

Key Concepts

Midpoint Rule

The midpoint rule is a numerical technique used to approximate the area under a curve, which is often required when dealing with complex functions. In numerical integration, we use this method by considering subintervals within the interval of interest.
For instance, if we want to approximate the integral of a function over an interval \[ [a, b] \], we divide this interval into smaller, equal-width subintervals. Middle points of these subintervals serve as the points where we evaluate the function.

• The formula for the midpoint rule when using \( n \) subintervals can be expressed as:
• \[ \Delta x = \frac{b-a}{n} \]
• The midpoint \( x_i \) of each subinterval is \( a + (i-0.5)\Delta x \).
• Each subinterval results in a rectangle and the function value at \( x_i \) gives its height.
• Total area \( A \) is: \[ A \approx \Delta x \sum_{i=1}^{n} f(x_i) \]

This rule provides a fairly accurate estimation and is popular due to its simplicity and ease of implementation, especially with functions that may not have straightforward antiderivatives.

Exponential Functions

Exponential functions are a class of mathematical functions that are characterized by their constant growth rate. They have the general form \( f(x) = e^{x} \), where \( e \) is a mathematical constant approximately equal to 2.71828.
Understanding these functions is crucial because they frequently appear in various fields, such as physics, finance, and biology. Their unique properties make them essential for modeling growth processes and describing phenomena such as compound interest, population growth, and radioactive decay.
Some important properties of exponential functions include:

• The derivative of \( e^{x} \) is \( e^{x} \).
• \( e^{0} = 1 \), meaning the function value is 1 when \( x = 0 \).
• The graph of \( e^{x} \) is always rising, reflecting its continuous growth.
• The growth rate is directly proportional to the value of the function, an attribute making it unique among mathematical functions.

These fundamental principles are what make exponential functions both fascinating and practical in solving real-world problems.

Approximation Techniques

Approximation techniques in mathematics are used to find numerical solutions to problems that might be difficult or impossible to solve exactly. These methods are extraordinarily valuable in calculus, especially where integrals or derivatives of functions are complex or not easily computed analytically.
Some common techniques include:

• Riemann Sums: Utilizes rectangles to approximate the area under curves.
• Trapezoidal Rule: Averages the function's value at each subinterval's endpoints.
• Simpson's Rule: Uses parabolic arcs instead of line segments to achieve a better approximation.

These techniques can significantly simplify the computational process:

• The midpoint rule is a specific form of Riemann Sum, using midpoints to better approximate the integral.
• Such methods help cope with the limitations of direct calculations, offering practical solutions to complex integrals.
• When implementing these techniques, one needs to balance between precision and computational efficiency.

Thus, numerical approximation is a powerful tool, extending the reach of calculus beyond just analytical expressions and into practical, real-world problem-solving.