Problem 1 - A Graphical Approach to Precalculus with Limits

Question

Let \(f(x)=2 x+1, x_{1}=0, x_{2}=2, x_{3}=4, x_{4}=6,\) and \(\Delta x=2\) (a) Find \(\sum_{i=1}^{4} f\left(x_{i}\right) \Delta x\) (b) The sum in part (a) approximates a definite integral by using rectangles. The height of each rectangle is given by the value of the function at the left endpoint. Write the definite integral that the sum approximates.

Step-by-Step Solution

Answer

The sum is 56, and it approximates \(\int_{0}^{8} (2x + 1) \, dx\).

Step 1: Calculate Function Values

For each given point \(x_i\) where \(x_1 = 0\), \(x_2 = 2\), \(x_3 = 4\), and \(x_4 = 6\), calculate the function value \(f(x_i)\). - \(x_1 = 0\): \(f(x_1) = 2(0) + 1 = 1\) - \(x_2 = 2\): \(f(x_2) = 2(2) + 1 = 5\) - \(x_3 = 4\): \(f(x_3) = 2(4) + 1 = 9\) - \(x_4 = 6\): \(f(x_4) = 2(6) + 1 = 13\)

Step 2: Sum of Products

Calculate \(\sum_{i=1}^{4} f(x_i) \Delta x\), where \(\Delta x = 2\).- Compute each product: - \(f(x_1) \Delta x = 1 \times 2 = 2\) - \(f(x_2) \Delta x = 5 \times 2 = 10\) - \(f(x_3) \Delta x = 9 \times 2 = 18\) - \(f(x_4) \Delta x = 13 \times 2 = 26\) - Sum: \(2 + 10 + 18 + 26 = 56\)

Step 3: Approximate the Definite Integral

The sum \(\sum_{i=1}^{4} f(x_i) \Delta x\) approximates the definite integral of \(f(x)\) over the interval from the starting point of \(x_1 = 0\) to the ending point after four increments of \(\Delta x = 2\). Therefore, the interval is \([0, 8]\). The definite integral this approximates is \[ \int_{0}^{8} (2x + 1) \, dx \]

Key Concepts

Rectangular Approximation Method

The Rectangular Approximation Method is a technique used to estimate the area under a curve, which in mathematics translates to approximating a definite integral. Imagine you have a curve defined by a function, such as the one in our exercise, and you want to calculate the area between the curve and the x-axis over a certain interval. Instead of calculating the exact area, which can be complex, we break the area into simpler shapes—rectangles in this case—and sum their areas to get an approximation.

• The height of each rectangle represents the value of the function at a specific point.
• The width is a fixed interval along the x-axis, noted as \( \Delta x \).
• In this scenario, each rectangle's height is determined by the function value at the left endpoint of each subinterval.

This method provides a good estimation and is particularly useful in scenarios where calculating the exact integral is challenging or impossible.

Function Evaluation

Function Evaluation is a foundational step in the Rectangular Approximation Method. In our exercise, we have a linear function \(f(x) = 2x + 1\). When evaluating this function, we plug in the given points \(x_1, x_2, x_3, \) and \(x_4\).

• For \(x_1 = 0\), the function value is \(f(0) = 2(0) + 1 = 1\).
• For \(x_2 = 2\), we find \(f(2) = 2(2) + 1 = 5\).
• Continuing with \(x_3 = 4\), the function gives \(f(4) = 2(4) + 1 = 9\).
• Finally, for \(x_4 = 6\), \(f(6) = 2(6) + 1 = 13\).

By calculating these values, we determine the heights of our rectangular approximations, using the left endpoint of each interval for maximum accuracy within the method.

Sum of Products

Sum of Products is a critical step in calculating an approximate area under a curve using the Rectangular Approximation Method. Once we have evaluated the function at specific points and determined \( \Delta x \), we compute the areas of our rectangles.Each rectangle's area is the product of the function value (height) and \( \Delta x \) (width).

• For \(x_1 = 0\), the product is \(1 \times 2 = 2\).
• For \(x_2 = 2\), we calculate \(5 \times 2 = 10\).
• For \(x_3 = 4\), it becomes \(9 \times 2 = 18\).
• For \(x_4 = 6\), the product is \(13 \times 2 = 26\).

Finally, adding these products results in the total approximated area \(2 + 10 + 18 + 26 = 56\). This sum gives us the total "lower" area estimate under the curve.

Interval Approximation

Interval Approximation involves choosing the correct interval over which you want to approximate the definite integral. In the exercise, we start from \(x_1 = 0\) and take four equal steps of \( \Delta x = 2 \) to cover an entire interval. This creates an interval from 0 to 8.The goal is to estimate the area under the function \(f(x) = 2x + 1\) from \(x = 0\) to \(x = 8\) using the width of each rectangle as \( \Delta x = 2\). By doing so, it facilitates a perspective on how the definite integral \( \int_{0}^{8} (2x + 1) \, dx \) is approximated using finite sub-intervals or steps.This division of the interval is crucial as it determines the resolution of your approximation. A smaller \( \Delta x \) increases accuracy but also requires more computations, while a larger \( \Delta x \) results in a broader and potentially less accurate approximation.