Problem 55 - A Graphical Approach to Precalculus with Limits

Question

Evaluate the terms of \(\sum_{i=1}^{4} f\left(x_{i}\right) \Delta x,\) with \(x_{1}=0, x_{2}=2, x_{3}=4, x_{4}=6,\) and \(\Delta x=0.5,\) for each function. $$f(x)=4 x-7$$

Step-by-Step Solution

Answer

The evaluated sum is 10.

Step 1: Understand the Task

We need to evaluate the sum \(\sum_{i=1}^{4} f\left(x_{i}\right) \Delta x\), which means calculating \(f(x_i)\) for each \(x_i\) and then multiplying it by \(\Delta x\), and finally summing all the terms.

Step 2: Evaluate f at x1

Here, apply the function \(f(x) = 4x - 7\) for \(x_1 = 0\): \[ f(x_1) = 4(0) - 7 = -7 \] Then multiply by \(\Delta x = 0.5\): \[ f(x_1) \Delta x = (-7) \times 0.5 = -3.5 \]

Step 3: Evaluate f at x2

Apply the function \(f(x) = 4x - 7\) for \(x_2 = 2\): \[ f(x_2) = 4(2) - 7 = 1 \] Then multiply by \(\Delta x = 0.5\): \[ f(x_2) \Delta x = 1 \times 0.5 = 0.5 \]

Step 4: Evaluate f at x3

Apply the function \(f(x) = 4x - 7\) for \(x_3 = 4\): \[ f(x_3) = 4(4) - 7 = 9 \] Then multiply by \(\Delta x = 0.5\): \[ f(x_3) \Delta x = 9 \times 0.5 = 4.5 \]

Step 5: Evaluate f at x4

Apply the function \(f(x) = 4x - 7\) for \(x_4 = 6\): \[ f(x_4) = 4(6) - 7 = 17 \] Then multiply by \(\Delta x = 0.5\): \[ f(x_4) \Delta x = 17 \times 0.5 = 8.5 \]

Step 6: Evaluate the Sum

Now sum all the terms: \[ \sum_{i=1}^{4} f(x_i) \Delta x = (-3.5) + 0.5 + 4.5 + 8.5 = 10 \]

Key Concepts

Function Evaluation

In this exercise, we are tasked with evaluating a function at specific points. The function given is a linear one, defined as \( f(x) = 4x - 7 \). Evaluating a function means substituting the variable \( x \) with given values (denoted as \( x_i \)) to find an output. This output gives us a numerical result specific to each input.

• First, identify all \( x_i \) values where the function needs to be evaluated. Here, \( x_1 = 0, x_2 = 2, x_3 = 4, x_4 = 6 \).
• For each \( x_i \), plug it into the function \( f(x) \).
• Calculate to find \( f(x_1), f(x_2), f(x_3), \) and \( f(x_4) \).

For instance, for \( x_1 = 0 \), the calculation is straightforward: \[f(x_1) = 4(0) - 7 = -7\]Repeat this process for each provided \( x_i \). This allows us to understand how the function behaves at these key points.

Definite Integral Approximation

Riemann Sums are a concept used to approximate the area under a curve between two points, leading us to the concept of a definite integral. In this exercise, the sum \[\sum_{i=1}^{4} f(x_i) \Delta x\]is a simple example of such an approximation. Here, \( \Delta x = 0.5 \) signifies the width of each sub-interval over the integration range.

• The width \( \Delta x \) is constant, reflecting equal partitioning of the range.
• The function values \( f(x_i) \) approximate the height of rectangles, representing slices of the area under the curve.

This approach helps understand integral calculus by breaking down the complex curve into manageable and calculable steps. By multiplying each function value \( f(x_i) \) by \( \Delta x \), and then summing them up, we approximate the area under the curve from \( x_1 \) to \( x_4 \). It is like calculating the total shaded region using rectangle areas where:\[sum = (f(x_1) \times \Delta x) + (f(x_2) \times \Delta x) + (f(x_3) \times \Delta x) + (f(x_4) \times \Delta x)\] This sum gives a numerical approximation equivalent to using rectangles to mimic the true shape of the curve.

Step-by-Step Solution

Following a methodical, step-by-step process ensures clarity and understanding. This exercise unravels the approximation through a sequence of evaluations and multiplications:1. **Calculate Function Values:**
Evaluate \( f(x) \) at each specified \( x_i \): - For \( x_1 = 0 \), \( f(x_1) = -7 \) - For \( x_2 = 2 \), \( f(x_2) = 1 \) - For \( x_3 = 4 \), \( f(x_3) = 9 \) - For \( x_4 = 6 \), \( f(x_4) = 17 \)2. **Multiply by \( \Delta x \):**
Multiply each function output by \( \Delta x = 0.5 \): - \( f(x_1) \times \Delta x = -3.5 \) - \( f(x_2) \times \Delta x = 0.5 \) - \( f(x_3) \times \Delta x = 4.5 \) - \( f(x_4) \times \Delta x = 8.5 \)3. **Add All Results:**
Finally, sum up these products: \[ (-3.5) + 0.5 + 4.5 + 8.5 = 10 \]This step-by-step process transforms theoretical principles into practical calculations. It not only supports function evaluation but also unfolds the logic of Riemann sums for approximating definite integrals.