Problem 54 - A Graphical Approach to College Algebra

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. $$f(x)=6+2 x$$

Step-by-Step Solution

Answer

Evaluate each term: 3, 7, 11, 15; then sum: 36.

Step 1: Identify the function and increment

The given function is \( f(x) = 6 + 2x \) and the increment \( \Delta x = 0.5 \).

Step 2: Identify the operation required

Determine what computation or simplification is needed.

Step 3: Perform the calculation

Execute the computation step by step.

Step 4: Simplify and state the result

Simplify and write the final answer.

Step 5: Conclude with the answer

Evaluate each term: 3, 7, 11, 15; then sum: 36.

Key Concepts

Definite Integral

A definite integral represents the area under a curve from one point to another on the x-axis. Imagine it as summing up countless tiny rectangles beneath the curve of a function, between two specified limits. In our exercise, we're given specific values and segments along the x-axis. Each segment has a width, called an increment, denoted by \( \Delta x = 0.5 \). The definite integral is central to calculus as it helps calculate areas, volumes, and even solve complex problems involving rates of change.

• When you see a definite integral, you're observing a process of summation. Each small piece, or slice of the total area under the curve, corresponds to a specific value from your function.
• This concept forms the building block for numerous calculations, allowing you to grasp changes over specific intervals on the x-axis.

Executing these steps involves evaluating a function at specific points—the key to understanding how these small areas accumulate into a larger total.

Piecewise Calculation

Piecewise calculations refer to evaluating a function in segments, especially useful when dealing with non-continuous functions or specific intervals with different rules. However, in our exercise, we're handling continuous, regular intervals. Each segment is evaluated separately, which simplifies understanding how the total sum or integral comes together.

• For each \( x_i \) given, you calculate the value of the function. This represents the height of a rectangle in a Riemann sum.
• Multiply the result by \( \Delta x \), the width, creating small rectangle areas representing part of the function's area.

Continuing this way allows for the complete sum of these small calculated areas to approximate the total area under the function.

Function Evaluation

Function evaluation involves plugging specific values into a mathematical expression. In this exercise, you're given a linear function \( f(x) = 6 + 2x \), and you'll substitute each \( x_i \) to find the function value for those points. This simple substitution is incredibly important—it determines the height of your rectangles in a Riemann sum, directly impacting your total calculated area under the curve.

• Evaluate at each \( x_i \): calculate \( f(x_1), f(x_2), f(x_3), \) and \( f(x_4) \) using your function formula.
• This yields essential data for assessing each slice of your definite integral.

Such evaluations help you visualize the changes and understand step-by-step what this function represents and how it behaves at different points along the x-axis. With consistent practice, interpreting and calculating through function values becomes second nature.