Problem 53 - 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. $$f(x)=4 x-7$$

Step-by-Step Solution

Answer

The value of the summation is 10.

Step 1: Understand the Summation

The problem asks us to evaluate the summation \( \sum_{i=1}^{4} f(x_i) \Delta x \). This means we need to calculate the value of \( f(x_i) \Delta x \) for each \( x_i \) and then sum all these values together.

Step 2: Determine Function Values

Next, for each value \( x_i \), we will compute \( f(x_i) \), where \( f(x) = 4x - 7 \). For this calculation: - \( f(x_1) = f(0) = 4(0) - 7 = -7 \) - \( f(x_2) = f(2) = 4(2) - 7 = 1 \) - \( f(x_3) = f(4) = 4(4) - 7 = 9 \) - \( f(x_4) = f(6) = 4(6) - 7 = 17 \)

Step 3: Multiply Each Function Value by \( \Delta x \)

Now, multiply each \( f(x_i) \) by \( \Delta x = 0.5 \): - \( f(x_1) \Delta x = -7 \times 0.5 = -3.5 \) - \( f(x_2) \Delta x = 1 \times 0.5 = 0.5 \) - \( f(x_3) \Delta x = 9 \times 0.5 = 4.5 \) - \( f(x_4) \Delta x = 17 \times 0.5 = 8.5 \)

Step 4: Compute the Sum

Finally, add up all the products from the previous step to complete the summation: \[ \sum_{i=1}^{4} f(x_i) \Delta x = (-3.5) + 0.5 + 4.5 + 8.5 = 10 \]

Key Concepts

Function Evaluation

When tackling a problem that involves function evaluation, the key is to substitute each value into the function and find the result. Let's take the function given here: \( f(x) = 4x - 7 \). To find the function value for given \( x_i \) values, substitute each \( x_i \) into the function:

• For \( x_1 = 0 \), \( f(x_1) = 4(0) - 7 = -7 \).
• For \( x_2 = 2 \), \( f(x_2) = 4(2) - 7 = 1 \).
• For \( x_3 = 4 \), \( f(x_3) = 4(4) - 7 = 9 \).
• For \( x_4 = 6 \), \( f(x_4) = 4(6) - 7 = 17 \).

By computing these, you have found the result of the function at each specific point or \( x_i \). This step of hitting each \( x_i \) helps us bridge the input to output calculation.
Once these values are identified, they will be used in further steps towards solving the problem.

Summation

Summation is a fancy mathematical term that just means adding numbers together. In this exercise, we are asked to evaluate a summation expression \( \sum_{i=1}^{4} f(x_i) \Delta x \). This requires multiplying the value of the function at each point \( x_i \) with a small change in x (denoted as \( \Delta x \)), and then adding them all up.
In simpler terms, the summation asks us to:

• Calculate each \( f(x_i) \Delta x \).
• Add all these products together.

In our case, the products we calculated were:

• \( f(x_1) \Delta x = -3.5 \)
• \( f(x_2) \Delta x = 0.5 \)
• \( f(x_3) \Delta x = 4.5 \)
• \( f(x_4) \Delta x = 8.5 \)

Adding them all results in a total sum of 10. Always remember, the purpose of the summation is to total up these calculated amounts to achieve one overall value.

Delta x

The concept of \( \Delta x \) represents the change or interval in the x-values and plays a crucial role in calculations involving Riemann Sums. In this exercise, \( \Delta x = 0.5 \) represents a uniform space or interval separating each consecutive \( x_i \).
The purpose of \( \Delta x \) is to simulate how we can approximate changes over a stretch by breaking them into smaller parts. When working with Riemann Sums, multiplied by each \( f(x_i) \), \( \Delta x \) helps to estimate the area under the curve or other changes we want to compute.
For instance, multiplying the function value by \( \Delta x \) essentially accounts for this effect by scaling the functional output (which approximates the height in geometrical interpretations) over the small interval \( \Delta x \). This way, it balances the computation, summing up these small portions to get insight or estimate cumulative effects over a larger interval, as done in the example task.