Problem 56 Evaluate the terms of \(\sum_{i=... [FREE SOLUTION]

91影视

A Graphical Approach to Precalculus with Limits

John Hornsby, Margaret L. Lial, Gary Rockswold

$Math Studyset 91影视 Explanations$ Math

7 Edition

Chapter 12: Problem 56

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

Short Answer

Expert verified

The evaluated sum is 24.

Step by step solution

Understand the Problem

We have a summation of the form $\sum_{i=1}^{4} f(x_i) \Delta x$ where we need to evaluate $f(x)$ at specific values of $x$ and multiply each by $\Delta x$. The function given is $f(x) = 6 + 2x$.

Evaluate f(x) at x1

Substitute $x_1 = 0$ into $f(x) = 6 + 2x$. Thus, $f(0) = 6 + 2(0) = 6$.

Evaluate f(x) at x2

Substitute $x_2 = 2$ into $f(x) = 6 + 2x$. Thus, $f(2) = 6 + 2(2) = 10$.

Evaluate f(x) at x3

Substitute $x_3 = 4$ into $f(x) = 6 + 2x$. Thus, $f(4) = 6 + 2(4) = 14$.

Evaluate f(x) at x4

Substitute $x_4 = 6$ into $f(x) = 6 + 2x$. Thus, $f(6) = 6 + 2(6) = 18$.

Multiply Each Term by 螖x

Multiply each evaluated $f(x_i)$ by $\Delta x = 0.5$. Thus, $f(0)\Delta x = 6 \times 0.5 = 3$, $f(2)\Delta x = 10 \times 0.5 = 5$, $f(4)\Delta x = 14 \times 0.5 = 7$, $f(6)\Delta x = 18 \times 0.5 = 9$.

Sum the Results

Add up all the results from Step 6: $3 + 5 + 7 + 9 = 24$.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Function Evaluation

Function evaluation is a fundamental step when working with Riemann sums. It helps you understand how the function behaves at specific points within the interval. In our exercise, the function $ f(x) = 6 + 2x $ needs to be evaluated at several specific values of $ x $. This means we substitute the given $ x_i $ values into the function to find $ f(x_i) $. Let's look at how this is done:

For $ x_1 = 0 $, substitute into the function to get $ f(0) = 6 + 2(0) = 6 $.
Similarly, for $ x_2 = 2 $, substitute to find $ f(2) = 6 + 2(2) = 10 $.
Proceed in this manner for $ x_3 = 4 $ and $ x_4 = 6 $ to get $ f(4) = 14 $ and $ f(6) = 18 $ respectively.

This step helps in preparing each term that will be later multiplied by $ \Delta x $. Each function evaluation gives you a piece of the puzzle needed to form the Riemann sum.

Delta x

$ \Delta x $ is a small change in $ x $ that represents the width of each subinterval in a partitioned interval when performing a Riemann sum. In the context of calculating an integral with Riemann sums, $ \Delta x $ plays a crucial role in determining the approximation of the area under a curve. Initially, choose your points for $ x $ values, for instance, $ x_1, x_2, x_3, $ and $ x_4 $ from the interval:

In our problem, $ \Delta x $ is given as 0.5, which means each segment between consecutive $ x_i $ values (e.g., from $ x_1 $ to $ x_2 $) will be 0.5 units wide.
Every function evaluation from earlier must now be multiplied by this $ \Delta x $.

This width $ \Delta x $ is a significant feature in forming each term of the Riemann sum, allowing us to approximate the area efficiently.

Summation

Summation is the process of adding a sequence of numbers, which in this case, are the function evaluations multiplied by $ \Delta x $. It's the final and crucial step in calculating a Riemann sum. Once each of the $ f(x_i) $ values has been appropriately evaluated and adjusted for $ \Delta x $, the resulting numbers are summed up:

Our calculations reveal these values: $ f(0) \times \Delta x = 3 $, $ f(2) \times \Delta x = 5 $, $ f(4) \times \Delta x = 7 $, and $ f(6) \times \Delta x = 9 $.
Adding these gives us the total: $ 3 + 5 + 7 + 9 = 24 $.

Summation in Riemann sums brings together all evaluated pieces over the interval to approximate the integral of a function.

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91影视

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

Short Answer

Step by step solution

Understand the Problem

Evaluate f(x) at x1

Evaluate f(x) at x2

Evaluate f(x) at x3

Evaluate f(x) at x4

Multiply Each Term by 螖x

Sum the Results

Key Concepts

Function Evaluation

Delta x

Summation

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