Sure! Let's work through the problem step-by-step.
### Problem Statement:
We need to find the values of [tex]\( y \)[/tex] for the function [tex]\( y = 2x^2 + 3x + 2 \)[/tex] within the interval [tex]\([-3, 3)\)[/tex].
### Step-by-Step Solution:
1. Define the Interval and Step Size:
- We need to consider the range of [tex]\( x \)[/tex] values from [tex]\(-3\)[/tex] to [tex]\( 2.9 \)[/tex]. The interval is [tex]\([-3, 3)\)[/tex], where [tex]\( x \)[/tex] ranges from [tex]\(-3\)[/tex] up to (but not including) [tex]\( 3 \)[/tex].
- We'll choose a step size of [tex]\( 0.1 \)[/tex] for [tex]\( x \)[/tex] values for a detailed evaluation.
2. List of [tex]\( x \)[/tex] Values:
The [tex]\( x \)[/tex] values in the specified range with a step of [tex]\( 0.1 \)[/tex] are:
[tex]\[
x = [-3.0, -2.9, -2.8, -2.7, -2.6, -2.5, -2.4, -2.3, -2.2, -2.1, -2.0, -1.9, -1.8, -1.7, -1.6, -1.5, -1.4, -1.3, -1.2, -1.1, -1.0, -0.9, -0.8, -0.7, -0.6, -0.5, -0.4, -0.3, -0.2, -0.1, 0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9, 2.0, 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 2.8, 2.9]
\][/tex]
3. Compute the Corresponding [tex]\( y \)[/tex] Values:
Using the function [tex]\( y = 2x^2 + 3x + 2 \)[/tex]:
- For [tex]\( x = -3.0 \)[/tex]:
[tex]\[
y = 2(-3.0)^2 + 3(-3.0) + 2 = 18 - 9 + 2 = 11.0
\][/tex]
- For [tex]\( x = -2.9 \)[/tex]:
[tex]\[
y = 2(-2.9)^2 + 3(-2.9) + 2 = 16.82 - 8.7 + 2 = 10.12
\][/tex]
- For [tex]\( x = -2.8 \)[/tex]:
[tex]\[
y = 2(-2.8)^2 + 3(-2.8) + 2 = 15.68 - 8.4 + 2 = 9.28
\][/tex]
- Continue this pattern for the rest of the [tex]\( x \)[/tex] values.
4. Full List of [tex]\( y \)[/tex] Values:
After computing for all [tex]\( x \)[/tex] values up to [tex]\( 2.9 \)[/tex], the corresponding [tex]\( y \)[/tex] values are:
[tex]\[
y = [11.0, 10.12, 9.28, 8.48, 7.72, 7.0, 6.32, 5.68, 5.08, 4.52, 4.0, 3.52, 3.08, 2.68, 2.32, 2.0, 1.72, 1.48, 1.28, 1.12, 1.0, 0.92, 0.88, 0.88, 0.92, 1.0, 1.12, 1.28, 1.48, 1.72, 2.0, 2.32, 2.68, 3.08, 3.52, 4.0, 4.52, 5.08, 5.68, 6.32, 7.0, 7.72, 8.48, 9.28, 10.12, 11.0, 11.92, 12.88, 13.88, 14.92, 16.0, 17.12, 18.28, 19.48, 20.72, 22.0, 23.32, 24.68, 26.08, 27.52]
\][/tex]
### Summary:
We evaluated the function [tex]\( y = 2x^2 + 3x + 2 \)[/tex] at intervals of [tex]\( 0.1 \)[/tex] within the range [tex]\([-3, 3)\)[/tex], leading to the series of [tex]\( x \)[/tex] values:
[tex]\[
x = [-3.0, -2.9, -2.8, \ldots, 2.8, 2.9]
\][/tex]
And their corresponding [tex]\( y \)[/tex] values are:
[tex]\[
y = [11.0, 10.12, 9.28, \ldots, 26.08, 27.52]
\][/tex]
