Find the successive difference constant for the table below. What type of function is described here? Explain how you know. X 10 1 234 y -10 -1 18 47 86
Find the successive difference constant for the table below. What type of function is described here? Explain how you know. X 10 1 234 y -10 -1 18 47 86
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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### Educational Website Transcription
#### Geometry
**Rectangle and Box Dimensions**
- **Rectangle**
- **Area (A)**: \( A = lw \)
- \( l \): length
- \( w \): width
- Diagram: A rectangle is shown with sides labeled as \( l \) (length) and \( w \) (width).
- **Box**
- **Volume (V)**: \( V = lwh \)
- \( l \): length
- \( w \): width
- \( h \): height
- Diagram: A box (rectangular prism) is shown with dimensions labeled as \( l \), \( w \), and \( h \).
#### Linear Equations
- **Slope Formula**:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
- \( m \) is the slope of the line through points \((x_1, y_1)\) and \((x_2, y_2)\).
- **Point-Slope Formula**:
\[
(y - y_1) = m(x - x_1)
\]
- Used to find the equation of a line given a point \((x_1, y_1)\) and slope \( m \).
- **Slope-Intercept Formula**:
\[
y = mx + b
\]
- \( m \) is the slope and \( b \) is the y-intercept.
- **Standard Equation of a Line**:
\[
Ax + By = C
\]
- A linear equation in standard form where \( A \), \( B \), and \( C \) are constants.
#### Arithmetic Properties
- **Additive Inverse**:
\( a + (-a) = 0 \)
- **Multiplicative Inverse**:
\( a \cdot \frac{1}{a} = 1 \)
- **Commutative Property**:
- Addition: \( a + b = b + a \)
- Multiplication: \( a \cdot b = b \cdot a \)
- **Associative Property**:
- Addition: \( (a + b) + c = a](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe0228723-8b7c-4bd5-a03d-f7c714ce5bf0%2F42d0b0ab-e7ca-4044-8100-4847d7cd44dd%2Fbnagrl8_processed.png&w=3840&q=75)
Transcribed Image Text:---
### Educational Website Transcription
#### Geometry
**Rectangle and Box Dimensions**
- **Rectangle**
- **Area (A)**: \( A = lw \)
- \( l \): length
- \( w \): width
- Diagram: A rectangle is shown with sides labeled as \( l \) (length) and \( w \) (width).
- **Box**
- **Volume (V)**: \( V = lwh \)
- \( l \): length
- \( w \): width
- \( h \): height
- Diagram: A box (rectangular prism) is shown with dimensions labeled as \( l \), \( w \), and \( h \).
#### Linear Equations
- **Slope Formula**:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
- \( m \) is the slope of the line through points \((x_1, y_1)\) and \((x_2, y_2)\).
- **Point-Slope Formula**:
\[
(y - y_1) = m(x - x_1)
\]
- Used to find the equation of a line given a point \((x_1, y_1)\) and slope \( m \).
- **Slope-Intercept Formula**:
\[
y = mx + b
\]
- \( m \) is the slope and \( b \) is the y-intercept.
- **Standard Equation of a Line**:
\[
Ax + By = C
\]
- A linear equation in standard form where \( A \), \( B \), and \( C \) are constants.
#### Arithmetic Properties
- **Additive Inverse**:
\( a + (-a) = 0 \)
- **Multiplicative Inverse**:
\( a \cdot \frac{1}{a} = 1 \)
- **Commutative Property**:
- Addition: \( a + b = b + a \)
- Multiplication: \( a \cdot b = b \cdot a \)
- **Associative Property**:
- Addition: \( (a + b) + c = a
Transcribed Image Text:Find the successive difference constant for the table below. What type of function is described here? Explain how you know.
| x | y |
|----|-----|
| 0 | -10 |
| 1 | -1 |
| 2 | 18 |
| 3 | 47 |
| 4 | 86 |
**Explanation:**
The table provides values of a function with inputs \(x\) and corresponding outputs \(y\). To determine the type of function, observe the differences between successive \(y\)-values.
1. Calculate the first differences:
- \( -1 - (-10) = 9 \)
- \( 18 - (-1) = 19 \)
- \( 47 - 18 = 29 \)
- \( 86 - 47 = 39 \)
2. Calculate the second differences:
- \( 19 - 9 = 10 \)
- \( 29 - 19 = 10 \)
- \( 39 - 29 = 10 \)
Since the second differences are constant, the function is quadratic. Quadratic functions have a constant second difference, indicating a parabolic relationship.
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