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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
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
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
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.
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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