Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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The image is a graph of the function \( f(x) = x^3 - 2x^2 + 3 \). ### Graph Details: - **Axes:** The graph contains a vertical axis (y-axis) and a horizontal axis (x-axis). Both axes are clearly labeled with increments. - **Curve:** The function is represented by a smooth purple curve. - **Behavior:** - The curve starts from the left, decreasing slightly and then increasing as it moves from \( x = -1 \) to \( x = 3 \). - The graph demonstrates a point of inflection near the origin, where the concavity changes. - As \( x \) increases, the curve rises steeply beyond \( x = 2 \). ### Function Characteristics: - **Equation:** \( f(x) = x^3 - 2x^2 + 3 \) - **Type:** Cubic polynomial - **General Form:** The equation is in the standard form of a cubic function: \( ax^3 + bx^2 + cx + d \), where \( a = 1 \), \( b = -2 \), \( c = 0 \), and \( d = 3 \). The graph visually represents how the cubic function behaves across the given range, capturing changes in direction and inflection points to illustrate polynomial growth.
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Transcribed Image Text:The image is a graph of the function \( f(x) = x^3 - 2x^2 + 3 \). ### Graph Details: - **Axes:** The graph contains a vertical axis (y-axis) and a horizontal axis (x-axis). Both axes are clearly labeled with increments. - **Curve:** The function is represented by a smooth purple curve. - **Behavior:** - The curve starts from the left, decreasing slightly and then increasing as it moves from \( x = -1 \) to \( x = 3 \). - The graph demonstrates a point of inflection near the origin, where the concavity changes. - As \( x \) increases, the curve rises steeply beyond \( x = 2 \). ### Function Characteristics: - **Equation:** \( f(x) = x^3 - 2x^2 + 3 \) - **Type:** Cubic polynomial - **General Form:** The equation is in the standard form of a cubic function: \( ax^3 + bx^2 + cx + d \), where \( a = 1 \), \( b = -2 \), \( c = 0 \), and \( d = 3 \). The graph visually represents how the cubic function behaves across the given range, capturing changes in direction and inflection points to illustrate polynomial growth.
### Approximating the Integral of a Function Using Midpoint Sums #### Problem Statement: a) Given the following graph of \( f(x) = x^3 - 2x^2 + 3 \) on the interval \([-1, 3]\). Approximate \( \int_{-1}^{3} f(x) \, dx \) by using the midpoint sums where \( n = 4 \). Shade the area that represents the midpoint sums. b) Find the absolute error for using the midpoint sums. (Recall: Absolute error = | True value - Approximation |) #### Explanation: - **Midpoint Rule:** - Divide the interval \([-1, 3]\) into 4 equal subintervals. - Identify the midpoint of each subinterval. - Calculate the area of rectangles using these midpoints and sum them up for an approximation of the integral. - **Absolute Error:** - Determine the true value of the integral. - Subtract the approximated value from the true value. - Take the absolute value of the result to find the absolute error. This exercise helps in understanding numerical integration techniques and error analysis.
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Transcribed Image Text:### Approximating the Integral of a Function Using Midpoint Sums #### Problem Statement: a) Given the following graph of \( f(x) = x^3 - 2x^2 + 3 \) on the interval \([-1, 3]\). Approximate \( \int_{-1}^{3} f(x) \, dx \) by using the midpoint sums where \( n = 4 \). Shade the area that represents the midpoint sums. b) Find the absolute error for using the midpoint sums. (Recall: Absolute error = | True value - Approximation |) #### Explanation: - **Midpoint Rule:** - Divide the interval \([-1, 3]\) into 4 equal subintervals. - Identify the midpoint of each subinterval. - Calculate the area of rectangles using these midpoints and sum them up for an approximation of the integral. - **Absolute Error:** - Determine the true value of the integral. - Subtract the approximated value from the true value. - Take the absolute value of the result to find the absolute error. This exercise helps in understanding numerical integration techniques and error analysis.
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