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The paraboloid z = 8-x-x²-3y2 intersects the plane x = 2 in a parabola. Find parametric equations in terms of t for the tangent line to this parabola at the point (2, 1, -1). Use a computer to graph the paraboloid, the parabola, and the tangent line on the same screen. (Enter your answers as comma-separated lists of equations. Use t as your parameter.)

The paraboloid z = 8-x-x²-3y2 intersects the plane x = 2 in a parabola. Find parametric equations in terms of t for the tangent line to this parabola at the point (2, 1, -1). Use a computer to graph the paraboloid, the parabola, and the tangent line on the same screen. (Enter your answers as comma-separated lists of equations. Use t as your parameter.)

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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**Transcription for Educational Website:** --- **Problem Statement:** The paraboloid \( z = 8 - x - x^2 - 3y^2 \) intersects the plane \( x = 2 \) in a parabola. Find parametric equations in terms of \( t \) for the tangent line to this parabola at the point \( (2, 1, -1) \). Use a computer to graph the paraboloid, the parabola, and the tangent line on the same screen. (Enter your answers as comma-separated lists of equations. Use \( t \) as your parameter.) --- **Detailed Explanation:** 1. **Paraboloid Equation**: \( z = 8 - x - x^2 - 3y^2 \) 2. **Plane Equation**: \( x = 2 \) **Intersection of Paraboloid and Plane**: By substituting \( x = 2 \) into the paraboloid equation, we can describe the parabola formed by their intersection. 3. **Given Point on Parabola**: \( (2, 1, -1) \) **Objective**: - **Find parametric equations** for the tangent line to the parabola at the point \( (2, 1, -1) \). - **Graph** the paraboloid, parabola, and the tangent line using a computer. This approach enables visualization of the 3D structure and understanding of how the tangent behaves at the given point of intersection. The parametric equations should be determined by calculating the gradient and finding the direction vectors at the specified point \( (2, 1, -1) \). **Graph Contents**: - A 3D **paraboloid surface** represented by the equation \( z = 8 - x - x^2 - 3y^2 \). - A **parabola** as the intersection curve within the plane \( x = 2 \). - The **tangent line** to the parabola at the point \( (2, 1, -1) \), depicted using its parametric equations. --- **Tools Required**: - Use of graphing software or a programming environment with 3D graphing capabilities (e.g., MATLAB, GeoGebra, Python with Matplotlib). By visualizing these components, deeper insight can be gained into the geometric properties and
Transcribed Image Text:**Transcription for Educational Website:** --- **Problem Statement:** The paraboloid \( z = 8 - x - x^2 - 3y^2 \) intersects the plane \( x = 2 \) in a parabola. Find parametric equations in terms of \( t \) for the tangent line to this parabola at the point \( (2, 1, -1) \). Use a computer to graph the paraboloid, the parabola, and the tangent line on the same screen. (Enter your answers as comma-separated lists of equations. Use \( t \) as your parameter.) --- **Detailed Explanation:** 1. **Paraboloid Equation**: \( z = 8 - x - x^2 - 3y^2 \) 2. **Plane Equation**: \( x = 2 \) **Intersection of Paraboloid and Plane**: By substituting \( x = 2 \) into the paraboloid equation, we can describe the parabola formed by their intersection. 3. **Given Point on Parabola**: \( (2, 1, -1) \) **Objective**: - **Find parametric equations** for the tangent line to the parabola at the point \( (2, 1, -1) \). - **Graph** the paraboloid, parabola, and the tangent line using a computer. This approach enables visualization of the 3D structure and understanding of how the tangent behaves at the given point of intersection. The parametric equations should be determined by calculating the gradient and finding the direction vectors at the specified point \( (2, 1, -1) \). **Graph Contents**: - A 3D **paraboloid surface** represented by the equation \( z = 8 - x - x^2 - 3y^2 \). - A **parabola** as the intersection curve within the plane \( x = 2 \). - The **tangent line** to the parabola at the point \( (2, 1, -1) \), depicted using its parametric equations. --- **Tools Required**: - Use of graphing software or a programming environment with 3D graphing capabilities (e.g., MATLAB, GeoGebra, Python with Matplotlib). By visualizing these components, deeper insight can be gained into the geometric properties and
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