Math Problem Statement
Solution
Let's start by solving the problem step-by-step.
We are given the surface equation: and asked to find the equations of the tangent plane and the normal line at the point .
1. Tangent Plane Equation:
The general formula for the equation of the tangent plane to a surface at a point is: Where , , and are the partial derivatives of the surface equation with respect to , , and , respectively.
However, the equation is already in the form , so we only need to compute the partial derivatives of the right-hand side with respect to and .
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Find partial derivatives:
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Evaluate these derivatives at the point :
Now, the tangent plane has the form: Substituting , , and :
- Simplify this equation to make the coefficient of equal to 1:
2. Normal Line Equation:
The direction vector for the normal line is given by the gradient of the function: So, the parametric equation for the normal line is:
Thus, the equations for the normal line are:
Would you like more details or clarifications? Here are 5 related questions for further practice:
- How would you find the tangent plane if the surface was implicitly given, such as ?
- Can you generalize the process to find the tangent plane to any surface ?
- What would happen to the normal line if we chose a different point on the surface?
- How does the method change if the surface equation is more complex?
- What is the geometric interpretation of the tangent plane and normal line in 3D space?
Tip: Always remember that the direction of the normal line is parallel to the gradient of the surface at a given point.
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Math Problem Analysis
Mathematical Concepts
Differentiation
Tangent Plane
Normal Line
Gradient
Formulas
Tangent plane equation: F_x(x_0, y_0, z_0)(x - x_0) + F_y(x_0, y_0, z_0)(y - y_0) + F_z(x_0, y_0, z_0)(z - z_0) = 0
Normal line equation: <x, y, z> = <x_0, y_0, z_0> + t<dF_x, dF_y, dF_z>
Theorems
Gradient theorem
Partial derivatives
Suitable Grade Level
Grades 11-12 or early university level
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