No. The point (5, 3 -3) has to satisfy both the equation of the plane and the equation of the surface.ktobrien said:Homework Statement
Find equations of the following.
x2-2y2+z2+yz=7, (5,3,-3)
(a) the tangent plane
(b) the normal line to the given surface at the point
Homework Equations
I know it involves fx, fy, fz
The Attempt at a Solution
I got 10x-15y-3z=7. Is this correct? Because its not true at the point (5,3,-3).
For one thing, this is not an equation, so there's no way to get an equation out of it. For another thing, if my memory is correct, the equation of the tangent plane is fx(5, 3, -3)(x - 5) + fy(5, 3, -3)(y - 3) + fz(5, 3, -3)(z - (-3)) = 0.ktobrien said:I got it by f(5,3,-3)+fx(5,3,-3)(x-5)+fy(5,3,-3)(y-3)+fz(5,3,-3)(z+3)
ktobrien said:As for the normal line I know the answers are
x=10t+5
y=-15+3
z=-3-3
The equation of the tangent plane is a mathematical representation of a plane that touches a curved surface at a specific point, known as the point of tangency. It is used to approximate the behavior of a surface at that point.
The equation of the tangent plane is calculated using the partial derivatives of the function representing the curved surface at the point of tangency. These derivatives are used to find the slope of the tangent line, which is then used to determine the equation of the tangent plane.
The equation of the tangent plane is used to approximate the behavior of a curved surface at a specific point. It is especially useful in the fields of mathematics, physics, and engineering, where it is used to analyze and solve problems involving curved surfaces.
Yes, the equation of the tangent plane can be applied to any type of surface, as long as the surface is differentiable at the point of tangency. This means that the surface must be smooth and continuous at that point.
The equation of the tangent plane has various applications in fields such as computer graphics and computer-aided design (CAD). It is also used in optimization problems, where it helps in finding the maximum or minimum values of a function at a specific point.