Find a Cartesian equation for the plane tangent to the hyperboloid x^2+y^2-z^2=25 at the point (

Find a Cartesian equation for the plane tangent to the...

Key Concepts

Tangent Plane

A tangent plane to a surface at a given point is the plane that just touches the surface at that point, matching its local behavior. For surfaces given by an implicit equation F(x, y, z) = 0, the tangent plane is determined using the normal vector at that point, which can be found by computing the gradient of F.

Gradient Vector

The gradient vector of a function F(x, y, z) is obtained by taking the partial derivatives with respect to x, y, and z. It points in the direction of the greatest rate of increase of the function and is orthogonal to the level surface defined by F(x, y, z) = 0, making it essential for formulating the equation of the tangent plane.

Implicit Differentiation and Level Surfaces

Surfaces defined implicitly by an equation F(x, y, z) = 0 are known as level surfaces. Implicit differentiation techniques rely on the gradient to extract information about rates of change on these surfaces, and are crucial for analyzing geometric properties like the tangent plane in the context of multivariable calculus.

Transcript

00:02 To find the targent plane, we can use the parameterization provided in exercise 34, where we had a equal to b, equal to c, equal to five.

00:18 So our parameterization, our teta u, it's equal to five, kosh u, because teta i, plus teta i, plus 5, push u sign theta j plus five sans y k where the interval for theta is from zero to two pi and u is from zero to two pi as well the point p which is x not y not z not x not y not in zero so the z component is zero the normal vector n is going to be r tita you have x not y0 zero cross r u to x not why not zero? so what do we do? we first need to find our theta, and this is equal to 5, negative 5, cosh u, sine, theta, i, plus 5, cosch, u .s, sign, cos.

02:27 So we differentiate sign and we have course cause tita 5 u so you have we have five course u cost u cause tita j then r u will be equal to five sange u cost teta i plus 5 u sign theta j plus five k u k okay so we notice that z equal to zero so where you have your z so for for z equal to zero implies that 5 such u .0 is equal to 0 so u .0 will be equal to 0 so then this implies that kosh u0 is equal to 1 then n so we have n n which is the cross products of this so you have i the i components of this so you have i the i components this is 5 sign theta notes this is 0 then you have your j components this is 5 cause theta not 0 then k components we have 0 5 this is equal to you have 25 cause tata i plus tata i plus 25 sign theta knot teta not j here.

05:24 So this is going to be, so you have 25.

05:33 This is equal to 25.

05:37 This will give us x not divided by square root of x not squared plus y0 squared plus y not squared i.

05:50 Plus 25 y.

05:54 Y not divided by square root of x not plus y not squared squared squared...

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