Tangent

Definition

Tangent Line

The tangent line to a plane curve at a given point is the straight line that just touches the curve at that point.

The tangent line at the point $P_0(x_0,y_0)$ on the level curve $f(x,y)=c$ of a differentiable differentiable function $f$ is the line through $P_0$ and normal to the gradient $\nabla f{|}_{p_0}$. $\nabla f(P_0{)}^{⊺}\cdot ⟨x-x_0,y-y_0⟩=0$ $f_x(P_0)(x-x_0)+f_y(P_0)(y-y_0)=0$

Tangent Plane

The tangent plane to a surface at a given point is the plane that just touches the surface at that point.

The tangent plane at the point $P_0(x_0,y_0,z_0)$ on the level surface $f(x,y,z)=c$ of a differentiable function $f$ is the plane through $P_0$ and normal to the gradient $\nabla f{|}_{p_0}$ $\nabla f(P_0{)}^{⊺}\cdot ⟨x-x_0,y-y_0,z-z_0⟩=0$ $f_x(P_0)(x-x_0)+f_y(P_0)(y-y_0)+f_z(P_0)(z-z_0)=0$

And the equation of the corresponding normal line is defined as $P_0+\nabla f(P_0)t,-\infty <t<\infty$

The plane tangent to the surface $z=f(x,y)$ of a differentiable function $f$ at the point $P_0(x_0,y_0,z_0)=(x_0,y_0,f(x_0,y_0))$ is $f_x(x_0,y_0)(x-x_0)+f_y(x_0,y_0)(y-y_0)-(z-z_0)=0$