Tangent Planes and Linear Approximations

Just as in 2-space we can visualize the line tangent to a curve at a point, in 3-space we can picture the plane tangent to a surface at a point.

Consider the surface given by z = f(x,y). Let (x0,y0,z0) be any point on this surface. If f(x,y) is differentiable at (x0,y0), then the surface has a (not vertical) tangent plane at (x0,y0,z0). The equation of the tangent plane at (x0,y0,z0) is given by

 fx(x0,y0)(x-x0)+fy(x0,y0)(y-y0)-(z-z0) = 0

Notes

• Dove una funzione è differenziabile essa è anche continua. Quindi se esistono le derivate parziali, dato che (vedi) la funzione non è detto che sia continua, non è detto neppure che sia differenziabile
• Recall that the equation of the plane containing a point (x0,y0,z0) and normal to the vector n = (a,b,c) is
 a(x-x0)+b(y-y0)+c(z-z0) = 0.
The derivation of the equation for the tangent plane just involves showing that the tangent plane is normal to the vector n = (fx(x0,yy), fy(x0,y0),-1).

• For surfaces F(x,y,z) = 0 that are not easily solved for z, the equation of the tangent plane at (x0,y0,z0) is
 Fx(x0,y0,z0)(x-x0) + Fy(x0,y0,z0)(y-y0) + Fz(x0,y0,z0)(z-z0) = 0
provided that ∇F(x0,y0,z0) ≠ 0. Note that if we let F(x,y,z) = f(x,y)-z, we obtain the equation given for the tangent plane to z = f(x,y) at (x0,y0,z0).
In breve, il piano tangente in P0 è l'insieme dei P tali che  ∇F(P0) · (P-P0) = 0  ossia il piano tangente è il piano perpendicolare al vettore gradiente.
[così come nel caso di una curva F(x,y)=0 ∇F è perpendicolare alla retta tangente]

#### Example

Let's find the equation of the plane tangent to the surface z = 4x3y2+2y at the point (1,-2,12).

Since f(x,y) = 4x3y2+2y ,

 fx(x,y) = 12x2y2 and fy(x,y) = 8x3y +2.
With x = 1 and y = -2,
 fx(1,-2)
 =
 12(1)2(-2)2 = 48
 fy(1,-2)
 =
 8(3)3(-2)+2 = -14.
Thus, the tangent plane has normal vector n = (48,-14,-1) at (1,-2,12) and the equation of the tangent plane is given by
 48(x-1)-14( y-(-2)) -(z-12) = 0.
Simplifying,
 48x-14y-z = 64

exploration : what do you notice about the surface and the tangent plane near the point?

The tangent plane to a surface at a point stays close to the surface near the point. In fact, if f(x,y) is differentiable at the point (x0,y0), the tangent plane to the surface z = f(x,y) at (x0,y0) provides a good approximation to f(x,y) near (x0,y0):

Solving fx(x0,y0)(x-x0) + fy(x0,y0)(y-y0)-(z-z0) = 0 for z,
 z = z0 +fx(x0,y0)(x-x0) + fy(x0,y0)(y-y0).

Since z0 = f(x0,y0), we have that
 z = f(x0,y0)+fx(x0,y0)(x-x0) + fy(x0,y0)(y-y0).

Near (x0,y0), the surface is close to the tangent plane. Thus,
 f(x,y) ≈ f(x0,y0)+fx(x0,y0)(x-x0) + fy(x0,y0)(y-y0)

We call this the linear approximation or local linearization of f(x,y) near (x0,y0).

Notes

• The linear approximation is really just the multivariable Taylor polynomial of degree 1 for f(x,y) about (x0,y0). It is only accurate near (x0,y0). Better approximations can be obtained by using higher-order Taylor polynomials.

• These concepts can be extended to functions of more than two variables:
 f(x,y,z) ≈ f(x0,y0,z0)+fx(x0,y0,z0)(x-x0) +fy(x0,y0,z0)(y-y0) + fz((x0,y0,z0)(z-z0)
where f(x,y,z) is differentiable at (x0,y0,z0).

• Δf(x,y) = f(x,y)−f(x0,y0) ≈ fx(x0,y0)(x-x0) + fy(x0,y0)(y-y0) fornisce una stima dell'errore che si commette calcolando f in (x,y) invece che in (x0,y0).
Nel caso di f(x,y) = x·y, Δ(x·y) ≈ y0Δx+x0Δy, da cui Δ(x·y)/(x0·y0) ≈ Δx/x0+Δy/y0, che possiamo esprimere dicendo che la precisione relativa di un prodotto è circa uguale alla somma delle precisioni relative dei due termini moltiplicati. Questo era un metodo (un po' grossolano) per stimare la precisione di un prodotto di dati approssimati usato prima della diffusione dei mezzi di calcolo. Usando polinomi do Taylor di grado maggiore del primo si potrebbero ottenere stime migliori dell'errore commesso.

#### Example

From our work in the previous example, the linear approximation to f(x,y) = 4x3y2+2y near x = 1, y = -2 is

 f(x,y) ≈ 48x-14y-64.
This is, of course, exact at x = 1, y = -2:
 f(1,-2) = 12 = 48(1)-14(-2)-64.
At x = 1.1 and y = -1.9, according to the linear approximation,
 f(1.1,-1.9) ≈ 48(1.1) -14(-1.9)-64 = 15.4,
which is very close to the exact value f(1.1,-1.9) = 15.41964!

Key Concepts [index]

• Tangent Plane to a Surface

Let (x0,y0,z0) be any point on the surface z = f(x,y). If f(x,y) is differentiable at (x0,y0), then the surface has a tangent plane at (x0,y0,z0) given by

 fx(x0,y0)(x-x0)+fy(x0,y0)(y-y0)-(z-z0) = 0

• Linear Approximation to a Surface

If f(x,y) is differentiable at (x0,y0), then near (x0,y0)

 f(x,y) ≈ f(x0,y0)+fx(x0,y0)(x-x0) + fy(x0,y0)(y-y0).