en
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math math-format
Tangent plane in coordinates
diff2-02-01
multiple
3240000
randRangeNonZero(-4,4) randRangeNonZero(-4,4) randRangeExclude(-8,8,[0,1]) randRangeExclude(-8,8,[0,1]) randRangeExclude(-8,8,[0,1]) randRangeExclude(-8,8,[0,1]) A *pow(X,3) + B *X*Y +C *X + D *Y*Y

Consider a function f: \mathbb R^2 \to \mathbb R with f(x,y) = A x^3 + Bx y+ Cx + D y^2 and a point P = (x_0,y_0)= (X,Y).

Find {\color{blue} a}, {\color{teal} b} und {\color{red} c} such that E: z = {\color{blue} a}x + {\color{teal} b} y + {\color{red} c} defines the tangent plane of f in (X,Y, f( X,Y)).

a {\color{blue} a} = 3*A*X*X + B*Y + C
b {\color{teal} b}= B*X+ Y* D*2
c {\color{red} c} = Z - X*(3*A*X*X + B*Y + C) -Y*( B*X+ Y* D*2)

If the tangent plane in (x_0,y_0) exists, it is graph of the function \ell with

\ell (x,y) = f(x_0,y_0) + f_x(x_0,y_0) (x-x_0) + f_y(x_0,y_0) (y-y_0).

Using the given values one gets z= \ell (x,y) = f(X,Y) + f_x(X,Y) (x-X) + f_y(X,Y) (y-Y).

We compute the missing numbers f_x(x,y)= 3*A x^2 + B y + C and f_y(x,y)= B x+ 2*D y.

The point P = (x_0,y_0)= (X,Y) delivers f_x(X,Y)= 3*A*X*X + B*Y + C and f_y(X,Y)= B*X+ 2* Y* D.

Combining this to z = Z + 3*A*X*X + B*Y + C (x-X) + B*X+ Y* D*2 (y-Y) and we must do some sorting and cleaning.

Eventually z = 3*A*X*X + B*Y + C x + B*X+ Y* D*2 y +Z - X*(3*A*X*X + B*Y + C) -Y*( B*X+ Y* D*2) and {\color{blue} a =3*A*X*X + B*Y + C}, \color{teal} b = B*X+ Y* D*2 and {\color{red}c = Z - X*(3*A*X*X + B*Y + C) -Y*( B*X+ Y* D*2)}.