The total differential approximation works in three-dimensional space, too. For example, consider the paraboloid
z=F(x,y)=(x^(2))/(2^(2))+(y^(2))/(5^(2))
at the point
1,5,(5)/(4)
. The total differential approximation to
z=F(x,y)
near
1,5
is given by the formula
F(x,y)~~(5)/(4)+(delF)/(delx)(1,5)(x-1)+(delF)/(dely)(1,5)(y-5)
In fact, this gives the equation a plane tangent to the surface
z=F(x,y)
at the point
1,5,(5)/(4)
. Now
(delF)/(delx)(1,5)=
(delF)/(dely)(1,5)=
Q 줨. So we have the linear approximation to
F(x,y)
near
1,5
:
F(x,y)~~
Use this formula to approximate (to 3 decimal places)
F(1.1,5.1)~~
囯 Compute directly from the definition (to 3 decimal places) the value
F(1.1,5.1)=
图