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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