Functions of several variables. Full differential

Theoretical questions:

1. Concept of functions of several variables; 2. Basic properties of functions of several variables; 3. Partial derivatives and total differential; 4. Partial derivatives and higher order differentials; 5. Tangent and surface normal; 6. Extrema of functions of several variables.

Classroom assignments:

1. Find the domain of existence of functions:

1. 2. 3. 4. . 5.

2. Find the first order partial derivatives for the function:

1. 2. 3.

4.

3. Find the second order partial derivatives:

1. 2. 3.

4. 5.

4. Find derivatives of functions defined implicitly:

1. 2. 3. 4. 5.

5. Find , if .

6. Find , if

7. Find the third order partial derivatives:

8. Find the third order partial derivatives:

1. 2.

3.

9. Write the equation of the tangent plane and normal to the curve , given by the equation in point :

1. 2.

10. Find the extremum of functions:

1. 2. 3.

4. 5. (x>0, y>0).

Homework:

Theoretical material: Antiderivative and indefinite integral.

Solve problems:

1. Find the second order partial derivatives:

1. 2. 3. 4. 5. .

2. Find the third order partial derivatives:

1. 2.

3. 4.

3. Find the extremum of functions:

1. . 2.

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