Practical class № 4. Function. The function limit. Fundamental theorems on limits

Theoretical questions:

1. Definition limit of function in point. 2. Limit of function in infinite. 3. Infinitely small and infinitely large functions (definition). 4. Basic theorems of limits. 5. Remarkable limits. 6. Definition and properties of infinitely function. 7. Comparing the orders of infinitely small. 8. Application of the infinitesimal to the calculation of limits. 9. One-sided limits. 10. Continuity of function. 11. Classification of points of discontinuity.

Classroom assignments:

1. Using the properties of limits of functions, find the following limits:

1) ; 2) ; 3) ;

4) ; 5) ;

2. Find limits: 1) ; 2) .

3. Find limits: 1) ; 2) ;

3) ;

4. Using equivalents infinite smalls, find the following limits:

1. 2. 3.

4. 5. 6.

5. Investigate continuity of the function and establish character of discontinuity points:

1. 2.

3. .

Homework:

Theoretical material: Derivative of function. Differential of function.

Solve problems:

1. Find limits of functions: а)

б) в)

г) д)

2. Investigate continuity of the function, find discontinuity points and establish character of discontinuity points.

а) б) .

Practical class № 5. The derivative of the function. Table of derivatives. The differential of a function

Theoretical questions:

1. Definition of derivative. 2. The geometric meaning of derivative. 3. The basic rules of finding derivatives. 4. Derivatives of the basic elementary functions. 5. Derivative of composite functions, implicit functions, the functions specified parametrically, the method of logarithmic differentiation. 6. Application of the differential to approximate calculations.

Classroom assignments:

1. Find derivatives of following functions:

1) ; 2) ; 3) ;

4) ; 5) ; 6) ;

7) ; 8) ; 9) ; 10) ;

11) ; 12) ; 13) .

2. Calculate derivatives of following implicit functions:

1. 2.

3. 4.

3. Calculate derivatives of following functions defined parametrically:

1. 2. 3. 4.

4. Using the method of logarithmic differentiation calculate derivatives of following functions:

1. 2. 3.

4. 5. 6.

5. Find differential of functions:

1. 2. 3.

4. 5.

6. Find the approximate value:

1. 2. 3. 4. 5. 6. 7.

Homework:

Theoretical material: L'Hopital's rule. Investigation of the function. Extremum of the function. Convexity, concavity and inflection points. Asymptote.

Solve problems:

1. Find derivatives of the following functions applying the formulas and rules of differentiation:

а) ; б) ;

в) ; г) ;

д) ; е) .

2. Calculate derivatives of functions:

1. 2. 3.

4. 5.

3. Show that the function satisfies the relation .

4. Find differentials of functions:

а) ; б) в)

5. Using a differential calculate approximately: а) ; б)

6. For function а) б)