This tutorial provides a detailed solution of problems of differential calculus of functions of one and several independent variables. Practical training prior to basic theoretical information, reference data and formulas. The book is intended for students of higher technical educational institutions. Practical tasks are distributed on the following topics:
1. Interval, interval, gap. The absolute magnitude of the number. The absolute values of the properties.
2. The values of constants and variables. Function. The region of existence functions. Basic elementary functions.
3. Plotting functions.
4. Plotting trigonometric and inverse trigonometric functions.
5. Plotting functions defined by several analytical expressions. Building a schedule sum, difference and product of several functions.
6. The solution of using graphs (Graphical solution of equations).
7. The reverse function and its graph. Periodic functions.
8. Sequences. Tensile sequence.
9. The infinitely small and infinitely large quantities.
10. The limit of a function. Exercise of finding the limit function.
11. Determination of the limits of trigonometric functions and exercises on
the use limit.
12. The number e.
13. Computation of limits expressions involving logarithms and exponentials.
14. Comparison of infinitesimals.
15. Continuity function. One-sided limits. break points and their classification.
16. Objectives, leading to the calculation of the derivative. A direct calculation of the derivative definition. Geometrical and mechanical meaning of derivative.
17. Differentiation of algebraic, trigonometric, inverse trigonometric functions.
18. Differentiation of logarithmic and exponential functions. Logarithmic differentiation.
19. Hyperbolic functions. Differentiation of Hyperbolic Functions. Differentiation of implicit functions.
20. Parametric representation of functions. Differentiation of functions,
defined parametrically.
21. Differential function.
22. Derivatives of higher orders Leibniz Formula.
23. The limit of the ratio of two infinitely small and infinitely large two
values (L'Hospital's rule).
24. Increase and decrease function. Determination of the maximum and minimum functions. The largest and smallest value of the function on the interval.
25. The inflection points. Asymptote.
26. A general investigation of the function.
27. Geometric applications derivative: the equation of the tangent and normal to the plane of the curve. The length of the tangent and normal. Subtangent and normal, and their length. The curvature radius of curvature. the center of curvature. The ratio between the curvature radius and the normal length. Evolute curve.
28. The functions of several independent variables. The region of existence. Partial derivatives. Total increment and total differential of the first order function of several independent variables.
29. Differentiation of composite functions of one and several independent variables.
30. Derivatives and differentials of higher order multiple functions
independent variables.
31. The lines and level surface. Derivative in a given direction. Gradient function.
32. Differentiation of implicit functions.
33. The extremum of a function of several independent variables. Maximum and minimum values of a function of two independent variables.
34. The tangent plane and normal to the surface.
1. Interval, interval, gap. The absolute magnitude of the number. The absolute values of the properties.
2. The values of constants and variables. Function. The region of existence functions. Basic elementary functions.
3. Plotting functions.
4. Plotting trigonometric and inverse trigonometric functions.
5. Plotting functions defined by several analytical expressions. Building a schedule sum, difference and product of several functions.
6. The solution of using graphs (Graphical solution of equations).
7. The reverse function and its graph. Periodic functions.
8. Sequences. Tensile sequence.
9. The infinitely small and infinitely large quantities.
10. The limit of a function. Exercise of finding the limit function.
11. Determination of the limits of trigonometric functions and exercises on
the use limit.
12. The number e.
13. Computation of limits expressions involving logarithms and exponentials.
14. Comparison of infinitesimals.
15. Continuity function. One-sided limits. break points and their classification.
16. Objectives, leading to the calculation of the derivative. A direct calculation of the derivative definition. Geometrical and mechanical meaning of derivative.
17. Differentiation of algebraic, trigonometric, inverse trigonometric functions.
18. Differentiation of logarithmic and exponential functions. Logarithmic differentiation.
19. Hyperbolic functions. Differentiation of Hyperbolic Functions. Differentiation of implicit functions.
20. Parametric representation of functions. Differentiation of functions,
defined parametrically.
21. Differential function.
22. Derivatives of higher orders Leibniz Formula.
23. The limit of the ratio of two infinitely small and infinitely large two
values (L'Hospital's rule).
24. Increase and decrease function. Determination of the maximum and minimum functions. The largest and smallest value of the function on the interval.
25. The inflection points. Asymptote.
26. A general investigation of the function.
27. Geometric applications derivative: the equation of the tangent and normal to the plane of the curve. The length of the tangent and normal. Subtangent and normal, and their length. The curvature radius of curvature. the center of curvature. The ratio between the curvature radius and the normal length. Evolute curve.
28. The functions of several independent variables. The region of existence. Partial derivatives. Total increment and total differential of the first order function of several independent variables.
29. Differentiation of composite functions of one and several independent variables.
30. Derivatives and differentials of higher order multiple functions
independent variables.
31. The lines and level surface. Derivative in a given direction. Gradient function.
32. Differentiation of implicit functions.
33. The extremum of a function of several independent variables. Maximum and minimum values of a function of two independent variables.
34. The tangent plane and normal to the surface.
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