This tutorial provides a detailed solution to problems in the differential calculus of functions of one and many independent variables. Practical lessons are preceded by basic theoretical information, reference data and formulas. The book is intended for students of higher technical educational institutions. Practical assignments are divided into the following topics:
1. Interval, segment, interval. The absolute value of the number. Properties of absolute values.
2. The quantities are constant and variable. Function. Function domain. Basic elementary functions.
3. Plotting functions.
4. Plotting trigonometric and inverse trigonometric functions.
5. Construction of graphs of functions given by several analytical expressions. Plotting the sum, difference and product of several functions.
6. Solving equations using graphs (Graphical solution of equations).
7. Inverse function and its graph. Periodic functions.
8. Sequences. Sequence limit.
9. Infinitesimal and infinitely large quantities.
10. Limit of function. Exercises to find the limit of a function.
11. Determining the limits of trigonometric functions and exercises on
use of the limit.
12. Number e.
13. Calculation of the limits of expressions containing logarithms and exponential functions.
14. Comparison of infinitesimal quantities.
15. Continuity of function. One-sided limits. Break points and their classification.
16. Tasks leading to the calculation of the derivative. Direct calculation of the derivative from the definition. The geometric and mechanical meaning of the 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.
20. Parametric representation of the function. Differentiation of functions specified parametrically.
21. Differential function.
22. Higher order derivatives Leibniz formula.
23. The limit of the ratio of two infinitesimal and two infinitely large
quantities (L'Hôpital's rule).
24. Increase and decrease of function. Determination of the maximum and minimum functions. The largest and smallest value of the function on the segment.
25. Points of inflection. Asymptotes.
26. General study of function.
27. Geometric applications of the derivative: equations of the tangent and normal to a plane curve. Tangent and normal lengths. Subtangent and normal and their lengths. Curvature, radius of curvature. Center of curvature. The relationship between the radius of curvature and the length of the normal. Evolution of the curve.
28. Functions of several independent variables. The area of ​​existence. Partial derivatives. Total increment and total first-order differential of a function of several independent variables.
29. Differentiation of a complex function from one or several independent variables.
30. Derivatives and differentials of higher orders of functions of several
independent variables.
31. Lines and surfaces of the level. Derivative of a function in a given direction. Function gradient.
32. Differentiation of implicit functions.
33. Extremum of a function of several independent variables. The largest and smallest values ​​of a function of two independent variables.
34. Tangent plane and normal to the surface.