Lecture II

1. Lecture II The elements of higher mathematics . The derivative of function

2. Lecture questions • Function • Representation of a function • Function derivative • Geometricinterpretation of function derivative • Some differentiation rules • Physical interpretation of function derivative • Second-order derivative, higher derivatives • Extremum of function

3. Definition of function. Independent and dependent variables. The domain and the range

4. A relationship between two variables, typically x and y, is called a function if there is a rule that assigns to each value of x one and only one value of y. When that is the case, we say that y is a function of argument of x.

5. The values that x may assume are called the domain of the function. We say that those are the values for which the function is defined. • Once the domain has been defined, then the values of y that correspond to the values of x are called the range.

6. Representation of a function There are many ways to represent or visualize functions: a function may be described by a formula, by a plot or graph, by an algorithm that computes it, by arrows between objects, or by a description of its properties. Sometimes, a function is described through its relationship to other functions (for example, inverse functions). In applied disciplines, functions are frequently specified by tables of values or by formulas. The equation y = ƒ(x) is viewed as a functional relationship between dependent and independent variables.

7. Differential calculus • Differential calculus, a field in mathematics, is the study of how functions change when their inputs change. The primary object of study in differential calculus is the derivative. A closely related notion is the differential.

8. Increments • The increment of a variable in changing from one numerical value to another is the difference found by subtracting the second value from the first. An increment of x is denoted by the symbol • And the corresponding increment of the function is

9. Function derivative • The derivative of a function is the limit of the ratio of the function increment to the argument increment, when the latter increment varies and approaches zero. If the limit exists, then f(x) is differentiable at x.

10. Derivative notations • Leibniz's notation • Lagrange's notation • Euler's notation • Newton's notation

11. General Rule for differentiation • First Step: In a function replace x by , giving a new value of the function, • Second Step: Subtract the given value of the function from the new value in order to find (the increment of the function). • Third Step: Divide the remainder by . • Fourth Step: Find the limit of this quotient, when varies and approaches the limit zero. This is the derivative required.

12. Geometrical interpretation of the derivative of a function

13. Geometrical interpretation of the derivative of a function • The value of the derivative at any point of a curve is equal to the slope of the line drawn tangent to the curve at that point.

14. Differentiation of • Constant • Power function • Exponential functions • Logarithmic functions

15. Differentiation of • Trigonometric functions

16. Rules of differentiation

17. Constant factor rule: • The derivative of the product of a constant and a function is equal to the product of the constant and the derivative of the function.

18. Sum rule: • The derivative of the algebraic sum of a finite number of functions is equal to the same algebraic sum of their derivatives.

19. Product rule: • The derivative of the product of two functions is equal to the derivative of the first function times the second function plus the derivative of the second function times the first function.

20. Quotient rule: • The derivative of a fraction is equal to the derivative of the numerator times the denominator, minus the derivative of the denominator times the numerator, all divided by the square of the denominator.

21. Chain rule: • If f is a function of g and g is a function of x, then the derivative of f with respect to x is equal to the derivative of f(g) with respect to g times the derivative of g(x) with respect to x.

22. Physical interpretation of the derivative of a function • a derivative of a coordinate (displacement or distance from the original position) with respect to time is a velocity. • Generalization: the derivative of a function at a given point is the instantaneous rate of change of the function with respect to its argument at a given point.

23. Second-order derivative, higher derivatives • The derivative of a function of x is in general also a function of x. This new function may also be differentiable, in which case the derivative of the first derivative is called the second derivative of the original function. • Similarly, the derivative of the second derivative is called the third derivative; and so on to the nth derivative.

24. Notations of the second derivative: of a higher order derivatives:

25. Applications of derivatives in investigation of functions • Domain interior points, in which a derivative of a function is equal to zero or doesn’t exist, are critical points of this function. • Critical points divide a function domain for intervals that within each of them a derivative saves a constant sign. • If at every point of an interval , then a function f(x) increases within this interval. • If at every point of an interval , then a function f(x) decreases within this interval.

26. Extreme points • Sufficient conditions of extreme: • if a derivative changes its sign from plus to minus at a critical point, then that is a point of maximum. If a derivative changes its sign from minus to plus at a critical point, then that is a point of minimum.

27. At B and R points, where a function is increasing, the tangent makes an acute angle with the axis of x, hence the slope is positive. At M and Q points, where a function is decreasing, the tangent makes an obtuse angle with the axis of x, therefore the slope is negative.C and P points are maxima. N point is a minimum.

28. To find intervals of function increase and decrease and the extreme points • Find the domain of a function. • Differentiate the function. • Obtain the roots of equation: . • Form open intervals with the zeros (roots) of the first derivative and the points of discontinuity (if any). • Take a value from every interval and find the sign they have in the first derivative. • Determine the intervals of function increase and decrease andextreme points, where the derivative changes its sign.

29. Thank you for your attention !