Mathematical analysis.

Workshop.

For university students in the specialty:

"State and municipal administration"

T.Z. Pavlova

Kolpashevo 2008

Chapter 1 Introduction to Analysis

1.1 Functions. General properties

1.2 Theory of limits

1.3 Continuity of function

2.1 Definition of derivative

2.4 Exploring functions

2.4.1 Full function study plan

2.4.2 Function study examples

2.4.3. The largest and smallest value of a function on a segment

2.5 L'Hospital's rule

3.1 Indefinite integral

3.1.1 Definitions and properties

3.1.2 Table of integrals

3.1.3 Basic integration methods

3.2 Definite integral

3.2.2 Methods for calculating a definite integral

Chapter 4

4.1 Basic concepts

4.2 Limits and continuity of functions of several variables

4.3.3 Total differential and its application to approximate calculations

Chapter 5

6.1 Utility function.

6.2 Indifference lines

6.3 Budget set

Homework assignments

1.1 Functions. General properties

A numerical function is defined on the set D of real numbers if each value of the variable is associated with some well-defined real value of the variable y, where D is the domain of the function.

Analytic representation of the function:

explicitly: ;

implicitly: ;

in parametric form:

different formulas in the domain of definition:

Properties.

Even function: . For example, the function is even, because .

Odd function: . For example, the function is odd, because .

Periodic function: , where T is the period of the function, . For example, trigonometric functions.

monotonic function. If for any of the domain of definition - the function is increasing, - decreasing. For example, - increasing, and - decreasing.

Limited feature. If there is a number M such that . For example, functions and , because .

Example 1. Find the scope of functions.

+ 2 – 3 +

1.2 Theory of limits

Definition 1. The limit of the function at is the number b, if for any ( is an arbitrarily small positive number) it is possible to find such a value of the argument starting from which the inequality is fulfilled.

Designation: .

Definition 2. The limit of the function at is the number b, if for any ( - arbitrarily small positive number) there is such a positive number that for all x values ​​satisfying the inequality the inequality is true.

Designation: .

Definition 3. The function is called infinitesimal for or , if or .

Properties.

1. The algebraic sum of a finite number of infinitesimal quantities is an infinitesimal quantity.

2. The product of an infinitesimal quantity and a bounded function (constant, another infinitesimal quantity) is an infinitesimal quantity.

3. The quotient of dividing an infinitesimal quantity by a function whose limit is different from zero is an infinitesimal quantity.

Definition 4. The function is called infinitely large for if .

Properties.

1. The product of an infinitely large quantity by a function whose limit is different from zero is an infinitely large quantity.

2. The sum of an infinitely large quantity and a bounded function is an infinitely large quantity.

3. The quotient of dividing an infinitely large quantity by a function that has a limit is an infinitely large quantity.

Theorem.(The relationship between an infinitesimal value and an infinitely large value.) If a function is infinitely small at (), then the function is an infinitely large value at (). And, conversely, if the function is infinitely large at (), then the function is an infinitely small value at ().

Limit theorems.

1. A function cannot have more than one limit.

2. The limit of the algebraic sum of several functions is equal to the algebraic sum of the limits of these functions:

3. The limit of the product of several functions is equal to the product of the limits of these functions:

4. The limit of the degree is equal to the degree of the limit:

5. The limit of the quotient is equal to the quotient of the limits, if the divisor limit exists:

.

6. The first remarkable limit.

Consequences:

7. Second remarkable limit:

Consequences:

Equivalent infinitesimal quantities at:

Calculation of limits.

When calculating the limits, the basic theorems on limits, the properties of continuous functions, and the rules that follow from these theorems and properties are used.

Rule 1 To find the limit at a point of a function that is continuous at this point, it is necessary to substitute its limit value instead of the argument x into the function under the limit sign.

Example 2. Find

Rule 2 If, when finding the limit of a fraction, the limit of the denominator is equal to zero, and the limit of the numerator is non-zero, then the limit of such a function is equal to .

Example 3. Find

Rule 3 If, when finding the limit of a fraction, the limit of the denominator is equal, and the limit of the numerator is non-zero, then the limit of such a function is equal to zero.

Example 4 Find

Often substitution of the limit value of an argument leads to undefined expressions of the form

.

Finding the limit of a function in these cases is called uncertainty disclosure. To reveal the uncertainty, it is necessary, before going to the limit, to carry out the transformation of this expression. Various techniques are used to uncover uncertainties.

Rule 4. The uncertainty of the form is revealed by transforming the sublimit function, so that in the numerator and denominator, select a factor whose limit is zero, and, reducing the fraction by it, find the limit of the quotient. To do this, the numerator and denominator are either factored or multiplied by the expressions conjugate to the numerator and denominator.

Rule 5 If the sublimit expression contains trigonometric functions, then the first remarkable limit is used to reveal the uncertainty of the form.

.

Rule 6. To reveal the uncertainty of the form at , the numerator and denominator of the sublimit fraction must be divided by the highest degree of the argument and then the limit of the quotient should be found.

Possible results:

1) the desired limit is equal to the ratio of the coefficients at the highest powers of the argument of the numerator and denominator, if these powers are the same;

2) the limit is equal to infinity if the degree of the numerator argument is higher than the degree of the denominator argument;

3) the limit is zero if the degree of the numerator argument is lower than the degree of the denominator argument.

a)

since

The degrees are equal, which means that the limit is equal to the ratio of the coefficients at higher degrees, i.e. .

b)

The degree of the numerator, the denominator is 1, which means that the limit is equal to

v)

The degree of the numerator is 1, the denominator is , so the limit is 0.

Rule 7. To reveal the uncertainty of the form , the numerator and denominator of the sublimit fraction must be multiplied by the conjugate expression.

Example 10

Rule 8. To reveal the uncertainty of the species, the second remarkable limit and its consequences are used.

It can be proved that

Example 11.

Example 12.

Example 13

Rule 9. When disclosing uncertainties, the sublimit function of which contains b.m.v., it is necessary to replace the limits of these b.m. to the limits of b.m., equivalent to them.

Example 14

Example 15

Rule 10 L'Hospital's rule (see 2.6).

1.3 Continuity of function

The function is continuous at the point if the limit of the function, when the argument tends to a, exists and is equal to the value of the function at this point.

Equivalent conditions:

1. ;

3.

Classification of break points:

rupture of the first kind

Removable - one-sided limits exist and are equal;

Fatal (jump) - one-sided limits are not equal;

discontinuity of the second kind: the limit of the function at a point does not exist.

Example 16. Establish the nature of the discontinuity of a function at a point or prove the continuity of a function at this point.

for , the function is not defined, so it is not continuous at this point. Because and correspondingly, , then is a discontinuity point of the first kind.

b)

compared to task (a), the function is extended at the point so that , so the given function is continuous at the given point.

When the function is not defined;

.

Because one of the one-sided limits is infinite, then is a discontinuity point of the second kind.

Chapter 2

2.1 Definition of derivative

Derivative Definition

The derivative or of a given function is the limit of the ratio of the increment of the function to the corresponding increment of the argument when the increment of the argument tends to zero:

Or .

The mechanical meaning of the derivative is the rate of change of the function. The geometric meaning of the derivative is the tangent of the slope of the tangent to the graph of the function:

2.2 Basic rules of differentiation

Name	Function	Derivative
Multiplication by a constant factor
Algebraic sum of two functions
The product of two functions
Quotient of two functions
Complex function

Derivatives of basic elementary functions

P / p No.	Function name	Function and its derivative
1	constant
2	power function special cases
3	exponential function special case
4	logarithmic function special case
5	trigonometric functions
6	reverse trigonometric

b)

2.3 Higher order derivatives

Second order derivative of a function

Second order derivative of the function:

Example 18.

a) Find the second order derivative of the function .

Solution. Let us first find the derivative of the first order .

From the derivative of the first order, we take the derivative again.

Example 19. Find the third order derivative of the function .

2.4 Exploring functions

2.4.1 Full function study plan:

Full Function Study Plan:

1. Elementary research:

Find the domain of definition and range of values;

Find out the general properties: even (odd), periodicity;

Find points of intersection with coordinate axes;

Determine areas of constancy.

2. Asymptotes:

Find vertical asymptotes if ;

Find oblique asymptotes: .

If any number, then are the horizontal asymptotes.

3. Research using:

Find the critical points, those. points at which or does not exist;

Determine the intervals of increase, those. intervals on which and decrease of the function - ;

Determine the extreme points: the points, when passing through which it changes sign from “+” to “-”, are the maximum points, from “-” to “+” - the minimum.

4. Research using:

Find points at which or does not exist;

Find areas of convexity, i.e. gaps, on which and concavities -;

Find inflection points, i.e. points at the transition through which the sign changes.

1. Individual elements of the study are plotted on the graph gradually, as they are found.

2. If there are difficulties with constructing a graph of a function, then the values ​​of the function are found at some additional points.

3. The purpose of the study is to describe the nature of the behavior of the function. Therefore, not an exact graph is built, but its approximation, on which the found elements (extrema, inflection points, asymptotes, etc.) are clearly marked.

4. It is not necessary to strictly adhere to the above plan; it is important not to miss the characteristic elements of the behavior of the function.

2.4.2 Function study examples:

1)

2) Function odd:

.

3) Asymptotes.

are the vertical asymptotes, since

Oblique asymptote .

5)

- inflection point.

2) Function odd:

3) Asymptotes: There are no vertical asymptotes.

Inclined:

are oblique asymptotes

4) - the function is increasing.

- inflection point.

Schematic graph of this function:

2) General function

3) Asymptotes

- no oblique asymptotes

is the horizontal asymptote at

- inflection point

Schematic graph of this function:

2) Asymptotes.

is the vertical asymptote, since

- no oblique asymptotes

, is the horizontal asymptote

Schematic graph of this function:

2) Asymptotes

is the vertical asymptote at , because

- no oblique asymptotes

, is the horizontal asymptote

3) – the function decreases on each of the intervals.

Schematic graph of this function:

To find the largest and smallest value of a function on a segment, you can use the scheme:

1. Find the derivative of the function.

2. Find the critical points of the function at which or does not exist.

3. Find the value of the function at critical points belonging to a given segment and at its ends and choose the largest and smallest of them.

Example. Find the smallest and largest value of the function on the given segment.

25. in between

2) - critical points

26. in between.

The derivative does not exist at , but 1 does not belong to this interval. The function decreases on the interval , which means that there is no maximum value, but the smallest value .

2.5 L'Hospital's rule

Theorem. The limit of the ratio of two infinitely small or infinitely large functions is equal to the limit of the ratio of their derivatives (finite or infinite), if the latter exists in the indicated sense.

Those. when disclosing uncertainties of the type or, you can use the formula:

.

27.

Chapter 3. Integral calculus

3.1 Indefinite integral

3.1.1 Definitions and properties

Definition 1. A function is called antiderivative for if .

Definition 2. The indefinite integral of a function f(x) is the set of all antiderivatives for this function.

Designation: , where c is an arbitrary constant.

Properties of the indefinite integral

1. Derivative of the indefinite integral:

2. Differential of the indefinite integral:

3. Indefinite integral of the differential:

4. Indefinite integral of the sum (difference) of two functions:

5. Taking the constant factor out of the sign of the indefinite integral:

3.1.2 Table of integrals

.1.3 Basic integration methods

1. Using the properties of the indefinite integral.

Example 29.

2. Bringing under the sign of the differential.

Example 30.

3. Variable replacement method:

a) replacement in the integral

where - a function that is easier to integrate than the original one; - function, inverse function ; - antiderivative of the function .

Example 31.

b) replacement in the integral of the form:

Example 32.

Example 33.

4. Integration by parts method:

Example 34.

Example 35.

Take separately the integral

Let's go back to our integral:

3.2 Definite integral

3.2.1 The concept of a definite integral and its properties

Definition. Let a continuous function be given on some interval. Let's plot it.

A figure bounded from above by a curve, from the left and right by straight lines and from below by a segment of the abscissa axis between points a and b, is called a curvilinear trapezoid.

S - area - curvilinear trapezoid.

Divide the interval by dots and get:

Integral sum:

Definition. The definite integral is the limit of the integral sum.

Properties of a definite integral:

1. A constant factor can be taken out of the integral sign:

2. The integral of the algebraic sum of two functions is equal to the algebraic sum of the integrals of these functions:

3. If the segment of integration is divided into parts, then the integral on the entire segment is equal to the sum of the integrals for each of the parts that have arisen, i.e. for any a, b, c:

4. If on the segment , then and

5. The limits of integration can be interchanged, and the sign of the integral changes:

6.

7. The integral at the point is equal to 0:

8.

9. (“about the mean”) Let y = f(x) be a function integrable on . Then , where , f(c) is the average value of f(x) on :

10. Newton-Leibniz formula

,

where F(x) is the antiderivative for f(x).

3.2.2 Methods for calculating a definite integral.

1. Direct Integration

Example 35.

a)

b)

v)

e)

2. Change of variables under the sign of a definite integral .

Example 36.

2. Integration by parts in a definite integral .

Example 37.

a)

b)

e)

3.2.3 Applications of the definite integral

Characteristic	Function type	Formula
	in Cartesian coordinates
curvilinear sector area	in polar coordinates
area of ​​a curved trapezoid	in parametric form
arc length	in Cartesian coordinates
arc length	in polar coordinates
arc length	in parametric form
body volume rotation	in Cartesian coordinates
body volume with a given transverse cross-section

Example 38. Calculate the area of ​​​​a figure bounded by lines: and .

Solution: Find the intersection points of the graphs of these functions. To do this, we equate the functions and solve the equation

So, the intersection points and .

Find the area of ​​the figure using the formula

.

In our case

Answer: the area is (square units).

4.1 Basic concepts

Definition. If each pair of independent numbers from a certain set is assigned one or more values ​​of the variable z according to some rule, then the variable z is called a function of two variables.

Definition. The domain of a function z is the set of pairs for which the function z exists.

The domain of a function of two variables is a certain set of points on the coordinate plane Oxy. The z-coordinate is called the applicate, and then the function itself is represented as some surface in the space E 3 . For instance:

Example 39. Find the scope of a function.

a)

The expression on the right side makes sense only when . This means that the domain of this function is the set of all points lying inside and on the boundary of a circle of radius R centered at the origin.

The domain of this function is all points of the plane, except for the points of lines, i.e. coordinate axes.

Definition. Function level lines are a family of curves on the coordinate plane described by equations of the form .

Example 40 Find feature level lines .

Solution. The level lines of a given function are a family of curves in the plane , described by the equation

The last equation describes a family of circles centered at the point О 1 (1, 1) of radius . The surface of revolution (paraboloid) described by this function becomes “steeper” as it moves away from the axis, which is given by the equations x = 1, y = 1. (Fig. 4)

4.2 Limits and continuity of functions of several variables.

1. Limits.

Definition. The number A is called the limit of the function as the point tends to the point, if for each arbitrarily small number there is such a number that the condition is true for any point, the condition is also true . Write down: .

Example 41. Find limits:

those. the limit depends on , which means that it does not exist.

2. Continuity.

Definition. Let the point belong to the domain of definition of the function . Then a function is called continuous at a point if

(1)

and the point tends to the point in an arbitrary way.

If condition (1) is not satisfied at any point, then this point is called the break point of the function. This may be in the following cases:

1) The function is not defined at the point .

2) There is no limit.

3) This limit exists, but it is not equal to .

Example 42. Determine if the given function is continuous at the point if .

Got that so this function is continuous at the point .

the limit depends on k, i.e. it does not exist at this point, which means that the function has a discontinuity at this point.

4.3 Derivatives and differentials of functions of several variables

4.3.1 Partial derivatives of the first order

The partial derivative of a function with respect to the argument x is the ordinary derivative of a function of one variable x for a fixed value of the variable y and is denoted:

The partial derivative of a function with respect to the argument y is the ordinary derivative of a function of one variable y for a fixed value of the variable x and is denoted:

Example 43. Find partial derivatives of functions.

4.3.2 Partial derivatives of the second order

Second order partial derivatives are partial derivatives of first order partial derivatives. For a function of two variables of the form, four types of second-order partial derivatives are possible:

Partial derivatives of the second order, in which differentiation is carried out with respect to different variables, are called mixed derivatives. The mixed second-order derivatives of a twice differentiable function are equal.

Example 44. Find partial derivatives of the second order.

4.3.3 Total differential and its application to approximate calculations.

Definition. The first order differential of a function of two variables is found by the formula

.

Example 45. Find the total differential for the function.

Solution. Let's find partial derivatives:

.

With small increments of the arguments x and y, the function receives an increment approximately equal to dz, i.e. .

The formula for finding the approximate value of a function at a point if its exact value at a point is known:

Example 46 Find .

Solution. Let ,

Then we use the formula

Answer. .

Example 47. Calculate approximately.

Solution. Let's consider a function. We have

Example 48. Calculate approximately.

Solution. Consider the function . We get:

Answer. .

4.3.4 Implicit function differentiation

Definition. A function is called implicit if it is given by an equation that is not solvable with respect to z.

The partial derivatives of such a function are found by the formulas:

Example 49. Find the partial derivatives of the function z given by the equation .

Solution.

Definition. A function is called implicit if it is given by an equation that is not solvable with respect to y.

The derivative of such a function is found by the formula:

.

Example 50. Find derivatives of these functions.

5.1 Local extremum of a function of several variables

Definition 1. The function has a maximum at the point if

Definition 2. The function has a minimum at the point if for all points sufficiently close to the point and distinct from it.

A necessary condition for an extremum. If the function reaches an extremum at the point , then the partial derivatives of the function vanish or do not exist at that point.

Points at which partial derivatives vanish or do not exist are called critical.

A sufficient sign of an extremum. Let the function be defined in some neighborhood of the critical point and have continuous second-order partial derivatives at this point

1) has a local maximum at the point if and ;

2) has a local minimum at the point if and ;

3) does not have a local extremum at the point if ;

Scheme of studying the extremum of a function of two variables.

1. Find the partial derivatives of the functions : and .

2. Solve the system of equations , and find the critical points of the function.

3. Find partial derivatives of the second order, calculate their values ​​at critical points and, using a sufficient condition, draw a conclusion about the presence of extrema.

4. Find the extrema of the function.

Example 51. Find extrema of a function .

1) Let's find partial derivatives.

2) Solve the system of equations

4) Find the partial derivatives of the second order and their values ​​at critical points: . At the point we get:

This means that there is no extremum at the point. At the point we get:

means at the minimum point.

5.2 Global extremum (largest and smallest value of the function)

The largest and smallest values ​​of a function of several variables, continuous on some closed set, are reached either at the extremum points or at the boundary of the set.

Scheme for finding the largest and smallest values.

1) Find the critical points lying inside the region, calculate the value of the function at these points.

2) Investigate the function on the boundary of the region; if the boundary consists of several different lines, then the study must be carried out for each section separately.

3) Compare the obtained values ​​of the function and choose the largest and smallest.

Example 52. Find the largest and smallest values ​​of a function in a rectangle.

Solution. 1) Find the critical points of the function, for this we find the partial derivatives: , and solve the system of equations:

We got the critical point A. The resulting point lies inside the given area,

The boundary of the region is made up of four segments: i. find the largest and smallest value of the function on each segment.

4) Let's compare the obtained results and get that at the points .

Chapter 6. The Consumer Choice Model

We will assume that there are n different goods. Then some set of goods will be denoted by the n-dimensional vector , where is the quantity of the i-th product. The set of all sets of goods X is called a space.

The choice of an individual consumer is characterized by a preference relation: it is believed that the consumer can say about any two sets which is more desirable, or he does not see a difference between them. The preference relation is transitive: if the set is preferred to the set and the set is preferred to the set, then the set is preferred to the set. We will assume that consumer behavior is fully described by the axiom of the individual consumer: each individual consumer makes a decision about consumption, purchases, etc., based on his system of preferences.

6.1 Utility function

On the set of consumer bundles X, the function , whose value on the consumer set is equal to the individual's consumer rating for this set. The function is called the consumer utility function or consumer preference function. Those. each consumer has its own utility function. But the whole set of consumers can be divided into certain classes of consumers (by age, property status, etc.) and each class can be assigned some, perhaps, an averaged utility function.

Thus, the function is the consumer assessment or the level of satisfaction of the needs of the individual when acquiring this set. If a set is preferable to a set for a given individual, then .

Utility function properties.

1.

The first partial derivatives of the utility function are called the marginal utilities of the products. From this property it follows that an increase in the consumption of one product with the same consumption of other products leads to an increase in consumer evaluation. Vector is the gradient of the function, it shows the direction of the greatest growth of the function. For a function, its gradient is a vector of marginal utilities of products.

2.

Those. The marginal utility of any good decreases as consumption increases.

3.

Those. the marginal utility of each product increases with the quantity of the other product.

Some kinds of utility functions.

1) Neoclassical: .

2) Square: , where the matrix is ​​negative definite and for .

3) Logarithmic function: .

6.2 Indifference lines

In applied problems and models of consumer choice, a special case of a set of two goods is often used, i.e. when the utility function depends on two variables. The indifference line is a line connecting consumer sets that have the same level of satisfaction of the needs of the individual. In essence, indifference lines are function level lines. Equations of indifference lines: .

Basic properties of indifference lines.

1. Lines of indifference corresponding to different levels of satisfaction of needs do not touch or intersect.

2. Lines of indifference decrease.

3. Lines of indifference are convex down.

Property 2 implies an important approximate equality .

This ratio shows how much an individual should increase (decrease) the consumption of the second product while decreasing (increasing) the consumption of the first product by one unit without changing the level of satisfaction of his needs. The ratio is called the rate of replacement of the first product by the second, and the value is called the marginal rate of replacement of the first product by the second.

Example 53. If the marginal utility of the first good is 6, and the second is 2, then with a decrease in the consumption of the first good by one unit, the consumption of the second good must be increased by 3 units at the same level of satisfaction of needs.

6.3 Budget set

Let is the vector of prices for a set of n products ; I is the income of the individual, which he is willing to spend on the purchase of a set of products. The set of bundles of goods costing at most I at given prices is called the budget set B. In this case, the set of bundles costing I is called the boundary G of the budget set B. Thus. the set B is bounded by the boundary G and natural constraints.

The budget set is described by the system of inequalities:

For the case of a set of two goods, the budget set B (Fig. 1) is a triangle in the coordinate system , bounded by the coordinate axes and the straight line .

6.4 Consumer demand theory

In the theory of consumption, it is assumed that the consumer always seeks to maximize his utility and the only limitation for him is the limited income I that he can spend on buying a set of goods. In general, the problem of consumer choice (the problem of rational consumer behavior in the market) is formulated as follows: find a consumer set , which maximizes its utility function given the budget constraint. Mathematical model of this task:

In the case of a set of two items:

Geometrically, the solution to this problem is the point of contact between the boundary of the budget set G and the indifference line.

The solution of this problem is reduced to solving the system of equations:

(1)

The solution of this system is the solution to the problem of consumer choice.

The solution to the consumer choice problem is called the demand point. This point of demand depends on prices and income I. I.e. the demand point is a function of demand. In turn, the demand function is a set of n functions, each of which depends on the argument:

These functions are called the demand functions of the respective goods.

Example 54. For a set of two goods on the market, known prices for them and income I, find the demand functions if the utility function has the form .

Solution. We differentiate the utility function:

.

We substitute the obtained expressions into (1) and obtain a system of equations:

In this case, the expenditure on each product will be half the income of the consumer, and the amount of the purchased product is equal to the amount spent on it, divided by the price of the product.

Example 55. Let the utility function for the first product , the second ,

the price of the first item, the price of the second. Income . How much of a good should a consumer purchase to maximize utility?

Solution. Find the derivatives of utility functions, substitute into system (1) and solve it:

This set of goods is optimal for the consumer in terms of utility maximization.

The control work must be completed in accordance with the option selected by the last digit of the record book number in a separate notebook. Each problem should contain a condition, a detailed solution, and a conclusion.

1. Introduction to calculus

Task 1. Find the domain of the function.

5.

Task 2. Find the limits of functions.

.

Task 3. Find function break points and determine their type.

1. 2. 3.

Chapter 2

Task 4. Find derivatives of these functions.

1. a); b) c) y = ;

d) y = x 6 + + + 5; e) y \u003d x tg x + ln sin x + e 3x;

f) y \u003d 2 x - arcsin x.

2. a) ; b) y = ; c) y = ; d) y \u003d x 2 - + 3; e) y = e cos ; f) y = .

3. a) y = lnx; b) y =; c) y = ln;

4. a) y = ; b) y \u003d (e 5 x - 1) 6; c) y = ; d) y = ; e) y = x 8 ++ + 5; f) y \u003d 3 x - arcsin x.

5. a) y \u003d 2x 3 - + e x; b) y = ; c) y = ;

d) y = ; e) y = 2 cos ; f) y = .

6. a) y = lnx; b) y =; c) y = ln;

d) y = ; e) y \u003d x 7 + + 1; f) y = 2.

7. a) ; b) y = ; c) y = ; d) y \u003d x 2 + xsinx +; e) y = e cos ; f) y = .

8. a) y = ; b) y \u003d (3 x - 4) 6; c) y = sintg;

d) y = 3x 4 - - 9+ 9; e) y = ;

e) y \u003d x 2 + arcsin x - x.

9. a); b) ; c) y = ; d) y \u003d 5 sin 3 x; e) y \u003d x 3 - - 6+ 3; f) y = 4x 4 + ln.

10. a) b) y = ; c) y = (3 x - 4) 6 ; d) y = ; e) y \u003d x 2 - x; f) y \u003d e sin 3 x + 2.

Task 5. Investigate a function and build its graph.

1. a) b) c).

2. a) b) v) .

3. a) b) v) .

4. b) v)

5. a) b) v) .

6. a) b) v) .

7. a) b) c).

8. a) b) c).

9. a) b) c).

10. a) b) v) .

Task 6. Find the largest and smallest value of the function on a given interval.

1. .

3. .

6. .

8. .

9. .

10. .

Chapter 3. Integral Calculus

Task 7. Find indefinite integrals.

1. a) b);

2. a) ;b) c) d).

4. G)

5. a) ; b); v) ; G).

6. a) ; b); v); G)

7. a) ; b) ; v) ; G)

8.a) ; b); v) ; G) .

9. a) ; b) c); G).

10. a) b) v) ; G) .

Task 8. Calculate definite integrals.

1.

2.

3.

4.

5.

6.

7. .

8.

9.

10.

Problem 9. Find improper integrals or prove that they diverge.

1. .

2. .

3. .

4. .

5. .

6. .

7. .

8. .

9. .

10. .

Problem 10. Find the area of ​​the area bounded by curves

1. .2. .

5. 6.

7. , .8..

10. , .

Chapter 4. Differential calculus of a function of several variables.

Task 11. Find the domain of the function (shown on the drawing).

Problem 12. Investigate the continuity of a function for

Task 13. Find the derivative of an implicitly given function.

Problem 14. Calculate approximately

1. a); b) ; v)

2. a) ; b) ; v) .

3. a) ; b) ; v) .

4. a) ; b) ; v) .

5. a); b) ; v) .

6. a); b) ; v) .

7. a); b) ; v) .

8. a) ;b) ; v)

9. a) ; b) ; v) .

10. a) ;b) ; v)

Problem 15. Investigate a function for extrema.

7. .

8. .

9. .

10. .

Problem 16. Find the largest and smallest value of a function in a given closed area.

1. in a rectangle

2.

3. in a rectangle

4. in the area bounded by a parabola

And the abscissa.

5. squared

6. in a triangle bounded by the coordinate axes and a straight line

7. in a triangle bounded by the coordinate axes and a straight line

8. in a triangle bounded by the coordinate axes and a straight line

9. in the area bounded by a parabola

And the abscissa.

10. in the area bounded by a parabola

And the abscissa.

The main

1. M.S. Crass, B.P. Chuprynov. Fundamentals of mathematics and its application in economic education: Textbook. - 4th ed., Spanish. – M.: Delo, 2003.

2. M.S. Crass, B.P. Chuprynov. Mathematics for economic specialties: Textbook. - 4th ed., Spanish. – M.: Delo, 2003.

3. M.S. Crass, B.P. Chuprynov. Mathematics for Bachelor of Economics. Textbook. - 4th ed., Spanish. – M.: Delo, 2005.

4. Higher mathematics for economists. Textbook for universities / N.Sh. Kremer, B.A. Putko, I.M. Trishin, M.N. Friedman; Ed. prof. N.Sh. Kremer, - 2nd ed., revised. and additional - M: UNITI, 2003.

5. Kremer N.Sh, Putko B.A., Trishin I.M., Fridman M.N. Higher mathematics for economic specialties. Textbook and Practicum (parts I and II) / Ed. prof. N.Sh. Kremer, - 2nd ed., revised. and additional - M: Higher education, 2007. - 893s. - (Fundamentals of Sciences)

6. Danko P.E., Popov A.G., Kozhevnikova T.Ya. Higher mathematics in exercises and tasks. M. high school. 1999.

Additional

1. I.I. Bavrin, V.L. Sailors. Higher Mathematics. "Vlados Humanitarian Publishing Center", 2002.

2. I.A. Zaitsev. Higher Mathematics. "High School", 1998.

3. A.S. Solodovnikov, V.A. Babaitsev, A.V. Brailov, I.G. Shandra. Mathematics in Economics / in two parts /. M. Finance and statistics. 1999.

for students medical, pediatric, dental

and medical and preventive faculties

to laboratory work

"Basic concepts of mathematical analysis"

1. Scientific and methodological substantiation of the topic:

The concepts of derivative and differential are among the basic concepts of mathematical analysis. The calculation of derivatives is necessary when solving many problems in physics and mathematics (finding speed, acceleration, pressure, etc.). The importance of the concept of derivative, in particular, is determined by the fact that the derivative of a function characterizes the rate of change of this function when its argument changes.

The use of the differential makes it possible to carry out approximate calculations, as well as to evaluate errors.

Methods for finding derivatives and differentials of functions and their application constitute the main task of differential calculus. The need for the concept of a derivative arises in connection with the formulation of the problem of calculating the speed of movement and finding the angle of the tangent to the curve. The inverse problem is also possible: determine the distance traveled by the speed, and find the corresponding function by the tangent of the slope of the tangent. Such an inverse problem leads to the concept of an indefinite integral.

The concept of a definite integral is used in a number of practical problems, in particular in problems of calculating the areas of plane figures, calculating the work done by a variable force, and finding the average value of a function.

In the mathematical description of various physical, chemical, biological processes and phenomena, equations are often used that contain not only the quantities under study, but also their derivatives of various orders of these quantities. For example, according to the simplest version of the law of bacterial reproduction, the rate of reproduction is proportional to the number of bacteria at a given time. If this number is denoted by N(t), then, in accordance with the physical meaning of the derivative, the rate of reproduction of bacteria is a derivative of N(t), and on the basis of the mentioned law, we can write the ratio N "(t) \u003d k∙N, where k\u003e 0 - coefficient of proportionality The resulting equation is not algebraic, since it contains not only the unknown function N(t), but also its first-order derivative.

2. Brief theory:

1. Problems leading to the concept of a derivative

1. The problem of finding the speed v of a material point. Let some material point make a rectilinear motion. At the point in time t 1 point is in position M 1. At the point in time t 2 pregnant M 2 . Denote the interval M 1 , M 2 across ∆S; t 2 – t 1 =Δt. The value is called the average speed of movement. To find the instantaneous speed of a point at a position M 1 necessary Δt head towards zero. Mathematically, this means that

, (1)

Thus, to find the instantaneous velocity of a material point, it is necessary to calculate the limit of the ratio of the increment of the function ∆S to the increment of the argument Δt provided that ∆t→0.

2. The problem of finding the angle of inclination of the tangent to the function graph.

Fig.1

Consider the graph of some function y=f(x). What is the angle of inclination
tangent drawn at a point M 1 ? At the point M 1 draw a tangent to the graph of the function. Pick an arbitrary point on the graph M 2 and draw a secant. It is tilted towards the axis OH at an angle α 1 . Consider ΔM 1 M 2 A:

, (2)

If the point M 1 fix and point M 2 approach M 1 , then the secant M 1 M 2 will become tangent to the graph of the function at the point M 1 and you can write:

, (3)

Thus, it is necessary to calculate the limit of the ratio of the increment of the function to the increment of the argument, if the increment of the argument tends to zero.

The limit of the ratio of the increment Δy of the function y=f(x) to the increment of the argument Δx at a given point x 0 as Δx tends to zero, is called the derivative of the function at a given point.

Derivative notation: y", f "(x), . By definition

, (4)

where Δx=х 2 -х 1 is the increment of the argument (the difference between two subsequent sufficiently close values ​​of the argument), Δy=y 2 -y 1 is the increment of the function (the difference between the values ​​of the function corresponding to these values ​​of the argument).

Finding the derivative of a given function is called its differentiation. Differentiation of the basic elementary functions is carried out according to ready-made formulas (see table), as well as using rules:

Derivative of an algebraic sum functions is equal to the sum of the derivatives of these functions:

(u+ υ )"= u" + υ "

2. The derivative of the product of two functions is equal to the sum of the products of the second function by the derivative of the first and the first function by the derivative of the second:

(u∙υ )"=u"υ + uυ "

3. Derivative of the quotient of two functions is equal to a fraction, the numerator of which is the difference between the products of the denominator and the derivative of the numerator and the numerator and the derivative of the denominator, and the denominator is the square of the denominator:

The physical meaning of the derivative. From the comparison of (4) and (1) it follows that the instantaneous speed of the rectilinear motion of a material point is equal to the derivative of the dependence of its coordinate on time.

The general meaning of the derivative of a function is that it characterizes rate (rapidity) of function change given the change in argument. The speed of physical, chemical and other processes, such as the rate of cooling of the body, the rate of a chemical reaction, the rate of reproduction of bacteria, etc., is also expressed using a derivative.

The geometric meaning of the derivative. The value of the tangent of the slope of the tangent drawn to the function graph is called in mathematics the slope of the tangent.

The slope of the tangent drawn to the graph of a differentiable function at some point is numerically equal to the derivative of the function at that point.

This statement is called geometric meaning of the derivative.

The content of the article

MATHEMATICAL ANALYSIS, a branch of mathematics that provides methods for the quantitative study of various processes of change; deals with the study of the rate of change (differential calculus) and the determination of the lengths of curves, areas and volumes of figures bounded by curved contours and surfaces (integral calculus). It is typical for problems of mathematical analysis that their solution is associated with the concept of a limit.

The beginning of mathematical analysis was laid in 1665 by I. Newton and (about 1675) independently by G. Leibniz, although important preparatory work was carried out by I. Kepler (1571–1630), F. Cavalieri (1598–1647), P. Fermat (1601– 1665), J. Wallis (1616–1703) and I. Barrow (1630–1677).

To make the presentation more lively, we will resort to the language of graphs. Therefore, it may be useful for the reader to look into the article ANALYTICAL GEOMETRY before reading this article.

DIFFERENTIAL CALCULUS

Tangents.

On fig. 1 shows a fragment of the curve y = 2x – x 2 enclosed between x= –1 and x= 3. Sufficiently small segments of this curve look straight. In other words, if R is an arbitrary point of this curve, then there is some straight line passing through this point and being an approximation of the curve in a small neighborhood of the point R, and the smaller the neighborhood, the better the approximation. Such a line is called a tangent to the curve at the point R. The main task of differential calculus is to construct a general method that allows you to find the direction of the tangent at any point on the curve where the tangent exists. It is easy to imagine a curve with a sharp break (Fig. 2). If R is the vertex of such a break, then it is possible to construct an approximating straight line PT 1 - to the right of the point R and another approximating line RT 2 - to the left of the point R. But there is no single line passing through the point R, which approached the curve equally well in the vicinity of the point P both on the right and on the left, hence the tangent at the point P does not exist.

On fig. 1 tangent FROM drawn through the origin O= (0,0). The slope of this straight line is 2, i.e. when the abscissa changes by 1, the ordinate increases by 2. If x and y are the coordinates of an arbitrary point on FROM, then moving away from O at a distance X units to the right, we move away from O on 2 y units up. Hence, y/x= 2, or y = 2x. This is the tangent equation FROM to the curve y = 2x – x 2 at point O.

It is now necessary to explain why, from the set of lines passing through the point O, the straight line is chosen FROM. What is the difference between a straight line with a slope of 2 and other straight lines? There is one simple answer, and it is hard for us to resist the temptation to give it using the analogy of a tangent to a circle: the tangent FROM has only one common point with the curve, while any other non-vertical line passing through the point O, crosses the curve twice. This can be verified as follows.

Since the expression y = 2x – x 2 can be obtained by subtracting X 2 of y = 2x(direct line equations FROM), then the values y for graphics there is less knowledge y for a straight line at all points, except for the point x= 0. Therefore, the graph is everywhere except for the point O, located below FROM, and this line and the graph have only one common point. In addition, if y = mx- the equation of some other straight line passing through the point O, then there must be two points of intersection. Really, mx = 2x – x 2 not only for x= 0, but also for x = 2 – m. And only when m= 2 both points of intersection coincide. On fig. 3 shows the case when m less than 2, so to the right of O there is a second intersection point.

What FROM is the only non-vertical line passing through the point O and having only one common point with the graph, which is not its most important property. Indeed, if we turn to other graphs, it will soon become clear that the property of the tangent we have noted is generally not satisfied. For example, from fig. 4 it can be seen that near the point (1,1) the plot of the curve y = x 3 is well approximated by a straight line RT, which, however, has more than one common point with it. However, we would like to consider RT tangent to this graph at the point R. Therefore, it is necessary to find some other way to highlight the tangent than the one that served us so well in the first example.

Let's assume that through the point O and an arbitrary point Q = (h,k) on the graph of the curve y = 2x – x 2 (Fig. 5) a straight line (called a secant) is drawn. Substituting in the equation of the curve the values x = h and y = k, we get that k = 2h – h 2 , therefore, the slope of the secant is equal to

At very small h meaning m close to 2. Moreover, choosing h close enough to 0, we can do m arbitrarily close to 2. We can say that m"goes to the limit" equal to 2 when h tends to zero, or what's the limit m equals 2 when h tending to zero. Symbolically it is written like this:

Then the tangent to the graph at the point O defined as a line passing through a point O, with a slope equal to this limit. This definition of a tangent is applicable in the general case.

We will show the advantages of this approach with one more example: we will find the slope of the tangent to the graph of the curve y = 2x – x 2 at an arbitrary point P = (x,y), not limited to the simplest case when P = (0,0).

Let Q = (x + h, y + k) is the second point on the graph, located at a distance h to the right of R(Fig. 6). It is required to find the slope coefficient k/h secant PQ. Dot Q is at a distance

over axis X.

Expanding the brackets, we find:

Subtracting from this equation y = 2x – x 2 , find the vertical distance from the point R to the point Q:

Therefore, the slope m secant PQ equals

Now that h tends to zero m tends to 2 - 2 x; we will take the last value for the slope of the tangent PT. (The same result will be obtained if h takes negative values, which corresponds to the choice of a point Q on the left of P.) Note that for x= 0 the result is the same as the previous one.

Expression 2 - 2 x is called the derivative of 2 x – x 2. In the old days, the derivative was also called "differential ratio" and "differential coefficient". If expression 2 x – x 2 designate f(x), i.e.

then the derivative can be denoted

In order to find out the slope of the tangent to the function graph y = f(x) at some point, it is necessary to substitute in fў ( x) value corresponding to this point X. So the slope fў (0) = 2 for X = 0, fў (0) = 0 for X= 1 and f¢ (2) = –2 at X = 2.

The derivative is also denoted atў , dy/dx, D x y and Do.

The fact that the curve y = 2x – x 2 near a given point is practically indistinguishable from its tangent at this point, allows us to speak of the slope of the tangent as the "slope of the curve" at the point of contact. Thus, we can assert that the slope of the curve we are considering has a slope of 2 at the point (0,0). We can also say that when x= 0 rate of change y relatively x equals 2. At point (2,0), the slope of the tangent (and the curve) is -2. (The minus sign means that as x variable y decreases.) At the point (1,1) the tangent is horizontal. We say the curve y = 2x – x 2 has a stationary value at this point.

Highs and lows.

We have just shown that the curve f(x) = 2x – x 2 is stationary at the point (1,1). Because fў ( x) = 2 – 2x = 2(1 – x), it is clear that when x, less than 1, fў ( x) is positive, and therefore y increases; at x, large 1, fў ( x) is negative, and therefore y decreases. Thus, in the vicinity of the point (1,1), indicated in Fig. 6 letter M, meaning at growing to a point M, stationary at the point M and decreases after the point M. Such a point is called a "maximum" because the value at at this point exceeds any of its values ​​in a sufficiently small neighborhood of it. Similarly, "minimum" is defined as the point around which all values y outweigh the value at at this very point. It may also happen that although the derivative of f(x) at some point and vanishes, its sign does not change in a neighborhood of this point. Such a point, which is neither a maximum nor a minimum, is called an inflection point.

As an example, let's find the stationary point of the curve

The derivative of this function is

and vanishes at x = 0, X= 1 and X= –1; those. at points (0,0), (1, –2/15) and (–1, 2/15). If X slightly less than -1, then fў ( x) is negative; if X slightly more than -1, then fў ( x) is positive. Therefore, the point (–1, 2/15) is the maximum. Similarly, it can be shown that the point (1, -2/15) is a minimum. But the derivative fў ( x) is negative both before the point (0,0) and after it. Therefore, (0,0) is an inflection point.

The study carried out on the shape of the curve, as well as the fact that the curve intersects the axis X at f(x) = 0 (i.e., for X= 0 or ) allow us to represent its graph approximately as shown in Fig. 7.

In general, if we exclude unusual cases (curves containing straight line segments or an infinite number of bends), there are four options for the relative position of the curve and the tangent in the vicinity of the tangent point R. (Cm. rice. 8, where the tangent has a positive slope.)

1) On both sides of the point R the curve lies above the tangent (Fig. 8, a). In this case, we say that the curve at the point R convex downward or concave.

2) On both sides of the point R the curve is located below the tangent (Fig. 8, b). In this case, the curve is said to be convex upwards or simply convex.

3) and 4) The curve is located above the tangent on one side of the point R and below - on the other. In this case R- inflection point.

Comparing values fў ( x) on both sides of R with its value at the point R, you can determine which of these four cases you have to deal with in a particular problem.

Applications.

All of the above finds important applications in various fields. For example, if a body is thrown vertically upward with an initial velocity of 200 feet per second, then the height s, on which they will be located through t seconds compared to the starting point will be

Proceeding in the same way as in the examples we have considered, we find

this value vanishes at s. Derivative fў ( x) is positive up to c and negative after this time. Hence, s increases to , then becomes stationary, and then decreases. This is the general description of the motion of a body thrown upwards. From it we learn when the body reaches its highest point. Next, substituting t= 25/4 in f(t), we get 625 feet, the maximum lift height. In this task fў ( t) has a physical meaning. This derivative shows the speed at which the body is moving at a time t.

Let's now consider another type of application (Figure 9). From a sheet of cardboard with an area of ​​75 cm 2, it is required to make a box with a square bottom. What should be the dimensions of this box in order for it to have the maximum volume? If X- side of the base of the box and h is its height, then the volume of the box is equal to V = x 2 h, and the surface area is 75 = x 2 + 4xh. Transforming the equation, we get:

Derivative of V turns out to be equal

and vanishes at X= 5. Then

and V= 125/2. Function graph V = (75x – x 3)/4 is shown in fig. 10 (negative values X omitted as having no physical meaning in this problem).

Derivatives.

An important task of differential calculus is the creation of methods that allow you to quickly and conveniently find derivatives. For example, it is easy to calculate that

(The derivative of the constant is, of course, zero.) It is not hard to deduce the general rule:

where n- any integer or fraction. For instance,

(This example shows how useful fractional exponents are.)

Here are some of the most important formulas:

There are also the following rules: 1) if each of the two functions g(x) and f(x) has derivatives, then the derivative of their sum is equal to the sum of the derivatives of these functions, and the derivative of the difference is equal to the difference of the derivatives, i.e.

2) the derivative of the product of two functions is calculated by the formula:

3) the derivative of the ratio of two functions has the form

4) the derivative of a function multiplied by a constant is equal to the constant multiplied by the derivative of this function, i.e.

It often happens that the values ​​of a function have to be calculated in stages. For example, to calculate sin x 2, we need to first find u = x 2 , and then already calculate the sine of the number u. We find the derivative of such complex functions using the so-called "chain rule":

In our example f(u) = sin u, fў ( u) = cos u, hence,

These and other similar rules make it possible to immediately write down the derivatives of many functions.

Linear approximations.

The fact that, knowing the derivative, we can in many cases replace the graph of a function near some point with its tangent at that point is of great importance, since straight lines are easier to work with.

This idea finds a direct application in the calculation of approximate values ​​of functions. For example, it is rather difficult to calculate the value for x= 1.033. But you can use the fact that the number 1.033 is close to 1 and that . Close x= 1 we can replace the tangent curve graph without making any serious mistake. The slope of such a tangent is equal to the value of the derivative ( x 1/3)ў = (1/3) x–2/3 for x = 1, i.e. 1/3. Since the point (1,1) lies on the curve and the slope of the tangent to the curve at this point is 1/3, the tangent equation has the form

On this straight line X = 1,033

Received value y should be very close to the true value y; and, indeed, it is only 0.00012 more than the true one. In mathematical analysis, methods have been developed that make it possible to improve the accuracy of such linear approximations. These methods ensure the reliability of our approximate calculations.

The procedure just described suggests one useful notation. Let P- the point corresponding to the graph of the function f variable X, and let the function f(x) is differentiable. Let's change the plot of the curve near the point R tangent to it at that point. If X change to value h, then the tangent ordinate will change by the value h H f ў ( x). If h very small, then the latter value is a good approximation to the true change in the ordinate y graphics. If instead h we will write a symbol dx(this is not a product!), but a change in the ordinate y denote dy, then we get dy = f ў ( x)dx, or dy/dx = f ў ( x) (cm. rice. eleven). Therefore, instead of Dy or f ў ( x) to denote the derivative, the symbol is often used dy/dx. The convenience of this notation depends mainly on the explicit appearance of the chain rule (the differentiation of a compound function); in the new notation, this formula looks like this:

where it is implied that at depends on u, a u in turn depends on X.

Value dy called differential at; actually it depends on two variables, namely: from X and increments dx. When increment dx very small, size dy is close to the corresponding change in the value y. But suppose that the increment dx little, no need.

Derivative of a function y = f(x) we denoted f ў ( x) or dy/dx. It is often possible to take the derivative of the derivative. The result is called the second derivative of f (x) and denoted f ўў ( x) or d 2 y/dx 2. For example, if f(x) = x 3 – 3x 2 , then f ў ( x) = 3x 2 – 6x and f ўў ( x) = 6x– 6. Similar notation is used for higher order derivatives. However, to avoid a large number of primes (equal to the order of the derivative), the fourth derivative (for example) can be written as f (4) (x), and the derivative n th order as f (n) (x).

It can be shown that the curve at a point is downward convex if the second derivative is positive and upward convex if the second derivative is negative.

If the function has a second derivative, then the change in the value y corresponding to the increment dx variable X, can be approximately calculated by the formula

This approximation is generally better than the one given by the differential fў ( x)dx. It corresponds to the replacement of part of the curve is no longer a straight line, but a parabola.

If the function has f(x) there are derivatives of higher orders, then

The remainder term has the form

where x- some number between x and x + dx. The above result is called the Taylor formula with a remainder. If f(x) has derivatives of all orders, then usually R n® 0 for n ® Ґ .

INTEGRAL CALCULUS

Squares.

The study of the areas of curvilinear plane figures opens up new aspects of mathematical analysis. Such problems were tried to solve even by the ancient Greeks, for whom determining, for example, the area of ​​a circle was one of the most difficult tasks. Great success in solving this problem was achieved by Archimedes, who also managed to find the area of ​​the parabolic segment (Fig. 12). Using very complex reasoning, Archimedes proved that the area of ​​a parabolic segment is 2/3 of the area of ​​the circumscribed rectangle and, therefore, in this case is equal to (2/3)(16) = 32/3. As we will see later, this result can be easily obtained by methods of mathematical analysis.

The predecessors of Newton and Leibniz, chiefly Kepler and Cavalieri, solved the problems of calculating the areas of curvilinear figures by a method that can hardly be called logically sound, but which proved to be extremely fruitful. When Wallis, in 1655, combined the methods of Kepler and Cavalieri with those of Descartes (analytical geometry) and took advantage of the newly born algebra, the stage for the emergence of Newton was fully prepared.

Wallis divided the figure, the area of ​​which was required to be calculated, into very narrow strips, each of which was approximately considered a rectangle. Then he added up the areas of the approximating rectangles and, in the simplest cases, obtained the value to which the sum of the areas of the rectangles tended to when the number of strips went to infinity. On fig. 13 shows rectangles corresponding to some striping of the area under the curve y = x 2 .

Main theorem.

The great discovery of Newton and Leibniz made it possible to eliminate the laborious process of passing to the limit of the sum of areas. This was done thanks to a new look at the concept of area. The bottom line is that we should represent the area under the curve as generated by the ordinate moving from left to right and ask how fast the area swept by the ordinates changes. We get the key to answering this question if we consider two special cases in which the area is known in advance.

Let's start with the area under the graph of the linear function y = 1 + x, since in this case the area can be calculated using elementary geometry.

Let A(x) is the part of the plane enclosed between the straight line y = 1 + x and segment OQ(Fig. 14). When driving QP right square A(x) increases. At what speed? It is not difficult to answer this question, since we know that the area of ​​a trapezoid is equal to the product of its height and half the sum of the bases. Hence,

Rate of area change A(x) is determined by its derivative

We see that Aў ( x) coincides with the ordinate at points R. Is it by chance? Let's try to check on the parabola shown in Fig. 15. Square A (x) under the parabola at = X 2 in the range from 0 to X is equal to A(x) = (1 / 3)(x)(x 2) = x 3/3. The rate of change of this area is determined by the expression

which exactly coincides with the ordinate at moving point R.

Assuming that this rule holds in the general case, so that

is the rate of change of the area under the graph of the function y = f(x), then this can be used for calculations of other areas. In fact, the ratio Aў ( x) = f(x) expresses a fundamental theorem that could be formulated as follows: the derivative, or the rate of change of the area as a function of X, is equal to the value of the function f (x) at the point X.

For example, to find the area under the graph of a function y = x 3 from 0 to X(Fig. 16), we set

A possible answer reads:

since the derivative of X 4/4 is really equal X 3 . Moreover, A(x) is zero for X= 0, as it should be if A(x) is indeed an area.

In mathematical analysis, it is proved that there is no other answer than the above expression for A(x), does not exist. Let us show that this statement is plausible using the following heuristic (non-rigorous) reasoning. Suppose there is some second solution V(x). If A(x) and V(x) “start” simultaneously from the zero value at X= 0 and change at the same rate all the time, then their values ​​will never X cannot become different. They must match everywhere; hence there is a unique solution.

How can you justify the ratio Aў ( x) = f(x) in general? This question can only be answered by studying the rate of area change as a function of X in general. Let m- the smallest value of the function f (x) in the interval from X before ( x + h), a M is the largest value of this function in the same interval. Then the area increment upon passing from X To ( x + h) must be enclosed between the areas of the two rectangles (Fig. 17). The bases of both rectangles are equal h. The smaller rectangle has a height m and area mh, larger, respectively, M and Mh. On a plot of area vs. X(Fig. 18) it can be seen that when the abscissa changes to h, the value of the ordinate (i.e. area) is increased by the amount between mh and Mh. The slope of the secant in this graph is between m and M. what happens when h goes to zero? If the graph of the function y = f(x) is continuous (i.e., does not contain discontinuities), then M, and m tend to f(x). Therefore, the slope Aў ( x) graph of the area as a function of X equals f(x). That was the conclusion that needed to be reached.

Leibniz proposed for the area under the curve y = f(x) from 0 to a designation

With a rigorous approach, this so-called definite integral must be defined as the limit of certain sums in the manner of Wallis. Given the result obtained above, it is clear that this integral is calculated under the condition that we can find such a function A(x), which vanishes when X= 0 and has a derivative Aў ( x) equal to f (x). Finding such a function is usually called integration, although it would be more appropriate to call this operation anti-differentiation, meaning that it is in a sense the inverse of differentiation. In the case of a polynomial, integration is easy. For example, if

which is easy to verify by differentiating A(x).

To calculate the area A 1 under the curve y = 1 + x + x 2 /2 enclosed between the ordinates 0 and 1, we simply write

and by substituting X= 1, we get A 1 = 1 + 1 / 2 + 1 / 6 = 5 / 3. Square A(x) from 0 to 2 is A 2 = 2 + 4 / 2 + 8 / 6 = 16 / 3. As can be seen from fig. 19, the area enclosed between the ordinates 1 and 2 is A 2 – A 1 = 11 / 3. It is usually written as a definite integral

Volumes.

Similar reasoning makes it surprisingly simple to calculate the volumes of bodies of revolution. Let's demonstrate this using the example of calculating the volume of a ball, another classical problem that the ancient Greeks, using the methods known to them, managed to solve with great difficulty.

Let's rotate a part of the plane enclosed inside a quarter of a circle of radius r, at an angle of 360° around the axis X. As a result, we get a hemisphere (Fig. 20), the volume of which we denote V(x). It is required to determine the rate at which the V(x) with increasing x. Going from X To X + h, it is easy to verify that the volume increment is less than the volume p(r 2 – x 2)h circular cylinder of radius and height h, and more than the volume p[r 2 – (x + h) 2 ]h cylinder radius and height h. Therefore, on the graph of the function V(x) the slope of the secant is enclosed between p(r 2 – x 2) and p[r 2 – (x + h) 2 ]. When h tends to zero, the slope tends to

At x = r we get

for the volume of the hemisphere, and therefore 4 p r 3/3 for the volume of the entire ball.

A similar method allows finding the lengths of curves and areas of curved surfaces. For example, if a(x) – arc length PR in fig. 21, then our task is to calculate aў( x). At the heuristic level, we will use a technique that allows us not to resort to the usual passage to the limit, which is necessary for a rigorous proof of the result. Let us assume that the rate of change of the function a(x) at the point R the same as it would be if the curve were replaced by its tangent PT at the point P. But from fig. 21 is directly visible, when stepping h to the right or left of the dot X along RT meaning a(x) changes to

Therefore, the rate of change of the function a(x) is

To find the function itself a(x), it is only necessary to integrate the expression on the right side of the equality. It turns out that integration is rather difficult for most functions. Therefore, the development of integral calculus methods is a large part of mathematical analysis.

Primitives.

Every function whose derivative is equal to the given function f(x), is called antiderivative (or primitive) for f(x). For instance, X 3 /3 - antiderivative for the function X 2 because ( x 3 /3)ў = x 2. Of course X 3/3 is not the only antiderivative of the function X 2 because x 3 /3 + C is also the derivative for X 2 for any constant WITH. However, in what follows we agree to omit such additive constants. In general

where n is a positive integer, since ( x n + 1/(n+ 1))ў = x n. Relation (1) is satisfied in an even more general sense if n replace with any rational number k, except for -1.

An arbitrary antiderivative function for a given function f(x) is usually called the indefinite integral of f(x) and denote it as

For example, since (sin x)ў = cos x, the formula

In many cases where there is a formula for the indefinite integral of a given function, it can be found in numerous widely published tables of indefinite integrals. The integrals of elementary functions are tabular (they include powers, logarithms, exponential function, trigonometric functions, inverse trigonometric functions, as well as their finite combinations obtained using addition, subtraction, multiplication and division). With the help of tabular integrals, integrals can also be calculated from more complex functions. There are many ways to calculate indefinite integrals; the most common of these is the variable substitution or substitution method. It consists in the fact that if we want to replace in the indefinite integral (2) x to some differentiable function x = g(u), then in order for the integral not to change, it is necessary x replaced by gў ( u)du. In other words, the equality

(substitution 2 x = u, whence 2 dx = du).

Let us present one more method of integration – the method of integration by parts. It is based on the well-known formula

After integrating the left and right sides, and taking into account that

This formula is called the integration-by-parts formula.

Example 2. Need to find . Since cos x= (sin x)ў , we can write that

From (5), assuming u = x and v= sin x, we get

And since (-cos x)ў = sin x we find that and

It should be emphasized that we have limited ourselves to a very brief introduction to a very extensive subject, in which numerous witty tricks have been accumulated.

Functions of two variables.

Due to the curve y = f(x), we considered two problems.

1) Find the slope of the tangent to the curve at a given point. This problem is solved by calculating the value of the derivative fў ( x) at the given point.

2) Find the area under the curve above the axis segment X bounded by vertical lines X = a and X = b. This problem is solved by calculating a definite integral.

Each of these problems has an analogue in the case of a surface z = f(x,y).

1) Find the tangent plane to the surface at a given point.

2) Find the volume under the surface above the part of the plane hu, bounded curve WITH, and on the side - perpendicular to the plane xy passing through the points of the boundary curve WITH (cm. rice. 22).

The following examples show how these problems are solved.

Example 4. Find the tangent plane to the surface

at the point (0,0,2).

A plane is defined if two intersecting lines lying in it are given. One of these lines l 1) we will get in the plane xz (at= 0), second ( l 2) - in the plane yz (x = 0) (cm. rice. 23).

First of all, if at= 0, then z = f(x,0) = 2 – 2x – 3x 2. Derivative with respect to X, denoted fў x(x,0) = –2 – 6x, at X= 0 has a value of -2. Straight l 1 given by the equations z = 2 – 2x, at= 0 - tangent to WITH 1 , lines of intersection of the surface with the plane at= 0. Similarly, if X= 0, then f(0,y) = 2 – y – y 2 , and the derivative with respect to at has the form

Because fў y(0.0) = -1, curve WITH 2 - line of intersection of the surface with the plane yz- has a tangent l 2 given by the equations z = 2 – y, X= 0. The desired tangent plane contains both lines l 1 and l 2 and is written by the equation

This is the equation of the plane. In addition, we get direct l 1 and l 2 , assuming, respectively, at= 0 and X = 0.

The fact that equation (7) really defines the tangent plane can be verified at a heuristic level if you notice that this equation contains first-order terms that appear in equation (6), and that second-order terms can be represented as -. Since this expression is negative for all values X and at, Besides X = at= 0, the surface (6) lies below the plane (7) everywhere, except for the point R= (0,0,0). We can say that the surface (6) is convex upward at the point R.

Example 5. Find the tangent plane to the surface z = f(x,y) = x 2 – y 2 at origin 0.

On surface at= 0 we have: z = f(x,0) = x 2 and fў x(x,0) = 2x. On the WITH 1 , intersection lines, z = x 2. At the point O the slope is fў x(0,0) = 0. On the plane X= 0 we have: z = f(0,y) = –y 2 and fў y(0,y) = –2y. On the WITH 2 , lines of intersection, z = –y 2. At the point O curve slope WITH 2 equals fў y(0,0) = 0. Since the tangents to WITH 1 and WITH 2 are axes X and at, the tangent plane containing them is the plane z = 0.

However, in the vicinity of the origin, our surface is not on the same side of the tangent plane. Indeed, the curve WITH 1 lies above the tangent plane everywhere, except for the point 0, and the curve WITH 2 - respectively below it. Surface intersects tangent plane z= 0 in straight lines at = X and at = –X. Such a surface is said to have a saddle point at the origin (Fig. 24).

Private derivatives.

In the previous examples, we used the derivatives of f (x,y) on X and by at. Let us now consider such derivatives in a more general way. If we have a function of two variables, for example, F(x,y) = x 2 – xy, then we can determine at each point two of its "partial derivatives", one - by differentiating the function with respect to X and fixing at, and differentiating the other with respect to at and fixing X. The first of these derivatives is denoted as fў x(x,y) or ¶ f/¶ x; the second is how f f y. If both mixed derivatives (by X and at, on at and X) are continuous, then ¶ 2 f/¶ x¶ y= ¶ 2 f/¶ y¶ x; in our example ¶ 2 f/¶ x¶ y= ¶ 2 f/¶ y¶ x = –1.

Partial derivative fў x(x,y) indicates the rate of change of the function f at point ( x,y) in the direction of increase X, a fў y(x,y) is the rate of change of the function f in ascending direction at. Function change rate f at point ( X,at) in the direction of the straight line constituting the angle q with positive axis direction X, is called the derivative of the function f towards; its value is a combination of two partial derivatives of the function f in the tangent plane is almost equal (for small dx and dy) true change z on the surface, but calculating the differential is usually easier.

The formula we have already considered from the change of variable method, known as the derivative of a complex function or the chain rule, in the one-dimensional case, when at depends on X, a X depends on t, looks like:

For functions of two variables, a similar formula has the form:

The concepts and notation of partial differentiation can be easily generalized to higher dimensions. In particular, if the surface is given implicitly by the equation f(x,y,z) = 0, the equation of the tangent plane to the surface can be given a more symmetrical form: the equation of the tangent plane at the point ( x(x 2 /4)], then integrates over X from 0 to 1. The final result is 3/4.

Formula (10) can also be interpreted as the so-called double integral, i.e. as the limit of the sum of volumes of elementary "cells". Each such cell has a base D x D y and a height equal to the height of the surface above some point of the rectangular base ( cm. rice. 26). It can be shown that both points of view on formula (10) are equivalent. Double integrals are used to find centers of gravity and numerous moments encountered in mechanics.

A more rigorous justification of the mathematical apparatus.

So far, we have presented the concepts and methods of mathematical analysis on an intuitive level and have not hesitated to resort to geometric figures. It remains for us to briefly consider the more rigorous methods that emerged in the 19th and 20th centuries.

At the beginning of the 19th century, when the era of assault and onslaught in the "creation of mathematical analysis" ended, questions of its justification came to the fore. In the works of Abel, Cauchy and a number of other outstanding mathematicians, the concepts of "limit", "continuous function", "convergent series" were precisely defined. This was necessary in order to introduce a logical order into the basis of mathematical analysis in order to make it a reliable research tool. The need for a thorough justification became even more obvious after the discovery in 1872 by Weierstrass of functions that are everywhere continuous but nowhere differentiable (the graph of such functions has a break at each of its points). This result made a stunning impression on mathematicians, since it clearly contradicted their geometric intuition. An even more striking example of the unreliability of geometric intuition was the continuous curve constructed by D. Peano, which completely fills a certain square, i.e. passing through all its points. These and other discoveries brought to life the program of "arithmetization" of mathematics, i.e. making it more reliable by substantiating all mathematical concepts with the help of the concept of number. The almost puritanical abstention from visualization in works on the foundations of mathematics had its historical justification.

According to modern canons of logical rigor, it is unacceptable to talk about the area under the curve y = f(x) and above the axis segment X, even f is a continuous function, without having previously determined the exact meaning of the term "area" and without establishing that the area defined in this way really exists. This problem was successfully solved in 1854 by B. Riemann, who gave a precise definition of the concept of a definite integral. Since then, the idea of ​​summation behind the concept of a definite integral has been the subject of many deep investigations and generalizations. As a result, today it is possible to give meaning to the definite integral, even if the integrand is discontinuous everywhere. New concepts of integration, to the creation of which A. Lebesgue (1875–1941) and other mathematicians made a great contribution, have increased the power and beauty of modern mathematical analysis.

It would hardly be appropriate to go into the details of all these and other concepts. We confine ourselves to giving rigorous definitions of the limit and the definite integral.

In conclusion, let us say that mathematical analysis, being an extremely valuable tool in the hands of a scientist and engineer, still attracts the attention of mathematicians today as a source of fruitful ideas. At the same time, modern development seems to indicate that mathematical analysis is increasingly absorbed by such dominant in the 20th century. branches of mathematics like abstract algebra and topology.

It is absolutely impossible to solve physical problems or examples in mathematics without knowledge about the derivative and methods for calculating it. The derivative is one of the most important concepts of mathematical analysis. We decided to devote today's article to this fundamental topic. What is a derivative, what is its physical and geometric meaning, how to calculate the derivative of a function? All these questions can be combined into one: how to understand the derivative?

Geometric and physical meaning of the derivative

Let there be a function f(x) , given in some interval (a,b) . The points x and x0 belong to this interval. When x changes, the function itself changes. Argument change - difference of its values x-x0 . This difference is written as delta x and is called argument increment. The change or increment of a function is the difference between the values ​​of the function at two points. Derivative definition:

The derivative of a function at a point is the limit of the ratio of the increment of the function at a given point to the increment of the argument when the latter tends to zero.

Otherwise it can be written like this:

What is the point in finding such a limit? But which one:

the derivative of a function at a point is equal to the tangent of the angle between the OX axis and the tangent to the graph of the function at a given point.

The physical meaning of the derivative: the time derivative of the path is equal to the speed of the rectilinear motion.

Indeed, since school days, everyone knows that speed is a private path. x=f(t) and time t . Average speed over a certain period of time:

To find out the speed of movement at a time t0 you need to calculate the limit:

Rule one: take out the constant

The constant can be taken out of the sign of the derivative. Moreover, it must be done. When solving examples in mathematics, take as a rule - if you can simplify the expression, be sure to simplify .

Example. Let's calculate the derivative:

Rule two: derivative of the sum of functions

The derivative of the sum of two functions is equal to the sum of the derivatives of these functions. The same is true for the derivative of the difference of functions.

We will not give a proof of this theorem, but rather consider a practical example.

Find the derivative of a function:

Rule three: the derivative of the product of functions

The derivative of the product of two differentiable functions is calculated by the formula:

Example: find the derivative of a function:

Solution:

Here it is important to say about the calculation of derivatives of complex functions. The derivative of a complex function is equal to the product of the derivative of this function with respect to the intermediate argument by the derivative of the intermediate argument with respect to the independent variable.

In the above example, we encounter the expression:

In this case, the intermediate argument is 8x to the fifth power. In order to calculate the derivative of such an expression, we first consider the derivative of the external function with respect to the intermediate argument, and then multiply by the derivative of the intermediate argument itself with respect to the independent variable.

Rule Four: The derivative of the quotient of two functions

Formula for determining the derivative of a quotient of two functions:

We tried to talk about derivatives for dummies from scratch. This topic is not as simple as it seems, so be warned: there are often pitfalls in the examples, so be careful when calculating derivatives.

With any question on this and other topics, you can contact the student service. In a short time, we will help you solve the most difficult control and deal with tasks, even if you have never dealt with the calculation of derivatives before.

On which we analyzed the simplest derivatives, and also got acquainted with the rules of differentiation and some techniques for finding derivatives. Thus, if you are not very good with derivatives of functions or some points of this article are not entirely clear, then first read the above lesson. Please tune in to a serious mood - the material is not easy, but I will still try to present it simply and clearly.

In practice, you have to deal with the derivative of a complex function very often, I would even say almost always, when you are given tasks to find derivatives.

We look in the table at the rule (No. 5) for differentiating a complex function:

We understand. First of all, let's take a look at the notation. Here we have two functions - and , and the function, figuratively speaking, is nested in the function . A function of this kind (when one function is nested within another) is called a complex function.

I will call the function external function, and the function – inner (or nested) function.

! These definitions are not theoretical and should not appear in the final design of assignments. I use the informal expressions "external function", "internal" function only to make it easier for you to understand the material.

To clarify the situation, consider:

Example 1

Find the derivative of a function

Under the sine, we have not just the letter "x", but the whole expression, so finding the derivative immediately from the table will not work. We also notice that it is impossible to apply the first four rules here, there seems to be a difference, but the fact is that it is impossible to “tear apart” the sine:

In this example, already from my explanations, it is intuitively clear that the function is a complex function, and the polynomial is an internal function (embedding), and an external function.

First step, which must be performed when finding the derivative of a complex function is to understand which function is internal and which is external.

In the case of simple examples, it seems clear that a polynomial is nested under the sine. But what if it's not obvious? How to determine exactly which function is external and which is internal? To do this, I propose to use the following technique, which can be carried out mentally or on a draft.

Let's imagine that we need to calculate the value of the expression with a calculator (instead of one, there can be any number).

What do we calculate first? First of all you will need to perform the following action: , so the polynomial will be an internal function:

Secondly you will need to find, so the sine - will be an external function:

After we UNDERSTAND with inner and outer functions, it's time to apply the compound function differentiation rule .

We start to decide. From the lesson How to find the derivative? we remember that the design of the solution of any derivative always begins like this - we enclose the expression in brackets and put a stroke at the top right:

First we find the derivative of the external function (sine), look at the table of derivatives of elementary functions and notice that . All tabular formulas are applicable even if "x" is replaced by a complex expression, in this case:

Note that the inner function has not changed, we do not touch it.

Well, it is quite obvious that

The result of applying the formula clean looks like this:

The constant factor is usually placed at the beginning of the expression:

If there is any misunderstanding, write down the decision on paper and read the explanations again.

Example 2

Find the derivative of a function

Example 3

Find the derivative of a function

As always, we write:

We figure out where we have an external function, and where is an internal one. To do this, we try (mentally or on a draft) to calculate the value of the expression for . What needs to be done first? First of all, you need to calculate what the base is equal to:, which means that the polynomial is the internal function:

And, only then exponentiation is performed, therefore, the power function is an external function:

According to the formula , first you need to find the derivative of the external function, in this case, the degree. We are looking for the desired formula in the table:. We repeat again: any tabular formula is valid not only for "x", but also for a complex expression. Thus, the result of applying the rule of differentiation of a complex function next:

I emphasize again that when we take the derivative of the outer function, the inner function does not change:

Now it remains to find a very simple derivative of the inner function and “comb” the result a little:

Example 4

Find the derivative of a function

This is an example for self-solving (answer at the end of the lesson).

To consolidate the understanding of the derivative of a complex function, I will give an example without comments, try to figure it out on your own, reason, where is the external and where is the internal function, why are the tasks solved that way?

Example 5

a) Find the derivative of a function

b) Find the derivative of the function

Example 6

Find the derivative of a function

Here we have a root, and in order to differentiate the root, it must be represented as a degree. Thus, we first bring the function into the proper form for differentiation:

Analyzing the function, we come to the conclusion that the sum of three terms is an internal function, and exponentiation is an external function. We apply the rule of differentiation of a complex function :

The degree is again represented as a radical (root), and for the derivative of the internal function, we apply a simple rule for differentiating the sum:

Ready. You can also bring the expression to a common denominator in brackets and write everything as one fraction. It’s beautiful, of course, but when cumbersome long derivatives are obtained, it’s better not to do this (it’s easy to get confused, make an unnecessary mistake, and it will be inconvenient for the teacher to check).

Example 7

Find the derivative of a function

This is an example for self-solving (answer at the end of the lesson).

It is interesting to note that sometimes, instead of the rule for differentiating a complex function, one can use the rule for differentiating a quotient , but such a solution will look like a perversion unusual. Here is a typical example:

Example 8

Find the derivative of a function

Here you can use the rule of differentiation of the quotient , but it is much more profitable to find the derivative through the rule of differentiation of a complex function:

We prepare the function for differentiation - we take out the minus sign of the derivative, and raise the cosine to the numerator:

Cosine is an internal function, exponentiation is an external function.
Let's use our rule :

We find the derivative of the inner function, reset the cosine back down:

Ready. In the considered example, it is important not to get confused in the signs. By the way, try to solve it with the rule , the answers must match.

Example 9

Find the derivative of a function

This is an example for self-solving (answer at the end of the lesson).

So far, we have considered cases where we had only one nesting in a complex function. In practical tasks, you can often find derivatives, where, like nesting dolls, one inside the other, 3 or even 4-5 functions are nested at once.

Example 10

Find the derivative of a function

We understand the attachments of this function. We try to evaluate the expression using the experimental value . How would we count on a calculator?

First you need to find, which means that the arcsine is the deepest nesting:

This arcsine of unity should then be squared:

And finally, we raise the seven to the power:

That is, in this example we have three different functions and two nestings, while the innermost function is the arcsine, and the outermost function is the exponential function.

We start to decide

According to the rule first you need to take the derivative of the outer function. We look at the table of derivatives and find the derivative of the exponential function: The only difference is that instead of "x" we have a complex expression, which does not negate the validity of this formula. So, the result of applying the rule of differentiation of a complex function next.