## How to Construct Graphs? – Shilov and Simplest Maxima Minima Problems – Natanson

In this post, we will see a double book How To Construct Graphs by G. E. Shilov And Simplest Maxima And Minima Problems by I. P. Natanson. These two books are part of the Topics in Mathematics series. The first part of this booklet. How to Construct Graphs by G. E. Shilov, presents simple methods of plotting graphs, first “by points” and then “by operations.”The latter method offers a means of constructing graphs of complicated functions by considering the function as a succession of operations performed on an initial quantity.
The second part, Simplest Maxima and Minima Problems by I. P. Natanson, shows how to solve certain maxima and minima problems by algebraic methods. This material is excellent prepa­ration for calculus, in which such problems are treated more gen­erally. (In order to relate this part to the preceding one, several paragraphs and Fig. A and Fig. B, not present in the Russian edition, have been added.)
This booklet can be read by anyone who has studied intermedi­ate algebra.

How to Construct Graphs  was translated from Russian by Jerome Kristian and Daniel A. Levine. Simplest Maxima and Minima Problems  was translated from Russian by C. Clark Kissinger and Robert B. Brown. The book was published in 1963.

You can get the book here.

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# HOW TO CONSTRUCT GRAPHS by G. E. Shilov 3

## CHAPTER l. Graphs “by Points” 3

1. Introduction 3
2. Coordinate system 3
3. Graph of an equation 4

## CHAPTER 2. Graphs “by Operations” 7

4. Graphs of first-degree equations 7
5. Graphs of second-degree equations 8
6. Graphs by multiplication 11
7. Graphs by division 13
8. Summary 18

Exercises and Solutions 20

# SIMPLEST MAXIMA AND MINIMA PROBLEMS by I. P. Natanson

Introduction 25

## CHAPTER l. The Fundamental Theorem on Quadratic Trinomials 26

1. Parabolas; minimum values 26
3. Maximum values 29
4. The Fundamental Theorem 30

## CHAPTER 2. Applications 33

5. Applications of the Fundamental Theorem 7 33
6. Applications of Problem 1 38

## CHAPTER 3. Further Theorems and Applications 40

7. Theorems derived from Problem 1 40
8. Generalization of Theorem l of section 42
9. Arithmetical applications 47
10. Geometrical applications 48
11. Summary 53 