Preface: Developing students' ability to find clues to problems on their own 003
This book's learning schedule 007

Lecture 1 Continuity of Functions
Mathematical Reasoning Core Concepts 012
Solving Continuity Problems 014
Discovery of Discontinuity 018
Proof of Basic Properties 026
Practice Essay Test Practice Problem 1: 028? Practice Problem 2: 029? Practice Problem 3: 030

Lecture 2 Differentiability
Mathematical Reasoning Core Concepts 032
Solving Problems Related to Differentiability 034
Practice Essay Exam Practice Problem 4: 039? Practice Problem 5: 040? Practice Problem 6: 041? Practice Problem 7: 042

Lecture 3 Absolute Inequalities
Mathematical Reasoning Core Concepts 044
Absolute inequalities and maximum and minimum 046
Geometric Approach 049
Basic Maximum and Minimum Problem 051
Practice Essay Exam Practice Problem 8 052? Practice Problem 9 053? Practice Problem 10 054? Practice Problem 11 056

Lecture 4 Proof of Inequality
Mathematical Reasoning Core Concepts 058
Triangle Inequality 059
Arithmetic Mean - Geometric Mean Inequality 061
Cauchy-Schwarz's Inequality 064
Proof of Inequalities Using Differential Calculus 065
Practice Essay Exam Practice Problem 12: 067? Practice Problem 13: 068? Practice Problem 14: 069? Practice Problem 15: 070? Practice Problem 16: 071? Practice Problem 17: 072

Lecture 5 Substitution Integrals and Partial Integrations
Mathematical Reasoning Core Concepts 074
Application of Integration by Parts 076
Application of Substitution Integration 078
081 When there are two or more functions whose integrals are unknown
Integration of Exponential and Logarithmic Functions 083
Integral around sine and cosine 085
Integral of tangent and secant centers 087
Complex Applications of Composite Functions 089
Calculating the integration by parts of the repeated form 091
Practice Essay Exam Practice Problem 18 093? Practice Problem 19 095? Practice Problem 20 096? Practice Problem 21 097

Lecture 6 Calculating Definite Integrators
100 Key Concepts in Mathematical Reasoning
Symmetry and Definite Integrals 102
Composition of Even Functions and Fractional Functions 104
Inverse Functions and Definite Integrals 106
Practice Essay Exam Practice Questions 22 110? Practice Questions 23 111? Practice Questions 24 112

Lecture 7 Functions and Existence
Mathematical Reasoning Core Concepts 114
Proof of the Existential Theorem 116
Application of the Existential Theorem to Continuity 118
Application of the Existence Theorem to Differentiability 119
Problem 122: Demonstrating Existence
Problem 123 showing more than one existence
Problem 125: Using the Conclusion of the Existential Theorem
Practice Essay Exam Practice Questions 25 126? Practice Questions 26 127? Practice Questions 27 128? Practice Questions 28 129

Lecture 8: Definite Integrals and Distinctive Quadratics
Mathematical Reasoning Core Concepts 132
Definite Integrators and Volume 134
Cavalieri's Principle 139
If 140
144 when n is finite

Lecture 9 Differential Equations
Mathematical Reasoning Core Concepts 148
Differential Equations in Practice 150
Practice Essay Exam Practice Problems 29 155? Practice Problems 30 156? Practice Problems 31 157? Practice Problems 32 158
Practice Problem 33 159


commentary


Commentary 1 Continuity of functions
Solving Continuity Problems 162
Discovery of Discontinuity 168
Practice Essay Test Practice Problem 1: 175? Practice Problem 2: 177? Practice Problem 3: 181

Commentary 2 Differentiability
Solving Problems Related to Differentiability 185
Practice Essay Exam Practice Problem 4: 191? Practice Problem 5: 195? Practice Problem 6: 198? Practice Problem 7: 200

Explanation 3 Absolute Inequality
Geometric Approach 204
Basic maximum and minimum problems 207
Practice Essay Exam Practice Problem 8 210? Practice Problem 9 213? Practice Problem 10 216? Practice Problem 11 221

Commentary 4 Proof of Inequality
Proof of Inequalities Using Differential Calculus 226
Practice Essay Exam Practice Problems 12 (231)? Practice Problems 13 (232)? Practice Problems 14 (237)? Practice Problems 15 (240)? Practice Problems 16 (243)? Practice Problems 17 (249)

Commentary 5 Substitution Integral and Partial Integration
Application of substitution integration 253
260 When there are two or more functions whose integrals are unknown
Integration of Exponential and Logarithmic Functions 263
Integral around sine and cosine 266
Integral of tangent and secant centers 270
Complex Applications of Composite Functions 275
Computation of Integration by Parts in Repeated Forms 278
Practice Essay Exam Practice Problem 18 (280)? Practice Problem 19 (282)? Practice Problem 20 (283)? Practice Problem 21 (286)

Commentary 6 Calculating definite integrals
Symmetry and Definite Integrals 289
Composition of Even Functions and Fractional Functions 294
Inverse Functions and Definite Integrals 296
Practice Essay Exam Practice Questions 22 302? Practice Questions 23 304? Practice Questions 24 306

Commentary 7 Functions and Existence
Problem 308: Demonstrating Existence
Problem 312 showing more than one existence
Problem 317: Using the Conclusion of the Existential Theorem
Practice Essay Exam Practice Questions 25 324? Practice Questions 26 325? Practice Questions 27 328? Practice Questions 28 3330

Commentary 8 Definite Integral and Distinctive Quadrature
Definite Integrators and Volume 332
In case 336
339 when n is finite

Commentary 9 Differential Equations
Differential Equations in Practice 343
Practice Essay Exam Practice Problems 29 345? Practice Problems 30 346? Practice Problems 31 347? Practice Problems 32 349? Practice Problems 33 350

Appendix: Success Story 354? Successful Essay Series 355? Success Story Contest 356

Mathematical Reasoning Pathfinder: A Classic Guide to Mathematical Reasoning, Focusing on Logic, Not Just Solving Problems
From the basics of mathematical reasoning to essential types and practical problems
The best self-study book in 3 volumes!

Even students who score high on the math section of the CSAT find the mathematical essay section difficult, as the types of questions on the mathematical essay section are more diverse and the logical structure is often more complex.
To help students understand the characteristics of mathematical reasoning and prepare for the exam on their own, "Mathematical Reasoning Pathfinder" consists of three volumes: the basics section, essential types section, and the key to a great turnaround section.
The focus is on developing problem-solving skills to approach unfamiliar problems, analyze their types and structures, and find answers on your own.
In addition, we provide specific examples of logical description methods and systematic answer writing techniques to help you achieve high scores in actual exams.


The three-volume series of “Math Argument Pathfinder” are as follows.


Volume 1: Basics_Argumentation Methods

Lecture 1 Argument 1
Lecture 2 Argument 2
Lecture 3 Argument 3
Lecture 4 Argument 4
Lecture 5: The Law of Abduction
Lecture 6 Mathematical Induction
It covers the correct way to think about mathematics, including problem-solving methods and proof methods, so that you can develop the basic skills necessary for any mathematical reasoning problem.
To help you learn these 'methods' without any difficulties, it is structured around the concepts of math, math ?, math ?.


Volume 2 Essential Types _ Essential Types of Math 1, Math 2, and Calculus
The mathematical reasoning questions were divided into categories.
Although unfamiliar to students, this book collects types of questions that frequently appear in mathematical reasoning tests and explains problem-solving approaches based on the core theories and principles of each type.
The focus was on developing students' ability to solve problems on their own.

Lecture 1 Continuity of Functions
Lecture 2 Differentiability
Lecture 3 Absolute Inequalities
Lecture 4 Quadratic Functions
Lecture 5 Algebraic Applications of Trigonometric Functions
Lecture 6 Limits of Sequences
Lecture 7 Level
Lecture 8 Plane Geometry
Lecture 9 Differential Calculus and Maximum and Minimum
Lecture 10 Limits of functions (0, infinitesimal)
Lecture 11 Limits of functions (∞, infinite)
Lecture 12 Functional Equations
Lecture 13 Roots and Intersections
Lecture 14 Street
Lecture 15 Planar Motion

The Key to a 3-Volume Reversal? The Key Types That Determine Success
It mainly covers types that can have a significant difference in scores and have a decisive impact on passing, and types that require prior study of types covered in Volumes 2 and 3.

＊ Probability, statistics, and geometry will be published separately to reflect recent exam trends.

Lecture 1 Analysis of Polynomials
Lecture 2 Number Theory
Lecture 3 Regularities and Sequences
Lecture 4 Proof of Inequality
Lecture 5 Substitution Integrals and Partial Integrations
Lecture 6 Calculating Definite Integrators
Lecture 7 Functions and Existence
Lecture 8: Definite Integrals and Distinctive Quadratics
Lecture 9 Differential Equations

“Why do you solve it that way?”

More important than how you solve a problem is why you solve it that way.
This book is not simply a collection of past exam questions and model explanations.
No matter how exemplary a solution method you know, it is meaningless if you cannot find it in the exam room.
All three volumes of the "Mathematics Pathfinder" series teach students how to identify problems and think in order through the "Window of Thought," thereby fostering their problem-solving skills to find clues on their own.
This is the biggest feature that sets it apart from other math essay study books.


Below is a description of each part of the book.

Window of Thought
Explained how to identify the problem and approach it fundamentally.

Problem explanation
The problem-solving process and thinking methods are explained as if in a lecture, so that students can study effectively on their own.

Example Answer
In cases where the problem explanation is long and contains a lot of content that does not necessarily need to be included when writing the answer in a real exam, example answers have been added.
A sample answer is an example of an answer that summarizes only the content that must be included in the answer in a real test.


Textbook concept
The concepts of the curriculum used in problem solving are summarized.

Concept
We have organized concepts that are helpful in practice or frequently used in mathematical reasoning.

This book was written with great effort to help students prepare for the essay test on their own in a limited amount of time, and to serve as a useful guide for teachers who teach mathematical essay writing.
What I want students to do is not to study by hoping for luck without thinking deeply or by looking at the answers and memorizing them.
Your mathematical reasoning skills will improve not by the number of problems you solve, but by the amount of time you spend thinking about them.
If you study each section carefully, you will be able to improve your skills sufficiently with just this one book.