1. Why Study Mathematics?
The answer is quite simple: mathematics is the foundational language of all sciences and is deeply intertwined with everyday life. From food, clothing, housing, and transportation to investing and personal finance, the social sciences, and cutting-edge fields like computer science, blockchain, big data, AI, and quantum computing—none of these can function without mathematics. Therefore, the primary objectives of studying mathematics are as follows:
1. To train one’s thinking and logical-reasoning skills.
2. To learn how to analyze problems and simplify complex situations.
3. To cultivate problem-solving abilities.
4. To help us understand the world and enrich our lives.
These critical thinking skills are most readily developed through the study of mathematics. With strong analytical and reasoning abilities, you’ll generate more ideas than others, devise better solutions when challenges arise, and enjoy a smoother, more creative, and more competitive life.
Many people claim that once you enter the workforce you never use mathematics or calculus, but this isn’t true. It’s not that math goes unused. Rather, most people simply enjoy the conveniences it brings without understanding how or where it’s applied. In a straightforward, everyday life you might not notice its presence, but without mathematics there would be no smartphones, computers, internet, or other conveniences in life. In-depth research and applications across all fields inevitably depend on math. Therefore, to boost personal or national competitiveness, one must master mathematics.
2. Features of This Book
◎ Detailed contents, step-by-step explanations, and thorough annotations.
◎ Includes Desmos dynamic visualization accessible via QR codes—simply scan with your smartphone for interactive learning.
Please Note:
◎ The Desmos system on mobile devices does not automatically adjust the size of expression labels. After scanning a QR code, you may occasionally see overlapping formulas; simply pinch-to-zoom to restore proper display.
◎ For a clearer view in the 3D Desmos environment on your phone, tap the tool icon at the top right of the screen and set the 3D workspace to “semi-transparent.” This makes the solid models to see clearly.
3. How to Excel in Calculus
Most people don’t really enjoy mathematics, and this usually comes down to two factors: innate ability and external influences. Innate ability refers to the mathematical talents inherited from one’s parents, while external influences include many factors—environment, family background, personal study methods, habits, and teaching style, to name a few. Even if your innate ability isn’t strong, with determination and effective study strategies there is plenty of room for improvement. Of course, finding a great teacher and cultivating good study habits will double your learning efficiency.
Below are some simple strategies and habits to help improve your mathematical thinking:
◎ Take meaningful notes. Don’t just copy what’s on the board—your notes should include annotations and explanations written for your own understanding. Record the instructor’s key ideas or insights (especially those not written on the board). Use at least three different colors to mark: (1) the most important points, (2) topics you find challenging, and (3) anything the teacher emphasizes specially.
◎ Identify the key points. After each lecture or chapter, reflect on which concepts are most critical, and mark them with special symbols in your notes.
◎ Break problems into steps. When you read a problem, think through how to divide it up for solution:
(1) Understand the problem statement.
(2) Determine the ultimate goal of the question.
(3) List the given conditions.
(4) Decide how to use those conditions to reach the answer.
(5) Identify the very first step.
(Further worked examples will show you how to structure each step in detail.)
Write these steps alongside the problem as you work through it.
◎ Practice deliberately. As soon as you grasp a type of problem, practice similar exercises. If your method is sound, you’ll improve rapidly; if it’s flawed, no amount of rote practice will help.
By applying these techniques consistently—meaningful note-taking, pinpointing key ideas, stepwise problem decomposition, and focused practice—you’ll significantly enhance your ability to learn and enjoy calculus.
I dedicate this book solely to my illiterate parents. If I can do it, I believe you can too.