## Understanding Calculus

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 Preface 1. Why Study   Calculus 2. Numbers 3. Functions 4. The Derivative 5. Differentiation 6. Applications 7. Free Falling   Motion 8. Understanding  Derivative 9. Derivative  Approximations 10. Integration   Theory 11. Understanding   Integration 12. Differentials

 Inverse Functions Exponents Exponential  Functions Applications of   Exponential  Functions Sine and Cosine   Function Sine Function Sine Function -   Differentiation and   Integration Oscillatory Motion Mean Value   Theorem Taylor Series More Taylor Series Integration   Techniques

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CHAPTER 3

### Chapter 3 - The Mathematical function and its Graph

#### Section 3.1 - The Scientific Method

Science is a rational method of observing situations and then forming hypotheses to understand them. The language of science is mathematics and it is through the mathematical function that a situation can be expressed in terms of the conditions that define it. The function is thus a fundamental concept in mathematics and Calculus is the branch of mathematics dealing with functions whose dimensions change. What then is a function and how is it expressed mathematically?

Before we can learn what a function is we need to understand how mathematics relates to science. The scientific method is an orderly and efficient way of analyzing physical situations. Its counterpoint, the trial and error method, is something you are probably more familiar with. Human nature inclines the mind more toward exploration and experimentation rather than systematic analysis. The trial and error method reflects the tangents that human logic takes when curiosity beckons.

The scientific method, on the other hand, provides an orderly path for the mind to follow in its passage from bewilderment to enlightenment. Without following the scientific method, an analysis will remain consistently complex and any theories formed will be too general to explain the phenomena accurately. Therefore an understanding of the scientific method is crucial to your understanding of mathematics and development as a scientist. Here are its main steps.

• Identify the problem or situation
• Narrow down the problem statement while being as specific as possible.
• Remove external factors to leave behind a few interacting conditions.
• Analyze remaining conditions along with their properties with respect to the entire system.
• Make reasonable assumptions about the controlled situation, such as mass/ energy/ people balance etc.
• Understand how the conditions come together to define the situation.
• Determine a logical relationship among the conditions.
• Select the most efficient solution that accurately explains the system through the interacting conditions.

Engineering and science both follow the same systematic analysis to solve problems and understand the world. Science deals more with objective and idealized situations while engineering is often concerned with subjective situations that require sound judgment. Situations are not always defined by fixed quantifiable conditions. The importance of the scientific method is its emphasis on clear, logical, and focused thinking that can analyze a specific situation in relation to the larger system that it is part of. The method is the quickest cure for frustration and desperation.

Mathematics is a reflection of the orderly analysis and conclusions of the scientific method. It is the language of science used to communicate ideas, theories and observations in a concise manner. From the scientific method, notice how the word, condition, appeared often. In mathematics conditions are referred to as dimensions. The following section will take a closer look at what a dimension represents.

Next section -> Section 3.2 - Nonsense Detection - The UnScientific Method

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