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Exponential functions are encountered frequently in calculus, representing processes that grow or decay at constant relative rates. The standard exponential function, noted as \(e^x\), has a number of important properties. One of these properties is its representation as a power series:

• \(e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \ldots\)

Each additional term in this series becomes smaller, particularly when \(x\) is a small positive number in the interval \((0,1)\). This series helps establish inequalities involving \(e^x\). For instance, the inequality \(e^x < 1 + x\) holds true because starting from the second term, each new term is significantly smaller than the term \(x\). Therefore, summing them maintains the inequality \(e^x = 1 + x + \, \text{small terms} \, < 1 + 2x\). This sort of reasoning is very common in calculus and helps understand the behavior of exponential functions mathematically.

Logarithmic functions are the inverses of exponential functions and play a crucial role in calculus. A key feature of the natural logarithm, \(\log(1+x)\), is its power series expansion:

• \(\log(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots\)

In the interval \(x \in (0,1)\), this series shows that each subsequent term becomes smaller in magnitude. For small positive values of \(x\), the major term in this series approximation is close to \(x - \frac{x^2}{2}\), which demonstrates that \(\log(1+x) < x\). The terms \(- \frac{x^2}{2} + \frac{x^3}{3} - \cdots\) are smaller corrections that ensure this inequality holds true. Understanding logarithms through their series expansion helps in grasping how these functions approximate growth related phenomena inversely to exponential functions.

Trigonometric functions, like \(\sin x\), are foundational in calculus and geometry. These functions describe periodic relationships and their behavior can be explored through calculus. In particular, for \(x\) in the interval \((0,1)\), \(\sin x\) is known to be greater than \(x\). This is because \(\sin x\) is an increasing function on this interval. While \(\sin x\) oscillates between -1 and 1 over longer intervals, in small regions such as \(0 < x < 1\), it closely follows a positive trend. The derivative \(\cos x\), which is positive in this interval, indicates \(\sin x\) is increasing, thereby establishing the inequality. This property illustrates how calculus helps in understanding the dynamic and oscillating nature of trigonometric functions.