91Ó°ÊÓ

Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This field constitutes a major part of modern mathematics education. It has two main branches: differential calculus concerning instantaneous rates of change and slopes of curves, and integral calculus concerning the accumulation of quantities and the areas under and between curves.

Linear approximation is a fundamental application of differential calculus. It allows us to approximate the value of a function near a certain point using the known value of the function and its derivative at that point. This technique is particularly useful when dealing with functions that are complex to evaluate precisely. The process involves finding the equation of the tangent line at a designated point, and then using this line to estimate values of the function near that point.

The natural logarithm, denoted as \(\ln(x)\), is the logarithm to the base \(e\), where \(e\) is an irrational and transcendental constant approximately equal to 2.71828. The natural logarithm is the inverse operation of taking \(e\) to the power of \(x\), expressed as \(e^x\).

In practical terms, \(\ln(x)\) measures the time needed to reach a certain level of continuous growth. It's commonly used in various fields such as mathematics, physics, and economics due to its unique properties, especially in dealing with growth processes and computing compound interest. In the given exercise, the approximation of \(\ln(1.5)\) using linear approximation leverages the simpler behavior of the log function near the point \(x=1\), thus facilitating easier computation.

Derivatives represent the rate at which a function is changing at any given point and are foundational in calculus. Formally, the derivative of a function at a certain point is the slope of the tangent line to the function's graph at that point. It gives us crucial information about the function's behavior, indicating whether it's increasing, decreasing, or at a turning point.

The derivative of the natural logarithm \(\ln(x)\) is \(\frac{1}{x}\), which implies that at \(x=1\), the slope of \(\ln(x)\) is 1. In the context of the problem, this property is used to find the linear approximation of \(\ln(x)\) around the point \(x=1\). The ability to calculate derivatives allows us to predict the function's value near that point (\(x=1.5\) in this case) without needing to work out the more complex original function at exactly that point.