Linear approximation and differentials from "summary" of Differential Calculus by S Balachandra Rao
Linear approximation and differentials are powerful tools in the study of calculus. They allow us to approximate the value of a function at a certain point by using the tangent line to that function at that point. This tangent line serves as an approximation to the function near the point of interest. To understand linear approximation better, let's consider a function f(x) and a point x=a. The tangent line to f(x) at x=a is given by the equation of the form y=f(a)+f'(a)(x-a). Here, f'(a) represents the derivative of the function f(x) evaluated at x=a. This equation represents the linear approximation to the function f(x) near the point x=a. Differentials, on the other hand, are closely related to linear approximation. The differential dy of a function y=f(x) is defined as dy=f'(x)dx. This means that the change in y (dy) can be approximated by the derivative of the function f(x) at x multiplied by the change in x (dx). By using differentials, we can approximate the change in the value of a function when the input changes by a small amount. This is useful in various applications, such as in physics, engineering, and economics, where small changes in variables can have significant effects on the overall system.- Linear approximation and differentials provide us with a way to simplify complex functions and understand their behavior near specific points. By approximating functions with simpler linear models, we can gain insights into the functions' properties and make predictions about their behavior without the need for complex calculations. These tools are essential in the study of calculus and play a crucial role in many real-world applications.
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