Find the linear approximation at x=0 to show that the...

Find the linear approximation at $x=0$ to show that the following commonly used approximations are valid for "small" $x .$ Compare the approximate and exact values for $x=0.01, x=0.1$ and $x=1$
$$\tan x \approx x$$

Key Concepts

Taylor Series Expansion

The Taylor series expansion is a way to represent a function as an infinite sum of terms calculated from the values of its derivatives at a single point. The linear approximation is essentially the first-order Taylor polynomial, capturing the function’s behavior near the expansion point and providing insight into the accuracy of the approximation.

Small Angle Approximation

The small angle approximation is a specific application of linear approximation used in trigonometry. It states that for angles close to zero (measured in radians), functions like sine, cosine, and tangent can be approximated by simpler expressions. For tangent, tan(x) is approximately equal to x when x is small, using the fact that higher order terms in its Taylor series have minimal impact near zero.

Linear Approximation

Linear approximation, or linearization, is a method in calculus where a function is approximated by a line (its tangent line) near a specific point. This technique uses the function’s value and slope at that point to predict values nearby, simplifying complex functions to manageable, linear forms for small changes in the input.

Derivative

The derivative of a function at a given point represents the slope of the tangent line at that point. In linear approximation, the derivative is crucial because it provides the rate of change of the function, which defines how the linearization behaves close to the point of interest.

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