Taylor's theorem

Taylor's theorem provides a way to approximate a smooth function near a specific point using a polynomial. This polynomial is built from the function's values and derivatives at that point, effectively capturing the function's behavior locally. In essence, it allows us to estimate the function's value close to a known point by summing terms that include the function itself, its slope, curvature, and higher-level changes. This is valuable because polynomials are easier to work with, and the approximation becomes more accurate as we include more terms.