This concept proves that the quadratic approximation of a product of two functions equals the quadratic approximation obtained by multiplying the individual quadratic approximations of each function and truncating terms beyond degree two, within single-variable calculus. The proof proceeds by expanding both sides symbolically using the product rule (for first and second derivatives of a product) and showing the resulting coefficients of the constant, linear, and quadratic terms match exactly, confirming that no information is lost by using componentwise approximation rather than differentiating the product function directly. This situates the topic within local polynomial (Taylor) approximation theory, establishing a compositional property that justifies a more practical computational method.
This concept proves that the quadratic approximation of a product of two functions equals the quadratic approximation obtained by multiplying the individual quadratic approximations of each function …