Applications
Applications [ edit ] Linear approximations [ edit ] Main article: Linear approximation Linear approximations are used to replace complicated functions with linear functions that are almost the same. Given a differentiable function f ( x , y ) with real values, one can approximate f ( x , y ) for ( x , y ) close to ( a , b ) by the formula � ( � , � ) ≈ � ( � , � ) + ∂ � ∂ � ( � , � ) ( � − � ) + ∂ � ∂ � ( � , � ) ( � − � ) . The right-hand side is the equation of the plane tangent to the graph of z = f ( x , y ) at ( a , b ) . Optimization [ edit ] Main article: Mathematical optimization For a continuously differentiable function of several real variables , a point P (that is, a set of values for the input variables, which is viewed as a point in R n ) is critical if all of the partial derivatives ...