Approximating Functions with Polynomials and Mean Value Theorems

From the geometric interpretation of the derivative, the derivative is equal to the slope of the tangent line to the graph of a function f at a point x 0 $$x_0$$ . In other words, at that point the function can be approximated by a tangent line, so the derivative shows by how much the functional value changes if the value of the independent variable x increases from its current value x 0 $$x_0$$ by 1, where instead of the actual change in the value of the function, we observe the change in the value on the tangent line. In the same manner, we may wonder by how much the functional value y = f ( x ) $$y=f(x)$$ will approximately change if we increase x from its current value x 0 $$x_0$$ by some arbitrarily small amount Δ x $$\Delta {x}$$ . We describe this change by introducing the term differential. Moreover, we show how to approximate functions of one variable by polynomials. We start from linear approximations and arrive to Taylor’s formula and the Taylor polynomial of degree n. Then we present the Bolzano-Weierstrass theorem. The importance of this theorem stems from the fact that it guarantees the existence of an equilibrium state or optimum for many economic models. We conclude by presenting two very important mean value theorems: Rolle’s theorem and Lagrange’s mean value theorem.