1. taylor approximations Taylor’s theorem shows that any function can be approximated in the vicinity of any convenient point by a series of terms involving the function and its derivatives. Here we...

1. taylor approximations Taylor’s theorem shows that any function can be approximated in the vicinity of any convenient point by a series of terms involving the function and its derivatives. Here we look at some applications of the theorem for functions of one and two variables.

a. Any continuous and differentiable function of a single variable,
f
(x), can be approximated near the point a by the formula
f
(x) =
f
(a) +
f
′(a) (x
−
a) + 0.5f
″(a) (x
−
a)2 + terms in
f
‴, f
⁗,…
Using only the first three of these terms results in a
quadratic
Taylor approximation. Use this approximation together with the definition of concavity given in Equation 2.85 to show that any concave function must lie on or below the tangent to the function at point a.

b. The quadratic Taylor approximation for any function of two variables,
f
(x,
y), near the point (a,
b) is given by
f
(x, y) =
f
(a, b) +
f1(a, b) (x
−
a) +
f2(a, b) (y
−
b) + 0.5[
f11(a, b) (x
−
a)2 + 2f12(a, b) (x
−
a) (y
−
b) +
f22(y
−
b)2] Use this approximation to show that any concave function (as defined by Equation 2.98) must lie on or below its tangent plane at (a,
b).