Given the function ex on 01 How many intervals do we need to

Given the function e^x on [0,1]. How many intervals do we need to approximate e^x by piece-wise quadratic function with error of 10^(-5)?

Solution

The Quadratic Approximation for a function y = f(x) based at a point x₀ is given by

Q(x) =f(x₀) + f\'(x₀) (x - x₀) + 1/2 f\'\'(x₀) (x - x₀)²

and approximately equals f(x) for x near x₀.

Notice that Q(x) = L(x) + 1/2 f\'\'(x₀) (x - x₀)²

where L(x) = f(x₀) + f\'(x₀) (x - x₀)

is the linear approximation. See p. 212, Stewart 5^th Edition, for a discussion of the Quadratic Approximations of functions of 1 variable.

The Quadratic Approximation for a function of two variables z = f(x,y) based at (x₀, y₀) is given by

Q(x,y) = f(x_0, y₀) + f_x (x_0, y₀) (x - x₀) + f_y (x_0, y₀) (y - y₀) +

1/2 { f_xx (x_0, y₀) (x - x₀)² + 2 f_xy (x_0, y₀) (x - x₀) (y - y₀) + f_yy (x_0, y₀) (y - y₀)² }

= L(x,y) + 1/2 { f_xx (x_0, y₀) (x - x₀)² + 2 f_xy (x_0, y₀) (x - x₀) (y - y₀) + f_yy (x_0, y₀) (y - y₀)² }

where L(x,y) is the linear approximation