Quadratic approximations


[Technology Exercise] e. Find the quadratic approximation to h(x) = √(1 + x) at x = 0. Graph h and its quadratic approximation together. Comment on what you see.

Verified step by step guidance

1. Identify the function for which you need the quadratic approximation: h(x) = √(1 + x).
2. Recall the formula for the quadratic approximation of a function f(x) at a point a: f(x) ≈ f(a) + f'(a)(x - a) + (f''(a)/2)(x - a)^2.
3. Calculate the first derivative of h(x) = √(1 + x) using the chain rule: h'(x) = (1/2)(1 + x)^(-1/2).
4. Evaluate the first derivative at x = 0: h'(0) = (1/2)(1 + 0)^(-1/2) = 1/2.
5. Calculate the second derivative of h(x): h''(x) = -(1/4)(1 + x)^(-3/2), and evaluate it at x = 0: h''(0) = -(1/4)(1 + 0)^(-3/2) = -1/4.

Key Concepts

Quadratic Approximation

Quadratic approximation is a method used to approximate a function using a second-degree polynomial. It is derived from the Taylor series expansion, where the function is approximated around a point, typically x = 0, using the function's value, first derivative, and second derivative at that point. This provides a more accurate approximation than a linear one, especially near the point of expansion.

Taylor Series

The Taylor series is an infinite sum of terms calculated from the values of a function's derivatives at a single point. For a function f(x), the Taylor series expansion around x = a is given by f(a) + f'(a)(x-a) + f''(a)(x-a)^2/2! + ..., which can be truncated to form polynomial approximations. The quadratic approximation is a specific case where the series is truncated after the second derivative term.

Graphing Functions

Graphing functions involves plotting the function's output values against its input values on a coordinate plane. This visual representation helps in understanding the behavior of the function and its approximations. When graphing h(x) = √(1 + x) and its quadratic approximation, one can observe how closely the approximation follows the actual function, particularly near the point of expansion, and where it starts to diverge.