Quadratic approximations


a. Let Q(x) = b₀ + b₁(x − a) + b₂(x − a)² be a quadratic approximation to f(x) at x = a with these properties:


i. Q(a) = f(a)
ii. Q′(a) = f′(a)
iii. Q″(a) = f″(a).


Determine the coefficients b₀, b₁, and b₂.

Verified step by step guidance

1. Start by understanding that a quadratic approximation is a polynomial that approximates a function f(x) around a point x = a. The approximation is given by Q(x) = b₀ + b₁(x − a) + b₂(x − a)².
2. The first condition Q(a) = f(a) implies that when x = a, Q(x) should equal f(x). Substitute x = a into Q(x) to get Q(a) = b₀. Therefore, b₀ = f(a).
3. The second condition Q′(a) = f′(a) means the derivative of Q(x) at x = a should equal the derivative of f(x) at x = a. Compute Q′(x) = b₁ + 2b₂(x − a). Substitute x = a to get Q′(a) = b₁. Therefore, b₁ = f′(a).
4. The third condition Q″(a) = f″(a) requires the second derivative of Q(x) at x = a to equal the second derivative of f(x) at x = a. Compute Q″(x) = 2b₂. Substitute x = a to get Q″(a) = 2b₂. Therefore, b₂ = f″(a)/2.
5. Summarize the coefficients: b₀ = f(a), b₁ = f′(a), and b₂ = f″(a)/2. These coefficients ensure that the quadratic approximation Q(x) matches the function f(x) and its first and second derivatives at x = a.

Key Concepts

Quadratic Approximation

Quadratic approximation is a method used to estimate a function near a point using a quadratic polynomial. It involves matching the function's value, first derivative, and second derivative at a specific point, ensuring the polynomial closely resembles the function's behavior around that point. This technique is useful for simplifying complex functions into more manageable forms for analysis.

Taylor Series

The Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its derivatives at a single point. For quadratic approximation, the Taylor series is truncated after the second derivative term, providing a polynomial that approximates the function near the point of expansion. This concept is essential for understanding how the coefficients in the quadratic approximation are derived.

Derivative Matching

Derivative matching involves ensuring that the derivatives of the approximating polynomial match those of the original function at a specific point. In the context of quadratic approximation, this means setting the polynomial's value, first derivative, and second derivative equal to those of the function at the point of interest. This ensures the polynomial accurately reflects the function's local behavior, allowing for precise approximation.