Quadratic approximations


[Technology Exercise] c. Graph f(x) = 1/(1 − x) and its quadratic approximation at x = 0. Then zoom in on the two graphs at the point (0,1). Comment on what you see.

Verified step by step guidance

1. Identify the function f(x) = \(\frac{1}{1-x}\) and recognize that we need to find its quadratic approximation at x = 0.
2. Recall that the quadratic approximation of a function f(x) at a point a is given by the formula: \(f(a) + f'(a)(x-a) + \frac{f''(a)}{2}(x-a)^2\).
3. Calculate the first derivative f'(x) of the function f(x) = \(\frac{1}{1-x}\) using the chain rule. The derivative is f'(x) = \(\frac{1}{(1-x)^2}\). Evaluate this at x = 0 to find f'(0).
4. Calculate the second derivative f''(x) of the function. Differentiate f'(x) = \(\frac{1}{(1-x)^2}\) again to get f''(x) = \(\frac{2}{(1-x)^3}\). Evaluate this at x = 0 to find f''(0).
5. Substitute f(0), f'(0), and f''(0) into the quadratic approximation formula to get the quadratic approximation of f(x) at x = 0. Graph both f(x) and its quadratic approximation, then zoom in on the point (0,1) to observe how closely the approximation matches the original function near this point.

Key Concepts

Quadratic Approximation

Quadratic approximation is a method used to approximate a function near a point using a quadratic polynomial. It is derived from the Taylor series expansion, where the function is approximated by a polynomial of degree two. This approximation provides a more accurate representation of the function near the point of interest compared to a linear approximation.

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) at x = a, the Taylor series is 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 using terms up to the second derivative.

Graphical Analysis

Graphical analysis involves visually examining the graphs of functions to understand their behavior and relationships. By graphing f(x) = 1/(1-x) and its quadratic approximation, one can observe how closely the approximation matches the original function near x = 0. Zooming in on the graphs at (0,1) helps to see the accuracy and limitations of the approximation in that region.