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.

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Identify the function f(x) = \(\frac{1}{1-x}\) and recognize that we need to find its quadratic approximation at x = 0.

Recall that the quadratic approximation of a function f(x) at a point a is given by the formula: \(f(a) + f'(a)(x-a) + \frac{f''(a)}{2}(x-a)^2\).

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).

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).

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.

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주요 개념

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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.

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.