Finding Taylor Polynomials
In Exercises 1–10, find the Taylor polynomials of orders 0, 1, 2, and 3 generated by f at a.
f(x) = √(1 − x),a = 0

Verified step by step guidance

1. Identify the function and the point about which the Taylor polynomial is generated: here, \( f(x) = \sqrt{1 - x} \) and \( a = 0 \).
2. Calculate the derivatives of \( f(x) \) up to the third order: find \( f'(x) \), \( f''(x) \), and \( f'''(x) \).
3. Evaluate the function and its derivatives at \( x = 0 \): compute \( f(0) \), \( f'(0) \), \( f''(0) \), and \( f'''(0) \).
4. Write the Taylor polynomials of orders 0, 1, 2, and 3 using the formula: \[ T_n(x) = \sum_{k=0}^n \frac{f^{(k)}(a)}{k!} (x - a)^k \], substituting \( a = 0 \) and the evaluated derivatives.
5. Express each polynomial explicitly: the 0th order is just \( f(0) \), the 1st order adds the linear term, the 2nd order adds the quadratic term, and the 3rd order adds the cubic term, all centered at \( x = 0 \).

Key Concepts

Taylor Polynomials

Taylor polynomials approximate a function near a point a by using a finite sum of derivatives of the function at a. The nth-order Taylor polynomial includes terms up to the nth derivative, providing increasingly accurate approximations as n increases.

Derivatives of Functions

Derivatives measure the rate of change of a function. To find Taylor polynomials, you must compute the function's derivatives at the point a, as these derivatives determine the coefficients of the polynomial terms.

Evaluating at the Expansion Point

The Taylor polynomial is centered at a specific point a, where the function and its derivatives are evaluated. This evaluation ensures the polynomial matches the function's value and slope behavior at that point, making the approximation accurate near a.