Matching functions with polynomials Match functions a–f with Taylor polynomials A–F (all centered at 0). Give reasons for your choices.


d. 1/(1 + 2x)


A. p₂(x)= 1 + 2x + 2x²
B. p₂(x) = 1 − 6x + 24x²
C. p₂(x) = 1 + x − x²/2
D. p₂(x) = 1 − 2x + 4x²
E. p₂(x) = 1 − x + (3/2)x²
F. p₂(x) = 1 − 2x + 2x²

Verified step by step guidance

1. Recall that the Taylor polynomial of a function \(f(x)\) centered at 0 (Maclaurin polynomial) up to degree 2 is given by: \[f(0) + f'(0)x + \frac{f''(0)}{2}x^2\]
2. Identify the function given: \[f(x) = \frac{1}{1 + 2x}\]
3. Calculate the value of the function at 0: \[f(0) = \frac{1}{1 + 0} = 1\]
4. Find the first derivative of \(f(x)\): \[f'(x) = -\frac{2}{(1 + 2x)^2}\] Then evaluate at 0: \[f'(0) = -2\]
5. Find the second derivative of \(f(x)\): \[f''(x) = \frac{8}{(1 + 2x)^3}\] Then evaluate at 0: \[f''(0) = 8\] Use these to write the Taylor polynomial: \[p_2(x) = 1 - 2x + \frac{8}{2}x^2 = 1 - 2x + 4x^2\] Match this with the given polynomials to find the correct one.

Key Concepts

Taylor Polynomials

Taylor polynomials approximate functions near a specific point (here, 0) using derivatives. The nth-degree Taylor polynomial uses derivatives up to order n to create a polynomial that matches the function's value and slope behavior at that point, providing a local approximation.

Derivatives and Their Role in Taylor Series

Derivatives of a function at the center point determine the coefficients of the Taylor polynomial. The first derivative gives the linear term coefficient, the second derivative gives the quadratic term coefficient (divided by 2!), and so on, reflecting the function's curvature and rate of change.

Geometric Series and Rational Functions

Functions like 1/(1 + 2x) can be expressed as a geometric series when |2x| < 1, allowing expansion into a power series. Recognizing this helps find the Taylor polynomial by truncating the series to the desired degree, linking rational functions to polynomial approximations.