Use a Symbolic Differentiation Utility to Find the Taylor Polynomials $f ( x ) = \frac { 1 } { 1 + x ^ { 2 } }$

question 4

Multiple Choice

Use a symbolic differentiation utility to find the Taylor polynomials (centred at zero) of degrees (a) 2, (b) 4, (c) 6, (d) 8. $f ( x ) = \frac { 1 } { 1 + x ^ { 2 } }$

• A) \(S _ { 2 } ( x ) = 1 - x ^ { 2 } , S _ { 4 } ( x ) = 1 - x ^ { 2 } + x ^ { 4 } , S _ { 6 } ( x ) = 1 - x ^ { 2 } + x ^ { 4 } - x ^ { 6 } , S _ { 8 } ( x ) = 1 - x ^ { 2 } + x ^ { 4 } - x ^ { 6 } + x ^ { 8 }\)
• B) \(S _ { 2 } ( x ) = 1 + x ^ { 2 } , S _ { 4 } ( x ) = 1 + x ^ { 2 } + x ^ { 4 } , S _ { 6 } ( x ) = 1 + x ^ { 2 } + x ^ { 4 } + x ^ { 6 } , S _ { 8 } ( x ) = 1 + x ^ { 2 } + x ^ { 4 } + x ^ { 6 } + x ^ { 8 }\)
• C) \(S _ { 2 } ( x ) = 1 - x ^ { 2 } , S _ { 4 } ( x ) = 1 - x ^ { 2 } - x ^ { 4 } , S _ { 6 } ( x ) = 1 - x ^ { 2 } - x ^ { 4 } - x ^ { 6 } , S _ { 8 } ( x ) = 1 - x ^ { 2 } - x ^ { 4 } - x ^ { 6 } - x ^ { 8 }\)
• D) \(S _ { 2 } ( x ) = 1 + x ^ { 2 } , S _ { 4 } ( x ) = 1 + x ^ { 2 } - x ^ { 4 } , S _ { 6 } ( x ) = 1 + x ^ { 2 } - x ^ { 4 } + x ^ { 6 } , S _ { 8 } ( x ) = 1 + x ^ { 2 } - x ^ { 4 } + x ^ { 6 } - x ^ { 8 }\)
• E) \(S _ { 2 } ( x ) = 1 - x ^ { 2 } , S _ { 4 } ( x ) = 1 - x ^ { 2 } + x ^ { 4 } , S _ { 6 } ( x ) = 1 - x ^ { 2 } + x ^ { 4 } + x ^ { 6 } , S _ { 8 } ( x ) = 1 - x ^ { 2 } + x ^ { 4 } + x ^ { 6 } + x ^ { 8 }\)

---

Definitions:

Emotional Act

Any action driven or significantly influenced by one's emotions rather than by logical reasoning.

Physical Act

An action or activity involving bodily movement, typically requiring effort and energy.

Proximodistal

A principle of development where growth starts from the center of the body and moves towards the extremities.

General-to-specific

A form of logical reasoning that begins with a general observation or set of observations and moves toward a more detailed and specific conclusion.