Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

c. Only even powers of x appear in the nth−order Taylor polynomial for f(x)=√(1+x²) centered at 0.

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Recall that the nth-order Taylor polynomial of a function \(f(x)\) centered at 0 (Maclaurin polynomial) is given by: \[T_n(x) = \sum_{k=0}^n \frac{f^{(k)}(0)}{k!} x^k,\] where \(f^{(k)}(0)\) is the \(k\)th derivative of \(f\) evaluated at 0.

Consider the function \(f(x) = \sqrt{1 + x^2}\). Notice that \(f(x)\) is an even function because \(f(-x) = \sqrt{1 + (-x)^2} = \sqrt{1 + x^2} = f(x)\).

Since \(f(x)\) is even, its Taylor series expansion around 0 will contain only even powers of \(x\). This is a general property: the Taylor series of an even function centered at 0 contains only even powers, and the Taylor series of an odd function contains only odd powers.

To confirm this, you can compute the first few derivatives of \(f(x)\) at 0 and observe that all derivatives of odd order vanish at 0, i.e., \(f^{(1)}(0) = 0\), \(f^{(3)}(0) = 0\), etc., which means the coefficients of odd powers are zero.

Therefore, the nth-order Taylor polynomial for \(f(x) = \sqrt{1 + x^2}\) centered at 0 contains only even powers of \(x\), making the statement true.

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Taylor Polynomial

A Taylor polynomial approximates a function near a specific point using derivatives at that point. The nth-order Taylor polynomial includes terms up to the nth derivative, expressed as powers of (x - a), where a is the center. It provides a polynomial approximation that matches the function's behavior locally.

Even and Odd Functions

An even function satisfies f(-x) = f(x), resulting in Taylor expansions with only even powers of x. An odd function satisfies f(-x) = -f(x), leading to only odd powers in its expansion. Understanding the symmetry of a function helps predict which powers appear in its Taylor series.

Derivatives of Composite Functions

Calculating Taylor polynomials often requires finding derivatives of composite functions like f(x) = √(1 + x²). Using the chain rule and recognizing patterns in derivatives helps determine which terms vanish or remain, influencing the presence of even or odd powers in the polynomial.

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.c. Only even powers of x appear in the nth−order Taylor polynomial for f(x)=√(1+x²) centered at 0.