Mathematics

Mathematics

Problem

Consider the function g(x)=1+x2g(x)=\sqrt{1+x^{2}}. A Taylor polynomial is to be constructed about the point x=1x=1. (a) Compute g(1)g(1), g′(1)g^{\prime}(1), and g′′(1)g^{\prime\prime}(1). Provide exact values, not decimal approximations. (b) Write the second–degree Taylor polynomial T2(x)T_2(x) for g(x)g(x) centered at x=1x=1. (c) Use T2(x)T_2(x) to approximate g(1.2)g(1.2). Give the result to five decimal places.
Intuitive explanation

The key is recognizing that Taylor polynomials capture local behavior through derivatives evaluated at a single point. Here, you'll compute g(1)g(1), g′(1)g'(1), g′′(1)g''(1), and so on, then build a polynomial centered at x=1x=1 using the formula g(x)≈g(1)+g′(1)(x−1)+g′′(1)2!(x−1)2+⋯g(x) \approx g(1) + g'(1)(x-1) + \frac{g''(1)}{2!}(x-1)^2 + \cdots. The challenge is finding those derivatives of 1+x2\sqrt{1+x^2} efficiently—chain rule and implicit differentiation are your friends.

Intuitive explanation

The key is recognizing that Taylor polynomials capture local behavior through derivatives evaluated at a single point. Here, you'll compute g(1)g(1), g′(1)g'(1), g′′(1)g''(1), and so on, then build a polynomial centered at x=1x=1 using the formula g(x)≈g(1)+g′(1)(x−1)+g′′(1)2!(x−1)2+⋯g(x) \approx g(1) + g'(1)(x-1) + \frac{g''(1)}{2!}(x-1)^2 + \cdots. The challenge is finding those derivatives of 1+x2\sqrt{1+x^2} efficiently—chain rule and implicit differentiation are your friends.

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