13.
Explanation
Approximation of a function by a Taylor series involves representing a function as an infinite sum of terms where each term is derived from the function's derivatives evaluated at a specific point (usually the center of expansion). The Taylor series provides a local approximation of the function near the center point, and it can be used to estimate the function's values for points close to the center.
The Taylor polynomial of degree 2 for a function f(x) centered at x=0 is also known as the second-degree Taylor polynomial or the Taylor series truncated at the second term. It can be expressed as:
2T2(x)=f(0)+f′(0)⋅x+2!f′′(0)⋅x2
Here, f(0) is the value of the function at the center point, f′(0) is the first derivative of the function at the center point, and f′′(0) is the second derivative of the function at the center point. The 2!2! in the denominator is just the factorial of 2.
Let's find the Taylor polynomial of degree 2 for the function f(x)=2+x1 centered at x=0:
-
Calculate the derivatives of f(x):
-
f(x)=2+x1
-
2f′(x)=(2+x)2−1
-
3f′′(x)=(2+x)32
-
-
Evaluate these derivatives at x=0:
-
if(0)=2+01=21
-
1f′(0)=(2+0)2−1=−1
-
2f′′(0)=(2+0)32=2
-
-
Plug these values into the Taylor polynomial formula:
2T2(x)=21−x+2!2⋅x2
Simplify the terms:
2T2(x)=21−x+x2
So, the Taylor polynomial of degree 2 for
f(x)=2+x1 centered at x=0 is
T2(x)=21−x+x2. This polynomial provides a good local approximation of the function near x=0.