Question

Taylor polynomials Find the nth-order Taylor polynomial for the following functions centered at the given point a.ƒ(x) = cosh x, n = 3, a = ln 2

Accepted Answer

Recall the definition of the nth-order Taylor polynomial of a function $$ f(x) $$ centered at $$ a $$:

$$ T_n(x) = \sum_{k=0}^n \frac{f^{(k)}(a)}{k!} (x - a)^k $$

where $$ f^{(k)}(a)  is $$the $$ k -th $$derivative of $$ f $$ evaluated at $$ a . $$Identify the function and the center: here, $$ f(x) = \cosh x $$, $$ n = 3 $$, and $$ a = \ln 2 . We $$need to find the derivatives of $$ \cosh x  up to $$order 3 and evaluate them at $$ x = \ln 2 . $$Compute the derivatives:

- $$ f(x) = \cosh x $$
- $$ f'(x) = \sinh x $$
- $$ f''(x) = \cosh x $$
- $$ f^{(3)}(x) = \sinh x $$

Then evaluate each at $$ x = \ln 2 $$:

$$ f(a), f'(a), f''(a), f^{(3)}(a) . $$Substitute the evaluated derivatives into the Taylor polynomial formula:

$$ T_3(x) = f(a) + \frac{f'(a)}{1!}(x - a) + \frac{f''(a)}{2!}(x - a)^2 + \frac{f^{(3)}(a)}{3!}(x - a)^3 $$

This gives the explicit polynomial up to degree 3 centered at $$ a = \ln 2 . $$Simplify the expression if desired by calculating factorials and writing the polynomial in standard form. This completes the construction of the 3rd-order Taylor polynomial for $$ \cosh x $$ centered at $$ \ln 2 .$$

Taylor polynomials Find the nth-order Taylor polynomial for the following functions centered at the given point a.
ƒ(x) = cosh x, n = 3, a = ln 2

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Key Concepts

Taylor Polynomials

Derivatives of Hyperbolic Functions

Evaluating at the Center Point