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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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 \( \ln 2 \).

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주요 개념

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Taylor Polynomials

A Taylor polynomial approximates a function near a point a by using the function's derivatives at that point. The nth-order Taylor polynomial includes terms up to the nth derivative, providing a polynomial that closely matches the function's behavior near a.

Derivatives of Hyperbolic Functions

Understanding the derivatives of hyperbolic functions like cosh(x) is essential, as these derivatives are used to construct the Taylor polynomial. For example, the derivative of cosh(x) is sinh(x), and these derivatives follow a predictable pattern.

Evaluating at the Center Point

To build the Taylor polynomial centered at a specific point a, you must evaluate the function and its derivatives at x = a. This step ensures the polynomial matches the function's value and slope behavior exactly at that point.

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