Taylor polynomials Find the nth-order Taylor polynomial for the following functions centered at the given point a.
ƒ(x) = sinh (-3x), n = 3, a = 0

Verified step by step guidance

1. 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 \( a \).
2. Identify the function and the center: here, \( f(x) = \sinh(-3x) \), \( n = 3 \), and \( a = 0 \). We will need to find \( f(0) \), \( f'(0) \), \( f''(0) \), and \( f^{(3)}(0) \).
3. Compute the derivatives of \( f(x) = \sinh(-3x) \) step-by-step: - First derivative: \( f'(x) = \frac{d}{dx} \sinh(-3x) \) - Second derivative: \( f''(x) = \frac{d}{dx} f'(x) \) - Third derivative: \( f^{(3)}(x) = \frac{d}{dx} f''(x) \) Remember to apply the chain rule carefully since the argument is \( -3x \).
4. Evaluate each derivative at \( x = 0 \): Calculate \( f(0) \), \( f'(0) \), \( f''(0) \), and \( f^{(3)}(0) \) by substituting \( x = 0 \) into each derivative expression.
5. Construct the Taylor polynomial of order 3 using the formula: \[ T_3(x) = f(0) + \frac{f'(0)}{1!} x + \frac{f''(0)}{2!} x^2 + \frac{f^{(3)}(0)}{3!} x^3 \] This polynomial approximates \( f(x) \) near \( x = 0 \).

Key Concepts

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.

Hyperbolic Sine Function (sinh)

The hyperbolic sine function, sinh(x), is defined as (e^x - e^(-x))/2. It is an odd function with derivatives that cycle between sinh and cosh, which is important when computing derivatives for the Taylor polynomial.

Evaluating Derivatives at the Center Point

To construct the Taylor polynomial centered at a, you must compute the function's derivatives at x = a. These values determine the coefficients of the polynomial terms, making accurate evaluation at the center essential.