Problem 20 - Calculus

Question

Find the third-degree Taylor polynomial for \(f(x)=\) $$x^{3}+7 x^{2}-5 x+1\( about \)x=0 .$$ What do you notice?

Step-by-Step Solution

Answer

The Taylor polynomial is identical to the original function, as both are third-degree polynomials.

Step 1: Understand Taylor Polynomial Formula

The third-degree Taylor polynomial for a function \( f(x) \) centered at \( x = a \) is given by \( T_3(x) = f(a) + f'(a)x + \frac{f''(a)x^2}{2!} + \frac{f'''(a)x^3}{3!} \). Since we are interested in \( x = 0 \), we substitute \( a = 0 \) into the formula.

Step 2: Compute Function and Derivatives at x = 0

First, calculate \( f(0) \), \( f'(0) \), \( f''(0) \), and \( f'''(0) \) for \( f(x) = x^3 + 7x^2 - 5x + 1 \). - First derivative: \( f'(x) = 3x^2 + 14x - 5 \), so \( f'(0) = -5 \).- Second derivative: \( f''(x) = 6x + 14 \), so \( f''(0) = 14 \).- Third derivative: \( f'''(x) = 6 \), so \( f'''(0) = 6 \).- Function at \( x=0 \): \( f(0) = 1 \).

Step 3: Apply Taylor Polynomial Formula

Now, apply the values into the Taylor polynomial formula:\[T_3(x) = f(0) + f'(0)x + \frac{f''(0)x^2}{2!} + \frac{f'''(0)x^3}{3!} = 1 - 5x + \frac{14x^2}{2} + \frac{6x^3}{6}.\]Simplify the expression to:\[T_3(x) = 1 - 5x + 7x^2 + x^3.\]

Step 4: Analyze the Result

The Taylor polynomial \( T_3(x) = 1 - 5x + 7x^2 + x^3 \) is exactly the same as the original polynomial \( f(x) \). This happens because \( f(x) \) is itself a third-degree polynomial, which means the Taylor polynomial of the same degree will match the original function entirely.

Key Concepts

third-degree polynomial

A third-degree polynomial is a mathematical expression of the highest degree term with exponent three. In simpler terms, it typically has the form:

• \( ax^3 + bx^2 + cx + d \)

where \( a, b, c, \) and \( d \) are constants. Polynomials are fundamental in algebra, especially when analyzing the behavior of functions. This specific polynomial has up to three real roots and can form various shapes when graphed. Often referred to as cubic polynomials, third-degree polynomials capture curves that feature bends or 'turns,' offering greater flexibility than linear or quadratic expressions. Understanding such polynomials is essential as they frequently appear in both theoretical and applied sciences.

derivatives

In calculus, derivatives represent the rate at which a function is changing at any given point. Think of them as the mathematical tool used for determining slopes of functions. For the function \( f(x) = x^3 + 7x^2 - 5x + 1 \), the derivative process involves

• First finding the first derivative: \( f'(x) = 3x^2 + 14x - 5 \).
• The second derivative: \( f''(x) = 6x + 14 \).
• The third derivative: \( f'''(x) = 6 \).

Derivatives play a crucial role in determining the Taylor polynomial, especially since they help predict how a function behaves close to a central point. Calculating derivatives is essential for finding those Taylor coefficients and formulating the polynomial itself.

polynomials

Polynomials are expressions in which variables are raised to whole-number exponents. They are often written in a standard form where the terms are ordered from the highest degree to the lowest. To illustrate, the polynomial expressed as \( x^3 + 7x^2 - 5x + 1 \) is already in its standard form.

• Each term in the polynomial (like \( x^3 \) or \( 7x^2 \)) is called a monomial.
• A polynomial can have one or more terms.

Understanding polynomials is critical throughout mathematics as they form the building blocks for more complex manipulations. They are crucial when calculating limits, derivatives, and integrals in calculus.

calculus

Calculus is a branch of mathematics focused on change and motion. It is split into two main branches: differential calculus and integral calculus. Derivatives are part of differential calculus and are integral in the development of Taylor polynomials. The exercise here centers around expanding a function using its Taylor series around a point, relying on its derivatives and coefficients.

• Differential calculus concerns calculating instantaneous rates of change (like slopes).
• Integral calculus focuses on finding areas under curves and accumulations.

Calculus is used widely in fields such as physics, engineering, and economics, making it essential for students to understand its basic principles and tools, like the Taylor polynomial in this problem.