Problem 49 - Calculus

Question

Give an example of: A function with a Taylor series whose third-degree term is zero.

Step-by-Step Solution

Answer

The function \( f(x) = x^4 \) has a Taylor series with a zero third-degree term at \( x=0 \).

Step 1: Introduction to Taylor Series

A Taylor series is an infinite sum of terms calculated from the values of a function's derivatives at a single point. It is expressed as: \[ f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \cdots \] where the coefficients involve the derivatives of the function \( f \) evaluated at the point \( a \).

Step 2: Identifying the Third-Degree Term

The third-degree term in the Taylor series expansion is \( \frac{f'''(a)}{3!}(x-a)^3 \). To make this term zero, we need \( f'''(a) = 0 \). This means the third derivative of the function at the point \( a \) must be zero.

Step 3: Choosing a Function

Let's consider the function \( f(x) = x^4 \). We know that the derivatives of \( f(x) \) are \( f'(x) = 4x^3 \), \( f''(x) = 12x^2 \), and \( f'''(x) = 24x \).

Step 4: Evaluating the Third Derivative

Evaluate \( f'''(x) \) at \( a = 0 \) to check if it's zero: \[ f'''(0) = 24 \times 0 = 0 \] Hence, the third-degree term in the Taylor series expanding around \( a = 0 \) is zero.

Step 5: Confirming the Taylor Series

The Taylor series expansion of \( f(x) = x^4 \) around \( x = 0 \) is: \( f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \cdots = 0 + 0 + 0 + 0 + \cdots + \frac{f^{(4)}(0)}{4!}x^4 + \cdots \), confirming the third-degree term is zero.

Key Concepts

Calculus

Calculus is a branch of mathematics that studies how things change. It provides tools to analyze continuous change and is foundational for understanding advanced mathematical concepts.
There are two major parts of calculus:

• Differential Calculus - This focuses on the concept of the derivative, which represents rates of change and slopes of curves.
• Integral Calculus - This deals with the concept of integration, which is about the accumulation of quantities and areas under curves.

Using calculus, we can model situations that involve change and predict future trends. It's essential for disciplines like physics, engineering, economics, and many more. Calculus serves as a foundation for understanding how functions behave and provides tools like Taylor series to approximate them.

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable in the expression that has a non-zero coefficient. It tells us a lot about the behavior of the polynomial function, especially as the input variable grows larger or smaller.

• A polynomial such as \( a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0 \) has degree \( n \), where \( a_n eq 0 \).
• The degree determines the maximum number of roots the polynomial can have.
• It also establishes the end behavior of the function.

In the context of Taylor series, knowing the degree helps in understanding which derivatives will influence the terms of expansion, such as in our example where the third-degree term was zero.

Derivative

A derivative represents the rate at which a function changes at any point. It is a central concept in differential calculus, providing insights into the slope of a function and how changes in input affect the output.

• The first derivative \( f'(x) \) gives the slope of the tangent line to the graph at any point \( x \).
• The second derivative \( f''(x) \) indicates concavity or convexity of the function and can be used to find points of inflection.
• The third derivative \( f'''(x) \), as shown in the solution, affects the third-degree term of the Taylor expansion.

At specific points, derivatives can be precisely calculated to determine behavior and properties of a function for complex analyses like Taylor series expansions.

Function Expansion

Expanding a function into a series allows us to approximate it using simpler polynomial expressions. This is crucial when dealing with complex functions that are hard to analyze directly.

• The Taylor series is one method of function expansion. It uses derivatives at a point to express the function as an infinite sum of terms.
• The third-degree polynomial component in a Taylor series provides information about the function's behavior close to a point, up to cubic changes.
• By choosing points where certain derivatives are zero, we can simplify expansions, as shown in the example where the third-degree term is zero.

Function expansions, through methods like Taylor series, are invaluable for approximating functions in fields like engineering and physics, where precision and simplicity are crucial.