Problem 73 - Calculus Single Variable

Question

Prove that any polynomial \(p(x)\) of degree \(k\) satisfies $$\begin{aligned}p(x)=& p(0)+p^{\prime}(0) \cdot x+\frac{p^{\prime \prime}(0)}{2 !} \cdot x^{2} \\\&+\cdots+\frac{p^{(k)}(0)}{k !} \cdot x^{k}\end{aligned}$$.

Step-by-Step Solution

Answer

Any polynomial of degree \(k\) satisfies the Taylor series form centered at zero up to \(x^k\).

Step 1: Understand the Question

We need to prove the statement given. We are asked to show that a polynomial function can be expressed in terms of its derivatives evaluated at zero, a form similar to a Taylor series expansion centered at zero.

Step 2: Recall the Taylor Series Formula

Recall the definition of the Taylor series for a function. For a polynomial function centered at zero, the formula is:\[ p(x) = \sum_{n=0}^{k} \frac{p^{(n)}(0)}{n!} x^n \]Here, \(p^{(n)}(0)\) represents the nth derivative of \(p(x)\) evaluated at zero.

Step 3: Verify the Polynomial Degree

A polynomial \(p(x)\) of degree \(k\) can be expressed in its general form as:\[ p(x) = a_k x^k + a_{k-1} x^{k-1} + \, \ldots \, + a_1 x + a_0 \]Derivatives of this polynomial will become zero when the order of the derivative exceeds the degree \(k\), which aligns with the given finite sum up to \(x^k\).

Step 4: Show Each Term Correspondence

Express each term of the polynomial based on its derivatives:- The constant term \(a_0\) corresponds to \(p(0)\).- The first derivative, evaluated at 0, gives \(a_1\).- The second derivative, evaluated at 0, divided by 2!, gives \(a_2\). And so forth for higher order terms. Each matches exactly with the terms \(\frac{p^{(n)}(0)}{n!} x^n\) of the Taylor series.

Step 5: Apply the Taylor Series Explanation

Apply the explanation for Taylor series expansion of a function \(p(x)\) centered at zero confirms that the function equals its series:\[ p(x) = p(0) + p^{\prime}(0)x + \frac{p^{\prime\prime}(0)}{2!}x^2 + \cdots + \frac{p^{(k)}(0)}{k!}x^k \]This equation holds because higher order terms are zero due to the polynomial being of degree \(k\).

Key Concepts

Polynomial Degree

The degree of a polynomial is a fundamental concept in polynomial algebra. It refers to the highest power of the polynomial's variable with a non-zero coefficient. For example, in the polynomial \(p(x) = 4x^3 + 3x^2 + 2x + 1\), the highest power of \(x\) is 3, making it a polynomial of degree 3.

Understanding the degree is important because it dictates the behavior and limits of polynomials. When you perform operations such as addition, subtraction, and multiplication on polynomials, the degree plays a crucial role in determining the result. For example, the degree of the sum of two polynomials cannot exceed the highest degree among them.

• Polynomial degree is determined by the term with the highest exponent.
• It impacts the number of solutions a polynomial can have.
• It guides the formation of its corresponding Taylor Series when expanded.

In our given problem, knowing that the polynomial \(p(x)\) is of degree \(k\) helps us confirm that the Taylor Series expansion will only contain terms up to \(x^k\). This is because once we take a derivative order higher than \(k\), the result is zero, hence no higher-order terms are present.

Derivatives of Polynomials

Derivatives are a key tool in calculus, allowing us to determine the rate of change of a function. For polynomials, derivatives are particularly straightforward to calculate. Given a polynomial, you can find its first derivative by multiplying the coefficient of each term by its exponent and then reducing the exponent by one.

For example, the first derivative of \(p(x) = a_k x^k + a_{k-1} x^{k-1} + \, \ldots \, + a_1 x + a_0\) is \(p'(x) = k a_k x^{k-1} + (k-1)a_{k-1} x^{k-2} + \, \ldots \, + a_1\). This process continues similarly for higher-order derivatives. However, past the degree of the polynomial, derivatives of higher order become zero.

• First derivative indicates the slope or rate of change.
• Second derivative is often associated with acceleration or curvature.
• Higher-order derivatives for a polynomial of degree \(k\) become zero.

Knowing how derivatives behave is crucial in our exercise because the Taylor series expansion uses derivatives evaluated at a specific point (in this case, zero) to construct an equivalent series representation of the polynomial.

Taylor Series Expansion

The Taylor Series is a powerful tool in calculus for representing functions as infinite sums. For a polynomial function, the Taylor series perfectly captures the function using a finite sum due to the polynomial's limited degree.

In practice, to form a Taylor series centered at zero (often called a Maclaurin series), one needs to evaluate the derivatives of the function at that center point, usually zero. Each derivative contributes a term to the series, weighted by the factorial of the order of the derivative. For instance, the nth term in the series is given by \(\frac{p^{(n)}(0)}{n!} x^n\).

• Offers an approach to approximate complicated functions.
• Centering at zero provides a Maclaurin series.
• Reflects polynomials exactly up to their degree.

In the exercise, this concept is critical because it shows that any polynomial can be perfectly represented by its Taylor series up to the degree \(k\), thereby proving the given equation and illustrating finite sum capabilities of the Taylor series for polynomials.

Function Centered at Zero

Centering a function at zero, frequently used in Maclaurin series, simplifies the analysis of the function by using the axis origin as a reference point. When we say a function is centered at zero, we mean that we're evaluating the function's derivatives at \(x = 0\).

For many functions, especially polynomials, this can lead to simpler calculations and cleaner expressions because higher order derivatives after a polynomial's degree vanish. Essentially, a Taylor series centered at zero provides a straightforward way to express and analyze the polynomial using its coefficients directly correlated with its derivatives evaluated at zero.

• Simplifies calculations and expressions.
• Directly connects to the function's coefficients when derivatives are evaluated at zero.
• Provides understanding of the function's behavior near the origin.

In this exercise, "function centered at zero" means we are looking at how each term of the polynomial \(p(x)\) is built from its derivatives at zero, creating the series representation that comprehensively reflects the polynomial itself.