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Magazine Chapter 7: Problem 2
Show that if \(f\) is a polynomial of degree \(n\), then it is its own Taylor polynomial of degree \(n\) with \(c=0\).
Short Answer
Expert verified
A polynomial of degree \(n\) is its own Taylor polynomial of degree \(n\) with \(c=0\), as evidenced by matching terms in both.
Step by step solution
01
Identify the Polynomial Expression
Let's consider a polynomial function \(f(x)\) of degree \(n\), which can be represented as \(f(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0\). Here \(a_n, a_{n-1}, \ldots, a_0\) are coefficients in the polynomial.
02
Define the Taylor Polynomial
The Taylor polynomial of degree \(n\) for a function \(f(x)\) at \(c=0\) is given by: \(P_n(x) = f(0) + \frac{f'(0)}{1!}x + \frac{f''(0)}{2!}x^2 + \cdots + \frac{f^{(n)}(0)}{n!}x^n\).
03
Calculate the Derivatives
Since \(f(x)\) is a polynomial, its derivatives \(f'(x)\), \(f''(x)\), etc., will each reduce the degree by one until zero is reached. For any \(n\)-degree polynomial, \(f^{(k)}(x)\) for \(k > n\) will be zero. The derivatives needed are up to \(f^{(n)}(x)\).
04
Evaluate Derivatives at \(c=0\)
Calculate each derivative of \(f(x)\) and evaluate them at \(x=0\). The \(k\)-th derivative evaluated at \(x=0\) will be the coefficient of \(x^k\) in the polynomial times \(k!\), i.e., \(f^{(k)}(0) = a_kk!\) if \(k\leq n\), otherwise zero.
05
Form the Taylor Polynomial
Substitute these evaluated derivatives back into the Taylor polynomial formula: \(P_n(x) = a_0 + \frac{a_1x}{1!} + \frac{a_2x^2}{2!} + \cdots + \frac{a_nx^n}{n!}\). Simplifying this, we observe each term becomes \(a_kx^k\), thus \(P_n(x) = f(x)\).
06
Conclude Polynomial as its Taylor Polynomial
The Taylor polynomial \(P_n(x)\) constructed is identical to the original polynomial \(f(x)\). Thus, the polynomial is its own Taylor polynomial of degree \(n\) at \(c=0\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Polynomial Function
A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. These functions are known for their smooth and continuous curves. Consider a polynomial function of degree \(n\) expressed as \(f(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0\).
Here are the key points to understand about polynomial functions:
Here are the key points to understand about polynomial functions:
- The highest power of the variable \(x\), which is \(n\) in this case, determines the degree of the polynomial.
- The coefficients \(a_n, a_{n-1}, \ldots, a_0\) are fixed numbers. They can be any real number, and they define the specific shape and position of the polynomial's graph.
- These coefficients provide insight into how each term in the polynomial behaves as \(x\) changes.
Degree of Polynomial
The degree of a polynomial is a vital concept that indicates the polynomial's complexity and behavior. It is defined as the largest exponent of the variable \(x\) with a non-zero coefficient.
In the polynomial \(f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0\), the degree is \(n\) if \(a_neq 0\).
Let's explore why the degree is important:
In the polynomial \(f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0\), the degree is \(n\) if \(a_neq 0\).
Let's explore why the degree is important:
- The degree tells us the maximum number of roots (or zeros) the polynomial can have.
- It gives the maximum flexibility or turns the polynomial graph can take.
- The end behavior of the polynomial function, or how it behaves as \(x\) moves towards positive or negative infinity, is mainly determined by the leading term \(a_nx^n\).
Derivatives
Derivatives are critical in calculus for understanding how functions change. For a polynomial function like \(f(x)\), its derivatives describe the rate of change of the function.
Here's how derivatives apply to polynomials:
Here's how derivatives apply to polynomials:
- The first derivative \(f'(x)\) gives the slope or rate of change at any point \(x\).
- Successive derivatives \(f''(x), f'''(x), \ldots\) continue to describe the changes in slope, each reducing the degree of the previous derivative by one.
- For a polynomial of degree \(n\), derivatives beyond \(f^{(n)}(x)\) become zero, as the function eventually becomes constant.
Coefficient Evaluation
Coefficient evaluation is the process of calculating the coefficients in a polynomial or series, such as the Taylor polynomial.
To evaluate coefficients:
To evaluate coefficients:
- Derivatives of the function are calculated at a specific point, usually the center \(c = 0\) for Taylor polynomials.
- These derivatives at \(c\) become the coefficients in the Taylor polynomial formula \(P_n(x)\).
- For instance, the \(k\)-th derivative evaluated at \(c=0\) is multiplied by the factorial of \(k\) as \(f^{(k)}(0) = a_kk!\).
