Question

Prove that the equation \(d^{m} x / d t^{n}=0\) is satisfied by all polynomials of degree less than \(n\), and only by these functions.

Short answer

Question: Prove that the equation \(d^n x / d t^n=0\) is satisfied by all polynomials of degree less than n, and only by these functions. Answer: The equation \(d^n x / d t^n=0\) is satisfied by all polynomials of degree less than n, as shown by the fact that the m-th derivative is zero for all polynomials of degree less than m. Moreover, the assumption of a non-polynomial function satisfying the equation leads to a contradiction, proving that the given equation can only be satisfied by polynomials of degree less than n.

Step-by-step solution

Write down the general form of a k-degree polynomial

The general form of a k-degree polynomial is: $$p(x) = a_0 + a_1 x + a_2 x^2 + ... + a_k x^k$$ where \(a_i\) are the coefficients of the polynomial and k is the degree of the polynomial.

Compute the m-th derivative of the k-degree polynomial

Using the power rule of derivatives, the m-th derivative of the polynomial will be: $$ \frac{d^m p(x)}{d x^m} = \sum_{i=m}^{k} \frac{a_i * i!}{(i-m)!} x^{i-m} $$

Set the m-th derivative equal to zero and analyze the result

Setting the m-th derivative equal to zero, we get: $$ \frac{d^m p(x)}{d x^m} = 0 = \sum_{i=m}^{k} \frac{a_i * i!}{(i-m)!} x^{i-m} $$ Now, this equation will be true for all x if and only if all the coefficients in the sum are zero. In other words, we must have: $$ a_i = 0 \quad \textrm{for} \quad i \geq m $$ This means that the highest degree with a non-zero coefficient in the polynomial is less than m. Since we're given that \(m = n\), we can conclude that the given equation \(d^n x / d t^n=0\) is satisfied by all polynomials of degree less than n.

Show that only these functions satisfy the given equation

To show that only polynomials of degree less than n satisfy the given equation, let us assume a function f(x) which is not a polynomial of degree less than n does satisfy the equation, i.e., $$ \frac{d^n f(x)}{d x^n} = 0 $$ Since f(x) is not a polynomial of degree less than n, taking the n-th derivative will not necessarily result in zero. This leads us to a contradiction, which means our assumption is incorrect. Therefore, the given equation can only be satisfied by polynomials of degree less than n. In conclusion, the equation \(d^{m} x / d t^{n}=0\) is satisfied by all polynomials of degree less than \(n\), and only by these functions.

Key concepts

Derivative

In calculus, a derivative represents how a function changes as its input changes. It's like finding the slope of a curve at any given point. The fundamental idea of derivatives is to capture the rate at which one quantity changes with respect to another.
You can think of derivatives as the infinitesimal change in a function concerning a tiny change in its input. For example, if you have a function \( f(x) \), its derivative \( f'(x) \) tells you the instantaneous rate of change of \( f \) with respect to \( x \).
One common way to write derivatives is using the Leibniz notation, such as \( \frac{d}{dx} \), which signifies the derivative of a function concerning \( x \).

• Derivatives play a crucial role in various fields, including physics, engineering, and economics.
• Understanding derivatives helps us analyze and approach problems involving changing quantities.

Polynomial

A polynomial is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. The form of a polynomial is usually written as \( p(x) = a_0 + a_1 x + a_2 x^2 + \ldots + a_k x^k \), where each \( a_i \) is a coefficient and \( x^k \) represents the variable raised to the power \( k \).
Polynomials are fundamental objects in algebra, and they appear in various forms throughout mathematics.
Here are some key points about polynomials:

• They can have constants, variables, and exponents.
• The exponents in a polynomial are whole numbers.
• Polynomials are smooth and continuous functions, which means they do not have breaks or sharp corners.

Polynomials are quite versatile and have many applications in scientific computations and modeling natural phenomena.

Degree of Polynomial

The degree of a polynomial is the highest power of the variable in that polynomial. Understanding this concept is crucial for solving many mathematical problems. If you look at a polynomial \( p(x) = a_0 + a_1 x + a_2 x^2 + \ldots + a_k x^k \), the degree of the polynomial is \( k \) as long as \( a_k eq 0 \).
The degree gives several insights into the polynomial's properties:

• It indicates the highest number of times you need to differentiate the polynomial before it becomes zero, which is crucial in the context of derivatives.
• It can tell you about the number of roots or solutions the polynomial might have.
• Polynomials of higher degree can have more complex shapes and behaviors when graphed.

In the given exercise, understanding that only polynomials of degree less than \( n \) will result in a zero \( n \)-th derivative is a key takeaway about the degree of a polynomial.

Power Rule of Derivatives

The power rule is a basic rule for finding the derivative of a function whose variable is raised to a power. If you have a function \( f(x) = x^n \), the power rule states that the derivative \( f'(x) \) is \( n \cdot x^{n-1} \).
Using this rule, you can quickly find the rate of change for simple polynomial functions. Here's how it works step-by-step:

• Multiply the current power by the coefficient.
• Subtract one from the power of the variable.
• Write the new term.

For example, the derivative of \( f(x) = 3x^4 \) is \( f'(x) = 12x^3 \). This rule is particularly useful because it simplifies the computation process and helps understand the differentiation of polynomials effectively. In the context of the exercise, the power rule is instrumental in demonstrating how the n-th derivative of a polynomial turns to zero once its degree is less than \( n \).