Problem 42 - Calculus Single Variable

Question

Find a polynomial whose derivative the given polynomial. \(x^{9}-2 x^{3}-1\)

Step-by-Step Solution

Answer

The polynomial is \(\frac{x^{10}}{10} - \frac{x^4}{2} - x + C\).

Step 1: Understand the Problem

We need to find the original polynomial whose derivative is given as \(x^9 - 2x^3 - 1\). The process involves finding the antiderivative or integral of the given polynomial.

Step 2: Apply the Power Rule for Integration

For each term in the polynomial, apply the reverse of the power rule. The power rule for differentiation states \(\frac{d}{dx} \left( x^n \right) = nx^{n-1}\). The reverse operation for integration is \(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\), where \(C\) is a constant of integration.

Step 3: Integrate Each Term

Integrate each term separately: - For \(x^9\), the integral is \(\frac{x^{10}}{10}\).- For \(-2x^3\), the integral is \(-2 \times \frac{x^4}{4} = -\frac{x^4}{2}\).- For \(-1\), the integral is \(-x\).

Step 4: Combine the Integrated Terms

Combine the expressions we have derived:\[ F(x) = \frac{x^{10}}{10} - \frac{x^4}{2} - x + C \]where \(C\) is the constant of integration.

Key Concepts

Polynomial

A polynomial is a mathematical expression made up of variables, coefficients, and exponents. It consists of terms that are added together. Each term in a polynomial expression has a variable raised to a power and multiplied by a constant coefficient. Polynomials are fundamental in algebra, representing a wide class of functions.

In this exercise, the given polynomial is \(x^9 - 2x^3 - 1\). This expression has multiple terms, each with its own degree, which refers to the highest power of the variable within the term. The degree of the polynomial itself is defined by its highest exponent, which in this case is 9.

• \(x^9\): A term with variable \(x\) raised to the power of 9 and no coefficient, as it's implied to be 1.
• \(-2x^3\): A term with a negative coefficient and the variable \(x\) raised to the power of 3.
• \(-1\): A constant term, which is a polynomial without any variable.

Polynomials can be ranked by these degrees and their terms can be manipulated through various algebraic operations, including integration.

Antiderivative

The antiderivative, also known as the indefinite integral, is essentially the reverse process of differentiation. Finding an antiderivative means finding a function that, when differentiated, results in the given function.

When given a polynomial to integrate, like \(x^9 - 2x^3 - 1\), our goal is to find a function, say \(F(x)\), such that \(F'(x) = x^9 - 2x^3 - 1\). This involves determining the original function that had a derivative equal to the given polynomial.

• Essentially, the antiderivative process rewinds the effect of the derivative by increasing the power of each term by one and dividing by the new power, following the rules for integration.
• An indefinite integral is represented by an integral symbol \(\int\) and includes a constant of integration, \(C\), because differentiating a constant results in zero, making the original derivative unable to reflect its presence.

Understanding the concept of an antiderivative is crucial for anyone delving into calculus, as it forms the basis for solving many real-world problems involving rates of change.

Power Rule

The power rule for integration is one of the simplest yet most powerful tools in calculus. It is used to integrate polynomials efficiently. When we apply the power rule, we're performing the reverse operation of the power rule for differentiation.

The differentiation power rule states that the derivative of \(x^n\) is \(nx^{n-1}\). Conversely, the power rule for integration tells us: \[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \] Here, \(n\) is a real number, and \(C\) is the constant of integration.

• The rule simply involves increasing the exponent by 1 and then dividing by this new exponent.
• It's essential for integrating basic terms of the polynomial, making it straightforward to handle polynomial equations of any degree.

In the context of our exercise, applying the power rule allowed us to integrate each term of the polynomial \(x^9 - 2x^3 - 1\) independently.

Constant of Integration

The constant of integration, represented by \(C\), is vital when dealing with indefinite integrals. When you take an antiderivative, there are infinitely many possible solutions because integrating a derivative results in losing constant information.

This happened because the derivative of a constant term is zero, meaning several different functions could have led to the same derivative result.

• This constant is what we add after integrating a function, symbolizing the missing piece due to the integration process not accounting for the constant term.
• In every indefinite integral, we must include \(C\) to encompass all potential original functions leading to the given derivative.

For our exercise, once all terms of \(x^9 - 2x^3 - 1\) are integrated, the function we arrive at is \(F(x) = \frac{x^{10}}{10} - \frac{x^4}{2} - x + C\). Remember, the constant makes this expression an entire family of functions, reflecting its inherent flexibility.