Problem 44 - Calculus Single Variable

Question

Find a polynomial whose derivative the given polynomial. \(10 x^{6}+x^{2}+4 x-3\)

Step-by-Step Solution

Answer

The required polynomial is \(\frac{10}{7}x^7 + \frac{1}{3}x^3 + 2x^2 - 3x + C\).

Step 1: Understand the Problem

We need to find a polynomial whose derivative matches the given polynomial. The provided polynomial is \(10x^6 + x^2 + 4x - 3\).

Step 2: Determine the Antiderivative

We will find the antiderivative of each term separately:1. For \(10x^6\), the antiderivative is \(\frac{10}{7}x^7\), based on the power rule for integration \(x^n \to \frac{x^{n+1}}{n+1}\).2. For \(x^2\), the antiderivative is \(\frac{1}{3}x^3\).3. For \(4x\), the antiderivative is \(2x^2\).4. For the constant term \(-3\), the antiderivative is \(-3x\).

Step 3: Combine the Antiderivatives

Combine the antiderivatives obtained in Step 2 to form the polynomial:\[\frac{10}{7}x^7 + \frac{1}{3}x^3 + 2x^2 - 3x + C\]Here, \(C\) is the constant of integration.

Step 4: Verify by Differentiation

Differentiate the combined polynomial to ensure it matches the original polynomial:1. The derivative of \(\frac{10}{7}x^7\) is \(10x^6\).2. The derivative of \(\frac{1}{3}x^3\) is \(x^2\).3. The derivative of \(2x^2\) is \(4x\).4. The derivative of \(-3x\) is \(-3\).Combine these derivatives: \(10x^6 + x^2 + 4x - 3\), which matches the given polynomial.

Key Concepts

Polynomial Differentiation

Polynomial differentiation is a key concept in calculus, particularly when dealing with functions composed of variables raised to various powers. Differentiation helps determine the rate at which a function's value changes with respect to a variable. In the context of polynomials, this simply involves applying the power rule.

• The process involves decreasing the power of a variable by one and multiplying the term by the original power.
• This is repeated for each term in the polynomial.

Suppose you have a polynomial like: \[ 10x^6 + x^2 + 4x - 3 \]Differentiating this would yield:

• For \(10x^6\), applying differentiation gives \(10 \times 6x^{6-1} = 60x^5\).
• For \(x^2\), it results in \(2x^{2-1} = 2x\).
• For \(4x\), it simplifies to simply \(4\).
• For the constant \(-3\), which does not change, so the derivative is \(0\).

Altogether, this explains how each term in a polynomial reacts to differentiation.

Power Rule for Integration

The power rule for integration is a straightforward method to find the antiderivative of polynomials. Unlike differentiation, which breaks down a function, integration builds it up, identifying the original function from its rate of change.The power rule for integration states:

• For any term \(x^n\), the antiderivative is \(\frac{x^{n+1}}{n+1}\).
• If \(neq -1\), this rule holds true, transforming each term by adding one to the exponent and dividing by the new exponent.

Let's see how it applies to the polynomial given:

• For \(10x^6\), the antiderivative is \(\frac{10}{7}x^7\).
• For \(x^2\), it becomes \(\frac{1}{3}x^3\).
• For \(4x\), the result is \(2x^2\).
• The constant \(-3\), once integrated, becomes \(-3x\).

By finding these antiderivatives, we can reconstruct the original function with an added constant.

Constant of Integration

When finding antiderivatives, or integrating a function, a constant of integration \(C\) is always added to the result. This is because differentiation of a constant yields zero, and thus any constant could have existed in the original function before differentiation.The constant of integration helps account for:

• Any potential vertical shifts in the function.
• Diverse solutions that still satisfy the original differential equation.

For the polynomial \( 10x^6 + x^2 + 4x - 3 \), after integrating, we obtain \[\frac{10}{7}x^7 + \frac{1}{3}x^3 + 2x^2 - 3x + C\]Here, \(C\) can be any real number, representing an infinite family of functions that differentiate back to the given polynomial. Always remember to include the constant of integration in indefinite integrals to express all possible original functions.