Problem 46 - Calculus Single Variable

Question

Find a function whose derivative is the given function. \(3 / x^{3}\)

Step-by-Step Solution

Answer

The function is \( f(x) = -\frac{3}{2x^2} + C \), where \( C \) is a constant.

Step 1: Understanding the Problem

We need to find a function, say \( f(x) \), such that when we take its derivative \( f'(x) \), we get the given function \( \frac{3}{x^3} \).

Step 2: Rewrite the Function for Integration

Rewrite the given function \( \frac{3}{x^3} \) using a negative exponent for simpler integration. This becomes \( 3x^{-3} \).

Step 3: Applying the Power Rule for Integration

Using the power rule for integration, integrate \( 3x^{-3} \). The power rule states that \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \) for any real number \( n eq -1 \).

Step 4: Integrate the Function

Integrate \( 3x^{-3} \) using the power rule: \[\int 3x^{-3} \, dx = 3 \cdot \frac{x^{-3+1}}{-3+1} + C = 3 \cdot \frac{x^{-2}}{-2} + C = -\frac{3}{2}x^{-2} + C\]Thus, the antiderivative is \( -\frac{3}{2}x^{-2} + C \).

Step 5: Converting Back to Original Form

Express the solution back in terms of positive exponents: \[ -\frac{3}{2x^2} + C \] This is the function \( f(x) \) whose derivative is \( \frac{3}{x^3} \).

Key Concepts

Understanding Integration

Integration is a fundamental concept in calculus that is essentially the reverse process of differentiation. When we integrate a function, we are looking to find a function whose derivative matches the given function.

Here are some key points on integration:

• It is used to calculate areas under curves, among many other applications.
• Integrating a function provides us with the "antiderivative," which includes a constant of integration, generally denoted as " + C".
• When integrating, we often use rules or formulas similar to how we use differentiation rules.

For example, in our problem, we want to find an antiderivative of the function \(\frac{3}{x^3}\). By integrating, we convert it into a new function whose derivative will give us back the original \(\frac{3}{x^3}\).

The Power Rule for Integration

The power rule for integration is a crucial tool that makes integrating terms like polynomials especially straightforward. It’s a formula that helps in finding antiderivatives of expressions where the variable is raised to a power.

Here's how it works:

• The formula is \(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\), where \(n\) is any real number except \(-1\).
• This rule is an extension of the power rule for differentiation but works in reverse.
• After applying the rule, remember to add the constant of integration \(C\), since integration leaves us with a family of functions.

In our example, we used the power rule to integrate \(3x^{-3}\). By applying the rule: \[\int 3x^{-3} \, dx = 3 \cdot \frac{x^{-2}}{-2} + C = -\frac{3}{2}x^{-2} + C\]This helped us find the antiderivative needed to solve our original problem.

Working with Negative Exponents in Integration

Negative exponents often confuse students when it comes to integration. However, they can be quite manageable once you understand how they manipulate the base variable.

Key aspects of negative exponents include:

• A negative exponent indicates that the base of the power is a reciprocal—meaning \(x^{-n} = \frac{1}{x^n}\).
• Rewriting functions using negative exponents can simplify the integration process.

For example, starting with the expression \(\frac{3}{x^3}\), we rewrote it as \(3x^{-3}\). This form allows us to directly use the power rule without needing to worry about the denominator explicitly.

Once we integrate and arrive at \(-\frac{3}{2}x^{-2}\), you can convert it back to a more familiar form using positive exponents: \(-\frac{3}{2x^2}\). Remember that consistency with negative and positive exponents can make integration more intuitive and less prone to mistakes.