Problem 20 - Applied Calculus

Question

Find an antiderivative. $$f(q)=5 q^{2}$$

Step-by-Step Solution

Answer

The antiderivative of \( f(q) = 5q^2 \) is \( \frac{5q^3}{3} + C \).

Step 1: Understand the Function

We need to find an antiderivative of the function \( f(q) = 5q^2 \). An antiderivative, also known as an indefinite integral, is a function whose derivative gives us the original function.

Step 2: Apply the Power Rule for Integration

The power rule for integration states that for any function \( q^n \) where \( n eq -1 \), the antiderivative is \( \frac{q^{n+1}}{n+1} \). We apply this rule to \( 5q^2 \).

Step 3: Integrate the Function

First, identify \( n \) for our term, which is 2. Using the power rule, the antiderivative of \( q^2 \) is \( \frac{q^{3}}{3} \). We then multiply by the constant coefficient, 5, giving us \( \frac{5q^3}{3} \).

Step 4: Add the Constant of Integration

Since we are looking for the general antiderivative, we need to add the constant of integration, \( C \), giving us the final antiderivative: \( \frac{5q^3}{3} + C \).

Key Concepts

Power Rule for Integration

The power rule for integration is a simple yet powerful tool used to find the antiderivatives of polynomials. It gives us a straightforward method to integrate functions of the form \( q^n \). Here's how it works:

• If you have a function like \( q^n \), where \( n eq -1 \), the antiderivative, also known as the indefinite integral, is \( \frac{q^{n+1}}{n+1} \).
• For instance, if you need the antiderivative of \( 5q^2 \), first focus on \( q^2 \).
• Apply the power rule: Increase the exponent by 1 (so it becomes 3), and divide by this new exponent, making it \( \frac{q^3}{3} \).
• Don't forget the coefficient 5. Multiply \( \frac{q^3}{3} \) by 5 to get \( \frac{5q^3}{3} \).

This resulting function is the essence of the power rule for integration, simplifying the process of finding antiderivatives.

Indefinite Integral

An indefinite integral, in simplest terms, is the antiderivative of a function. Unlike a definite integral, it does not compute a specific numerical value. Instead, it represents a family of functions. Here's what makes it special and how we apply it:

• The indefinite integral of a function \( f(q) \) is given by the function \( F(q) \) whose derivative is \( f(q) \).
• Using our example, the indefinite integral of \( f(q) = 5q^2 \) leads us to find \( F(q) = \frac{5q^3}{3} + C \).
• Through integration, you have essentially 'undone' the derivative, bringing you back to a more general form of the function.

Indefinite integrals are a fundamental concept in calculus, allowing you to reverse the process of differentiation.

Constant of Integration

When finding an indefinite integral, there's an important step you must not overlook—the constant of integration, denoted as \( C \). This constant is crucial for these reasons:

• It represents an infinite number of possible solutions. When differentiating, any constant disappears, so integrating adds it back to cater to all these possibilities.
• For \( 5q^2 \), after applying the integration process, we reached \( \frac{5q^3}{3} \). To encompass all solutions, we add \( C \), resulting in \( \frac{5q^3}{3} + C \).
• This \( C \) makes the result a family of functions, each differing by a constant value.

In the context of calculus, always remember to include the constant of integration when finding indefinite integrals.