Problem 1 - Calculus

Question

Find the general antiderivative \(F(x)+C\) for each of the following. $$ f(x)=5 $$

Step-by-Step Solution

Answer

The general antiderivative is \(F(x) = 5x + C\).

Step 1: Understanding the Basic Concept of Antiderivatives

The antiderivative, also known as the indefinite integral, is the reverse process of differentiation. To find the antiderivative of a constant function, we need to apply the basic rule that the antiderivative of a constant \(c\) is \(cx + C\), where \(C\) is the constant of integration.

Step 2: Identifying the Function

The function given is \(f(x) = 5\). Here, \(5\) is a constant function, which means it doesn't depend on \(x\).

Step 3: Applying the Antiderivative Rule

Using the rule from Step 1, the antiderivative of \(5\) is simply \(5x + C\), where \(C\) is the constant of integration. This function represents the family of all antiderivatives of \(f(x)\).

Step 4: Writing the General Antiderivative

The general antiderivative \(F(x)\) is therefore \(F(x) = 5x + C\), where \(C\) is any constant. It includes all possible functions whose derivative is \(f(x) = 5\).

Key Concepts

Indefinite Integral

The indefinite integral is a critical concept in calculus. It serves as the inverse of differentiation, helping us find the original function given its derivative. When you calculate an indefinite integral, you often see the notation \( \int f(x) \, dx \). This indicates you are integrating the function \( f(x) \) with respect to \( x \).

Unlike definite integrals, indefinite integrals do not have limits of integration and instead result in a family of functions—hence they are called 'indefinite.' This family of functions is expressed as a function plus a constant \( C \).

• **Purpose:** Reverses differentiation to find the original function.
• **Notation:** Uses the integral sign \( \int \) and adds +C, indicating the constant of integration.
• **Result:** Produces a general form of the antiderivative.

For example, finding the indefinite integral of \( f(x) = 5 \) results in \( F(x) = 5x + C \), representing all linear functions with slope 5.

Constant of Integration

The constant of integration \( C \) is crucial for understanding indefinite integrals. Whenever you integrate a function, you must include \( C \) to account for all possible vertical shifts of the antiderivative. This is because many functions can differ by just a constant and still have the same derivative.

In practice, when you differentiate the result of an indefinite integral, the constant disappears (i.e., the derivative of a constant is zero). Thus, without \( C \), you can't encapsulate the entire scope of solutions. For example:

• **Integration Example:** The integral of \( f(x) = 5 \) is \( 5x + C \).
• **Why it's Important:** Without \( C \), you'd miss out on all other functions that could differ from a base function by just a constant.
• **Real-life Analogy:** Imagine stacking identical books at different heights; the base stays the same, but the vertical position varies, just like how functions differ by a constant.

Include \( C \) each time to convey that there are infinite solutions sharing the same derivative.

Differentiation

Differentiation is another foundational concept opposite to integration. It involves calculating the rate at which a function changes. Differentiation takes you from a function to its derivative, focusing on finding the slope or steepness at any point.

When you differentiate a function, you see how it behaves locally, providing a dynamic picture of how function values alter. To illustrate, with the function \( g(x) = 5x + C \), the process of differentiation yields the derivative \( g'(x) = 5 \). This tells us that for every unit increase in \( x \), the function grows by 5 units. Notice how

• **Core Function:** Differentiation shapes our understanding of changes over time or space.
• **Signs of Change:** If the derivative is positive, the function ascends; if negative, it descends.
• **Application Example:** A constant derivative of 5 implies a linear relationship with a steady slope, as in \( 5x \).

Understanding differentiation allows us to better grasp how variables interact and create curves or lines, essential for analytical exploration.