Problem 2 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the general antiderivative \(F(x)+C\) for each of the following. $$ f(x)=x-4 $$

Step-by-Step Solution

Answer

The general antiderivative is \( F(x) = \frac{x^2}{2} - 4x + C \).

Step 1: Identify the function to integrate

The function given is \( f(x) = x - 4 \). We need to find the antiderivative of this expression.

Step 2: Find the antiderivative of each term

To find the antiderivative, integrate each term separately.1. The antiderivative of \( x \) is \( \frac{x^2}{2} \).2. The antiderivative of \( -4 \) is \( -4x \).

Step 3: Combine the antiderivatives and add the constant of integration

Combine the antiderivatives found in the previous step and include the constant of integration \( C \). Thus, the general antiderivative is:\[F(x) = \frac{x^2}{2} - 4x + C\]

Key Concepts

Antiderivative

An antiderivative, often known as an indefinite integral, is a function that reverses the process of differentiation. To find an antiderivative is to determine a function from its derivative. In calculus, if you know a derivative function like \( f(x) \), you can find a function \( F(x) \) whose derivative is \( f(x) \).
This is important because many problems in calculus are about reversing the process of differentiation. For example, if \( f(x) = x - 4 \), its antiderivative would be a function \( F(x) \) such that \( F'(x) = f(x) \). Finding the antiderivative answers questions about areas and accumulations.
Here’s something interesting: a function can have more than one antiderivative. This leads us to something called the constant of integration.

Integration

Integration is the process of finding the antiderivative or the integral of a function. When you integrate a function, you're essentially combining the individual terms' antiderivatives together. In calculus, integration is a crucial concept because it helps solve problems that involve area, volume, and the accumulation of quantities.
When dealing with a polynomial like \( f(x) = x - 4 \), each term in the polynomial is integrated separately and then combined:

• The integration of \( x \) gives \( \frac{x^2}{2} \).
• The integration of \( -4 \) gives \( -4x \).

The result is the antiderivative \( F(x) = \frac{x^2}{2} - 4x \). This final result represents all possible functions whose derivative would give the original function \( f(x) \).

Constant of integration

The constant of integration \( C \) is an essential part of calculating indefinite integrals or antiderivatives. When you conduct integration, there’s always an unknown constant because differentiation of a constant results in zero, and this information is lost during differentiation.
This means that when finding an antiderivative \( F(x) \), you are not only looking for one function, but all functions that could have been differentiated to produce \( f(x) \). This entire family of functions is represented as:

• \( F(x) = \frac{x^2}{2} - 4x + C \)
• "+ C" accounts for all possible vertical shifts of the antiderivative function on a graph.

Understanding the constant of integration is important in shaping the general solution of an indefinite integral.

Polynomial functions

Polynomial functions are expressions that include terms consisting of variables raised to whole-number exponents, such as \( f(x) = x - 4 \). These functions are straightforward and serve as excellent practice models for integration, as they demonstrate core principles without introducing complicating factors.
When it comes to finding the antiderivative or integrating polynomial functions, the power rule for integration is applied. The power rule states that for any term \( x^n \), its antiderivative is \( \frac{x^{n+1}}{n+1} \). This rule simplifies finding antiderivatives of polynomial terms.
For our example:

• The term \( x \) becomes \( \frac{x^2}{2} \).
• The constant term, \( -4 \), after integration becomes \( -4x \).

Integrating polynomial functions forms the basis of understanding calculus, making subsequent complex functions more approachable.