Problem 1 - Calculus Early Transcendentals: Pearson New International Edition

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: Understand the Antiderivative Concept

The antiderivative of a function is another function whose derivative is the original function. In essence, finding the antiderivative is the reverse process of differentiation.

Step 2: Identify the Function Type

The function given is a constant function, \(f(x) = 5\). The antiderivative of a constant \(k\) is \(kx + C\).

Step 3: Apply Antiderivative Rules for Constants

To find the antiderivative of a constant, multiply the constant by \(x\) and add the constant of integration \(C\). This gives us:\[F(x) = 5x + C\]

Step 4: Verification (Optional)

Differentiate the antiderivative obtained, \(F(x) = 5x + C\), to check if its derivative equals the original function. \[\frac{d}{dx}(5x + C) = 5\] This matches the original function \(f(x) = 5\), verifying the solution.

Key Concepts

Constant Functions

A constant function is expressed in the form of \(f(x) = c\), where \(c\) is a real number that does not change with respect to \(x\). This means no matter the value of \(x\), the function value remains constant. For instance, in the given exercise \(f(x) = 5\), the value of the function is always 5.
Constant functions have several interesting properties:

• They have a straight horizontal line graph, parallel to the \(x\)-axis.
• The slope of constant functions is always zero, implying no change in \(y\) value as \(x\) changes.
• In calculus, finding an antiderivative of a constant function involves reversing the process of differentiation, often turning straightforward calculations into clear results.

Understanding how constant functions work is essential as they lay the foundation for exploring deeper calculus concepts.

Reverse Differentiation

Reverse differentiation, better known as finding an antiderivative, involves determining a function whose derivative is the original function provided. This process turns the typical differentiation path around.
The antiderivative is denoted by \(F(x)\), and it includes a constant of integration, \(C\), because differentiation of a constant is zero, leaving us with an infinite number of possible functions. In simple terms, if you differentiate \(F(x) = 5x + C\), the result is \(f(x) = 5\), showcasing the reverse nature - "undoing" differentiation.
You might wonder why reverse differentiation matters. Well, it provides an integral part of solving problems related to area, displacement, and accumulation. Therefore, understanding this reverse process helps uncover many mathematical applications.

Antiderivative Rules

To find antiderivatives, certain rules come into play, making it easier to identify the corresponding function. These antiderivative rules guide us systematically in reverse differentiation.
For constant functions, the rule is straightforward: multiply the constant by \(x\) and add \(C\), the constant of integration. So for \(f(x) = c\), the antiderivative \(F(x)\) becomes \(cx + C\).
Here's why these rules are important:

• They reduce errors by providing a structured approach to finding antiderivatives.
• They aid in simplifying complex processes, transforming difficult computations into manageable steps.
• Recognizing these patterns allows for quicker problem-solving and better understanding of calculus as a whole.

So, applying these antiderivative rules is a fundamental skill in calculus, helping bridge the gap between algebraic manipulation and understanding calculus concepts.