Problem 17 - Applied Calculus

Question

Find an antiderivative. $$f(x)=5$$

Step-by-Step Solution

Answer

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

Step 1: Understand the Concept of Antiderivative

An antiderivative of a function is a function whose derivative is the original function. For our problem, we need to find a function whose derivative is 5.

Step 2: Recall Basic Derivative Rules

Remember that the derivative of a constant, such as \(5x\), is constant \(5\). Thus, \(5x\) is a potential antiderivative of \(f(x) = 5\).

Step 3: Consider the Constant of Integration

When finding antiderivatives, we must add a constant \(C\), because the derivative of any constant is zero. Therefore, the most general antiderivative is \(5x + C\).

Step 4: Write the Final Solution

The antiderivative of the function \(f(x) = 5\) is \(F(x) = 5x + C\), where \(C\) is any constant.

Key Concepts

Constant of Integration

When dealing with antiderivatives, a fundamental aspect to consider is the constant of integration. But what exactly is this mysterious constant, often represented by the letter \(C\)? In calculus, when you take the derivative of a function, every constant vanishes because the derivative of a constant is zero. Therefore, when you work backwards to find an antiderivative, there is an infinite number of possible solutions differing by a constant.To address this ambiguity, we include the constant of integration. In practice, this means the general solution to finding an antiderivative takes the form of a function plus \(C\). For example, if the derivative of a function \(F(x)\) is \(f(x) = 5\), then the antiderivative should be expressed as \(F(x) = 5x + C\). This ensures we account for all potential initial conditions and solutions. Including \(C\) is crucial when solving real-world problems in applied calculus.

Basic Derivative Rules

Understanding basic derivative rules is essential when you are working with antiderivatives, or integrals. These rules help simplify the process of differentiating and integrating functions. Let's explore a few key rules:

• Power Rule: The derivative of \(x^n\) is \(nx^{n-1}\)
• Constant Rule: The derivative of a constant \(c\), like in our original exercise, is zero.
• Coefficient Rule: The derivative of a constant times a function is equal to the constant times the derivative of the function.

For our exercise, the simplest derivative rule applies—\(f(x) = 5\) means the function is constant. Thus, an antiderivative of 5 is simply \(5x\), based on understanding these rules. Always remember to apply these foundational derivative principles, as they simplify the process of identifying and calculating antiderivatives.

Applied Calculus

Applied calculus is where the principles of calculus, including derivatives and antiderivatives, come into play in real-world scenarios.Whether it's engineering, physics, economics, or any field requiring mathematical modeling, integrals play a crucial role. For example, finding the antiderivative can help calculate area under a curve, determine velocity from acceleration, or predict total cost from marginal cost.It's essential to accurately express antiderivatives with a constant of integration \(C\), as this constant often represents initial conditions or specific solutions needed in practical applications.The ability to compute antiderivatives effectively allows professionals to solve differential equations and understand cumulative changes over time, making applied calculus an indispensable tool in both theoretical and practical contexts. Keeping these connections in mind helps illustrate why mastering calculus concepts is so pivotal.