Problem 1 - 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: Understanding the Problem

We are tasked with finding an antiderivative of the constant function \( f(x) = 5 \). The antiderivative of a function \( f(x) \) is a function \( F(x) \) such that \( F'(x) = f(x) \).

Step 2: Apply the Antiderivative Rule for Constants

The antiderivative of any constant \( a \) is given by \( a imes x + C \), where \( C \) is the constant of integration. Since our function is \( f(x) = 5 \), we apply this rule to get \( 5x + C \).

Step 3: Conclude the Solution

Thus, the antiderivative of \( f(x)=5 \) is \( F(x) = 5x + C \), where \( C \) is an arbitrary constant.

Key Concepts

Constant Function

In calculus, a constant function is one of the simplest types of functions you can encounter. It is represented by a function that does not change, regardless of the input value. For example, in our exercise, the constant function is given by \( f(x) = 5 \). This essentially means that for every value of \( x \), the function always outputs 5. The graph of a constant function is a horizontal line when plotted in the Cartesian plane.Constant functions have some unique properties:

• The derivative of a constant function is always zero because the rate of change of a constant is null. It doesn't increase or decrease.
• The antiderivative, or integral, of a constant function involves adding a variable, typically denoted as \( x \), to transform it into a linear function, plus an integration constant.

Understanding the concept of a constant function is crucial as it forms the foundation for more complex calculus concepts like derivatives and integrals.

Integration

Integration is the process of finding the antiderivative of a function, which essentially means finding the original function from its derivative. It is one of the two fundamental operations in calculus, with the other being differentiation.When we integrate a function, we are determining the area under the curve of that function with respect to a certain axis. The process of integration is symbolized by the integral sign \( \int \) and ultimately results in adding a constant of integration \( C \). This constant is crucial because it accounts for any missing vertical shifts in the function. In the case of our exercise, integrating the constant function \( f(x) = 5 \) results in:

• Applying the rule for the integral of a constant \( a \), which gives us \( F(x) = 5x + C \).
• Here, \( 5x \) represents the linear part of the function and \( C \) represents the constant that can shift the line up or down.

Integration is essential for solving many problems in physics, engineering, and other fields where determining the total accumulation of quantities is required.

Derivative

Differentiation is the opposite process of integration, concerning itself with finding the rate of change of a function. When you calculate the derivative, you are essentially determining how much a function changes as its input changes. For a constant function, like the \( f(x) = 5 \) from our exercise, the derivative is simple:

• The derivative of any constant, no matter its value, is zero. This is because a constant does not change and, as such, has no rate of change.
• Given \( f(x) = 5 \), the derivative \( f'(x) = 0 \).

Understanding derivatives is important in a multitude of applications, such as finding the slope of a tangent line to a curve at a point, or determining the speed of an engine. This foundation in calculus allows you to grasp more complex concepts as you progress in your studies.