Problem 30 - Single Variable Calculus

Question

Suppose \(f^{\prime}(x)=c\), where \(c\) is a constant, for all values of \(x\). Show that \(f\) must be a linear function of the form \(f(x)=c x+d\) for some constant \(d\). Hint: Use the corollary to Theorem \(3 .\)

Step-by-Step Solution

Answer

Given the derivative of a function \(f(x)\) is a constant, \(f'(x) = c\), we first integrate the derivative with respect to x: \[\int f'(x) dx = \int c dx\] \[c \int dx = cx + C\] We obtain the general form of the function, \(f(x) = cx + C\). Replacing the arbitrary constant \(C\) with another constant, say \(d\), we have: \[f(x) = cx + d\] This confirms that the function is a linear function.

Step 1: Integrate the derivative function

The given function is \(f'(x) = c\). Now, integrate this function with respect to x to find the original function. \[\int f'(x) dx = \int c dx\]

Step 2: Evaluate the integral

Since c is a constant, it can be pulled out of the integral. \[\int c dx = c \int dx\] Now, the integral of \(dx\) is simply the variable \(x\): \[c \int dx = cx + C\] where, \(C\) is a constant of integration.

Step 3: Write the function in form of \(f(x) = cx + d\)

We now have the general form of the function: \[f(x) = cx + C\] Since \(C\) is an arbitrary constant, we can replace it with another constant, say \(d\) (as per the original question) without changing the essence of the function: \[f(x) = cx + d\]

Step 4: Conclusion

So given that the derivative of a function \(f(x)\) is a constant, \(f'(x) = c\), the function must be of the form \(f(x) = cx + d\) where d is another constant. This confirms that the function is indeed a linear function.

Key Concepts

Integration

Integration is a fundamental concept in calculus, acting as the reverse process of differentiation. Given a derivative, integration helps to find the original function. When integrating a constant, such as in our function where we have \(f'(x) = c\), the integral is calculated with respect to variable \((x)\), giving the output as the original function plus an arbitrary constant.
The integral of a constant can be expressed as:

• \(\int c\, dx = cx + C\)

Here, \(c\) is the constant being integrated, \(x\) is the variable of integration, and \(C\) represents the constant of integration. This process effectively reconstructs the original equation which includes a linear function element, showing how integral calculus relates the derivative back to the function.

Constant of Integration

The constant of integration, represented as \(C\) in the integration process, is an essential aspect of finding indefinite integrals. When we integrate a function like \(f'(x) = c\), we end up with an expression \(f(x) = cx + C\). This \(C\) arises because integration does not specify the vertical shift of the antiderivative, so it represents all possible vertical shifts.
Key points about the constant of integration include:

• It accounts for any constant added to a function without altering its derivative.
• In linear functions, this constant is crucial as it determines the specific line on a coordinate plane.
• A function's graph can be shifted vertically by varying \(C\), producing parallel lines.

Thus, this constant explains why antiderivatives, and in our case the linear function, maintain all possible solutions.

Derivative of a Constant

Understanding the derivative of a constant is fairly straightforward. A constant function is a horizontal line on a graph, and its rate of change is zero. Therefore, the derivative of any constant is zero.
Here's the breakdown:

• Consider a constant \(d\) in a function \(f(x) = cx + d\).
• The derivative of \(d\) is 0 because changing \(x\) doesn't affect the value of \(d\).

When dealing with linear equations in calculus, the constant term \(d\) reflects the \(y\)-intercept in the function \(f(x) = cx + d\), and understanding its derivative as 0 helps to clarify why the shape remains unchanged except for position on the y-axis.

Linear Equation

A linear equation is fundamental to algebra and calculus, represented in the form \(f(x) = mx + b\) where \(m\) and \(b\) are constants. In the scenario of integration, the constant \(m\) is found as a result of integrating a constant derivative such as \(f'(x) = c\), resulting in a general linear form \(f(x) = cx + d\). Linear equations describe straight lines on a coordinate plane, which have:

• A slope denoted by \(c\) in our integrated function, which describes the steepness or incline of the line.
• An intercept \(d\) which indicates the point where the line crosses the \(y\)-axis.
• No higher powers or non-linear terms involved, meaning the change between \(f(x)\) values is constant.

By thoroughly understanding linear equations, one sees how integration of a constant leads naturally to this form, demonstrating its respect to the principles of constant rates of change and linearity.