Problem 97 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If \(F\) and \(G\) are antiderivatives of \(f\) on an interval \(I\), then \(F(x)=G(x)+C\) on \(I\).

Step-by-Step Solution

Answer

The statement is true. If \(F\) and \(G\) are antiderivatives of the same function \(f\) on the interval \(I\), then \(F(x) = G(x) + C\) on \(I\), for some constant \(C\). This is proven by defining \(H(x)=F(x)-G(x)\), showing that \(H'(x)=0\), and applying the constant function rule.

Step 1: Properties of Antiderivatives

We are given that \(F\) and \(G\) are antiderivatives of the function \(f\) on the interval \(I\). From the definition of antiderivatives, we know that if \(F'(x)=f(x)\) and \(G'(x)=f(x)\) on \(I\).

Step 2: Comparing Their Derivatives

Since we know that both antiderivatives \(F\) and \(G\) satisfy the condition \(F'(x)=f(x)\) and \(G'(x)=f(x)\) on the interval \(I\), we will now compare their derivatives: \[F'(x) = G'(x)\] This must be true for all values of \(x\) in the interval \(I\).

Step 3: Finding the Difference

Now, let's introduce a new function, say \(H(x)=F(x)-G(x)\). Now, differentiate this function with respect to \(x\): \[H'(x) = F'(x) - G'(x)\] Since we know \(F'(x) = G'(x)\), we can substitute this into the equation above: \[H'(x) = F'(x) - F'(x) = 0\] This means that the function \(H(x)\) has a constant derivative of 0.

Step 4: Constant Function Rule

Since we have shown that \(H'(x)=0\), we can now apply the constant function rule. The constant function rule states that if a function has a constant derivative of 0, then the function is a constant function. Thus, \(H(x)\) is constant on the interval \(I\). Consequently, we have: \[H(x) = F(x) - G(x) = C\Rightarrow F(x)=G(x) + C\] where \(C\) is a constant.

Step 5: Conclusion

We have proven that if \(F\) and \(G\) are antiderivatives of the same function \(f\), then \(F(x) = G(x) + C\) on the interval \(I\), for some constant \(C\). Therefore, the statement is true.

Key Concepts

Constant Function Rule

The constant function rule is a fundamental concept in calculus. It states that if the derivative of a function is zero across an interval, then the function itself is constant over that interval. This idea helps us understand situations involving antiderivatives. In the exercise mentioned, we see that two antiderivatives, \(F\) and \(G\), of a function \(f\) have their derivatives equal when compared:

• If \(F'(x) = G'(x)\) for all \(x\) in an interval \(I\), then the difference \(H(x) = F(x) - G(x)\) has a derivative of zero: \(H'(x) = 0\).
• This zero derivative implies that \(H(x)\) is a constant, thus \(F(x) = G(x) + C\) for a constant \(C\).

This property is useful for solving many problems involving antiderivatives, ensuring that any two antiderivatives of the same function on an interval differ only by a constant.

Interval Analysis

Interval analysis is important when working with functions in calculus, especially when dealing with derivatives and integrals. An interval is a set of real numbers lying between two numbers called endpoints. Analyzing a function over an interval is crucial because:

• It allows understanding how a function behaves between two points.
• Limits the region where certain properties apply, including constant derivatives or antiderivatives.

In the context of antiderivatives, like in the exercise, ensuring that \(F\) and \(G\) are antiderivatives in the same interval \(I\) is key. This shared interval guarantees that their derivatives satisfy \(F'(x) = f(x)\) and \(G'(x) = f(x)\) across all points within \(I\). Essentially, it ensures the continuity and well-behaved nature of \(f\) and its antiderivatives over \(I\). Whatever conclusions we draw about constants or differences also hold true consistently across this interval.

Differentiation

Differentiation is the process of finding the derivative of a function. It is a core operation in calculus and provides a measure of how a function changes as its input changes. Differentiating a function helps in understanding the rate of change or slope at any particular point.Key points about differentiation include:

• The derivative of a function \(f(x)\) at a point \(x\) is often denoted as \(f'(x)\) or \(\frac{df}{dx}\).
• In an antiderivative context, if \(F'(x) = f(x)\), then \(F(x)\) is an antiderivative of \(f\).

In the given problem, both \(F\) and \(G\) are differentiated to establish that they have the same derivative, \(f(x)\), across an interval. This equality of derivatives leads to them differing by a constant. Differentiation here is what allows us to apply the constant function rule, ultimately proving \(F(x) = G(x) + C\). Without differentiation, such relationships between functions would be difficult to determine.