Problem 66 - Essential Calculus

Question

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(f^{\prime}(x)=g(x),\) then \(\int g(x) d x=f(x)+C\).

Step-by-Step Solution

Answer

The statement is true. The result is one of the versions of the Fundamental Theorem of Calculus.

Step 1: Understand the statement and conditions

This statement posits that the derivative of \(f(x)\), denoted as \(f'(x)\), is equal to \(g(x)\). If this condition is true, then the integral of \(g(x)\) should be equal to \(f(x) + C\). According to the Fundamental Theorem of Calculus, this statement is generally true. The equation \(f'(x) = g(x)\) implies \(g(x)\) is the derivative of \(f(x)\). Consequently, \(f(x)\) should be the antiderivative (integral) of \(g(x)\). Taking the integral should yield \(f(x) + C\), where \(C\) is the constant of integration.

Step 2: Evaluate the statement

The integral of \(g(x)\) is indeed \(f(x)+C\). Given that \(f'(x) = g(x)\), we can reverse this operation by integrating \(g(x)\), which should yield the original function \(f(x)\) (up to a constant \(C\)), which is the constant of integration. This constant appears because the derivative of a constant is zero, making it impossible to know from the derivative alone whether such a constant was in the original function. Therefore, the statement 'If \(f'(x) = g(x)\), then \(\int g(x) dx = f(x) + C\), ' is indeed true.

Key Concepts

Understanding Derivatives

The concept of a derivative is fundamental in calculus as it represents the rate at which a function is changing at any given point. When you take the derivative of a function, denoted as \(f'(x)\), you are effectively finding the slope of the tangent line to the graph of the function at a particular point.

• For example, if \(f(x) = x^2\), then the derivative \(f'(x) = 2x\), which represents how quickly or slowly \(f(x)\) changes as \(x\) varies.
• Derivatives are widely used in various fields such as physics, engineering, and economics to model real-world phenomena.

In the context of the given problem, the derivative \(f'(x) = g(x)\) suggests that the function \(g(x)\) indicates how \(f(x)\) changes with respect to \(x\). Understanding the relationship between \(f'(x)\) and \(g(x)\) is crucial for applying the Fundamental Theorem of Calculus.

Exploring Indefinite Integrals

An indefinite integral, denoted as \(\int g(x) \, dx\), is essentially the reverse process of taking a derivative. It represents the collection of all antiderivatives of a function. When you integrate a function, you are finding another function \(f(x)\) such that its derivative brings you back to the original function \(g(x)\).

• For instance, if you have \(g(x) = 2x\), integrating it gives \(f(x) = x^2 + C\).
• The notation \(\int g(x) \, dx\) symbolizes the act of integration and is a key tool in both pure and applied mathematics.

The Fundamental Theorem of Calculus ties derivatives and integrals together, stating that if \(f'(x) = g(x)\), then \(\int g(x) \, dx = f(x) + C\), as it reverses the process of differentiation. This also confirms the assertion in the exercise that \(\int g(x) \, dx = f(x) + C\) when \(f'(x) = g(x)\).

Constant of Integration

When solving indefinite integrals, you’ll always encounter the "constant of integration," denoted by \(C\). This constant is crucial because the integral process can only reverse the differentiation of a function sans any constant values. Why? Because the derivative of any constant is zero, making it invisible if it was initially part of \(f(x)\).

• For example, both \(x^2\) and \(x^2 + 5\) have the same derivative: \(2x\).
• This reflects the idea that integrating \(2x\) can lead to many valid solutions, such as \(x^2, x^2 + 5, x^2 - 3\), etc.

Thus, we add \(C\) after every indefinite integral to account for this uncertainty and indicate that there is an infinite number of potential original functions. In the given exercise, the constant \(C\) ensures that any original constant lost through differentiation is properly acknowledged when performing the integration.