Answer
See the explanation.
Work Step by Step
Integration and differentiation are fundamental concepts in calculus, and they are often considered as "inverses" of each other due to their relationship in the context of calculus operations.
1. **Inverse Relationship:**
- **Differentiation:** This process involves finding the derivative of a function, which represents the rate of change of the function with respect to its independent variable. It helps in understanding how the function's output changes as the input varies.
- **Integration:** On the other hand, integration is the process of finding the antiderivative of a function. An antiderivative is a function whose derivative is the original function. Integration helps in determining the accumulated change or total quantity represented by a rate of change function.
2. **Fundamental Theorem of Calculus:**
- The Fundamental Theorem of Calculus establishes a deep connection between integration and differentiation. It states that if a function \(f(x)\) is continuous on a closed interval \([a, b]\) and \(F(x)\) is the antiderivative of \(f(x)\), then:
\[
\int_{a}^{b} f(x) \,dx = F(b) - F(a)
\]
This theorem links the process of finding the antiderivative (integration) to the process of evaluating the accumulated change over an interval.
3. **Inverses in Action:**
- When we differentiate a function and then integrate the result, or vice versa, we can end up with the original function (up to a constant of integration).
- If \(F(x)\) is the antiderivative of \(f(x)\), then differentiating \(F(x)\) yields \(f(x)\) again.
- Similarly, if \(f(x)\) is a continuous function, integrating \(f(x)\) and then differentiating the result yields \(f(x)\) again.
4. **Example:**
- Let's consider a simple example: Suppose \(f(x) = 2x\).
- If we integrate \(f(x)\), we get \(F(x) = x^2 + C\), where \(C\) is the constant of integration.
- If we then differentiate \(F(x)\), we obtain \(f(x) = 2x\) again.
