Question

Given below are two statements :
Statement (I) : Differentiation and Integration are just opposite of each other.
Statement (II) : Differentiation is the slope of any curve at any point, while integration is the area under that same curve specified between two points.
In light of the above statements, choose the most appropriate answer from the options given below.

• A. Both Statement (I) and Statement (II) are true.
• B. Both Statement (I) and Statement (II) are false.
• C. Statement (I) is true but Statement (II) is false.
• D. Statement (I) is false but Statement (II) is true.

Hint:

Keep the basic geometric meanings in mind:
- Derivative (\(dy/dx\)): Gives the instantaneous rate of change (Slope of tangent).
- Definite Integral (\(\int_a^b f(x)dx\)): Accumulates values over an interval (Area under the curve).
They are inverse operations of each other.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Calculus is fundamentally divided into differential calculus and integral calculus.
The relationship between these two branches is established by the Fundamental Theorem of Calculus, which proves they are inverse operations.

Step 2: Detailed Explanation:
Let us evaluate both statements:
- Statement (I): "Differentiation and Integration are just opposite of each other."
This is a true statement. According to the Fundamental Theorem of Calculus:
\[ \frac{d}{dx} \left( \int_{a}^{x} f(t) \, dt \right) = f(x) \]
This proves that differentiation reverses the process of integration.
Similarly, the indefinite integral of the derivative of a function returns the original function (plus a constant):
\[ \int f'(x) \, dx = f(x) + C \]
Thus, they are inverse operations.
- Statement (II): "Differentiation is the slope of any curve at any point, while integration is the area under that same curve specified between two points."
This is a true statement describing the geometric interpretations of both operations:
- Geometrically, the derivative \(\frac{dy}{dx}\) at any point represents the slope of the tangent line to the curve at that point.
- Geometrically, the definite integral \(\int_{a}^{b} f(x) \, dx\) represents the net signed area bounded by the curve \(y = f(x)\), the x-axis, and the vertical lines \(x = a\) and \(x = b\).
Therefore, both Statement (I) and Statement (II) are true.

Step 3: Final Answer:
The correct option is (A).