B.Sc 1st Semester Mathematics Practical Detailed
Verification of Rolle’s Theorem
Aim:
To verify Rolle’s Theorem for the function f(x) = x² - 4x + 3 in the closed interval [1,3] and to
understand its geometrical and analytical significance in Differential Calculus.
Apparatus Required:
Practical notebook, pen, pencil, ruler, graph paper.
Theory – Rolle’s Theorem:
Rolle’s Theorem is an important theorem of Differential Calculus which establishes a relation
between continuity, differentiability and stationary points of a function. It states that if a function f(x)
is continuous in the closed interval [a,b], differentiable in the open interval (a,b), and if f(a) = f(b),
then there must exist at least one point c in (a,b) such that the first derivative at that point becomes
zero, that is f'(c) = 0.
Conditions of Rolle’s Theorem:
1 The function must be continuous in the closed interval [a,b].
2 The function must be differentiable in the open interval (a,b).
3 The values of the function at the end points must be equal, i.e., f(a) = f(b).
Geometrical Interpretation:
Geometrically, Rolle’s theorem signifies that if a curve begins and ends at the same height on the
graph, then at some intermediate point the tangent to the curve must be horizontal. This point
represents a stationary point where the slope of the tangent is zero.
Given:
f(x) = x² − 4x + 3 Interval: [1,3]
Step 1: Verification of Continuity
The given function is a polynomial in x. All polynomial functions are continuous for every real value
of x. Therefore, f(x) is continuous in the interval [1,3].
Step 2: Verification of Differentiability
Since the function is polynomial, its derivative exists at every point. Hence, f(x) is differentiable in
the open interval (1,3).
Step 3: Verification of f(1) and f(3)
f(1) = (1)² − 4(1) + 3 = 1 − 4 + 3 = 0
f(3) = (3)² − 4(3) + 3 = 9 − 12 + 3 = 0
Thus, f(1) = f(3). The third condition is satisfied.
Step 4: Differentiation of the Function
f(x) = x² − 4x + 3
Differentiating with respect to x:
f'(x) = 2x − 4
Verification of Rolle’s Theorem
Aim:
To verify Rolle’s Theorem for the function f(x) = x² - 4x + 3 in the closed interval [1,3] and to
understand its geometrical and analytical significance in Differential Calculus.
Apparatus Required:
Practical notebook, pen, pencil, ruler, graph paper.
Theory – Rolle’s Theorem:
Rolle’s Theorem is an important theorem of Differential Calculus which establishes a relation
between continuity, differentiability and stationary points of a function. It states that if a function f(x)
is continuous in the closed interval [a,b], differentiable in the open interval (a,b), and if f(a) = f(b),
then there must exist at least one point c in (a,b) such that the first derivative at that point becomes
zero, that is f'(c) = 0.
Conditions of Rolle’s Theorem:
1 The function must be continuous in the closed interval [a,b].
2 The function must be differentiable in the open interval (a,b).
3 The values of the function at the end points must be equal, i.e., f(a) = f(b).
Geometrical Interpretation:
Geometrically, Rolle’s theorem signifies that if a curve begins and ends at the same height on the
graph, then at some intermediate point the tangent to the curve must be horizontal. This point
represents a stationary point where the slope of the tangent is zero.
Given:
f(x) = x² − 4x + 3 Interval: [1,3]
Step 1: Verification of Continuity
The given function is a polynomial in x. All polynomial functions are continuous for every real value
of x. Therefore, f(x) is continuous in the interval [1,3].
Step 2: Verification of Differentiability
Since the function is polynomial, its derivative exists at every point. Hence, f(x) is differentiable in
the open interval (1,3).
Step 3: Verification of f(1) and f(3)
f(1) = (1)² − 4(1) + 3 = 1 − 4 + 3 = 0
f(3) = (3)² − 4(3) + 3 = 9 − 12 + 3 = 0
Thus, f(1) = f(3). The third condition is satisfied.
Step 4: Differentiation of the Function
f(x) = x² − 4x + 3
Differentiating with respect to x:
f'(x) = 2x − 4
