![What is Mean Value Theorem ?
The Mean Value Theorem states that if a function f is continuous on the
closed interval [a,b] and differentiable on the open interval (a,b), then
there exists a point c in the interval (a,b) such that f'(c) is equal to the
function's average rate of change over [a,b]. In other words, the graph has a
tangent somewhere in (a,b) that is parallel to the secant line over [a,b].](https://image.slidesharecdn.com/meanvaluerollestheorem-240518104130-321b953d/85/Understanding-Mean-Value-Theorem-and-Rolle-s-Theorem-Key-Concepts-Applications-and-Practical-Examples-for-Mastering-Calculus-2-320.jpg)
![Expressions
Suppose f(x) is a function that satisfies both of the following.
• f(x) is continuous on the closed interval [a,b].
• f(x)is differentiable on the open interval (a,b).
Then there is a number c such that a < c < b and,
f’(c) =
𝑓 𝑏 −𝑓(𝑎)
𝑏−𝑎
Or,
f(b)−f(a)=f′(c)(b−a)](https://image.slidesharecdn.com/meanvaluerollestheorem-240518104130-321b953d/85/Understanding-Mean-Value-Theorem-and-Rolle-s-Theorem-Key-Concepts-Applications-and-Practical-Examples-for-Mastering-Calculus-3-320.jpg)
![Example 1
Determine all the numbers c which satisfy the conclusions of the Mean Value Theorem for the
following function.
f(x)=𝒙𝟑
+ 𝟐𝒙𝟐
+ 𝒙 on [−1,2]
𝑓′
𝑐 =
𝑓 2 −𝑓 −1
2−(−1)
𝑓′
𝑐
𝑓 𝑏 −𝑓 𝑎
𝑏−𝑎
3𝑐2
+ 4𝑐 − 1 =
14 − 2
3
= 4
3𝑐2
+ 4c − 5 = 0
𝑥 =
−4 ± 42 − 4(3)(−5)
2(3)](https://image.slidesharecdn.com/meanvaluerollestheorem-240518104130-321b953d/85/Understanding-Mean-Value-Theorem-and-Rolle-s-Theorem-Key-Concepts-Applications-and-Practical-Examples-for-Mastering-Calculus-5-320.jpg)
![Example 1
c =
−4 + 76
6
; c =
−4 + 76
6
c = 0.7863 ; c = -2.1196
Here only one value (c = 0.7863) belongs to interval [a,b] so we will exclude the
other value (since it isn’t in the interval),
C = 0.7863](https://image.slidesharecdn.com/meanvaluerollestheorem-240518104130-321b953d/85/Understanding-Mean-Value-Theorem-and-Rolle-s-Theorem-Key-Concepts-Applications-and-Practical-Examples-for-Mastering-Calculus-6-320.jpg)
![What is Rolle’s Theorem ?
Rolle's theorem states that "If a function f is defined in the closed
interval [a, b] in such a way that it satisfies the following condition: i) f
is continuous on [a, b], ii) f is differentiable on (a, b), and iii) f (a) = f
(b), then there exists at least one value of x, let us assume this value
to be c, which lies between a and b i.e. (a < c < b ) in such a way that
f‘(c) = 0."](https://image.slidesharecdn.com/meanvaluerollestheorem-240518104130-321b953d/85/Understanding-Mean-Value-Theorem-and-Rolle-s-Theorem-Key-Concepts-Applications-and-Practical-Examples-for-Mastering-Calculus-7-320.jpg)
![Expression
Rolle's theorem states that
If
• f(x) is continuous on [a, b]
• f(x) is differentiable on (a, b)
• f(a) = f(b)
then there will exists at least one c belongs to (a,b) where,
f’(c) = 0](https://image.slidesharecdn.com/meanvaluerollestheorem-240518104130-321b953d/85/Understanding-Mean-Value-Theorem-and-Rolle-s-Theorem-Key-Concepts-Applications-and-Practical-Examples-for-Mastering-Calculus-8-320.jpg)
Understanding Mean Value Theorem and Rolle’s Theorem: Key Concepts, Applications, and Practical Examples for Mastering Calculus
- 1. Mean Value & Rolle’s Theorem Muhammad Uzair Asif
- 2. What is Mean Value Theorem ? The Mean Value Theorem states that if a function f is continuous on the closed interval [a,b] and differentiable on the open interval (a,b), then there exists a point c in the interval (a,b) such that f'(c) is equal to the function's average rate of change over [a,b]. In other words, the graph has a tangent somewhere in (a,b) that is parallel to the secant line over [a,b].
- 3. Expressions Suppose f(x) is a function that satisfies both of the following. • f(x) is continuous on the closed interval [a,b]. • f(x)is differentiable on the open interval (a,b). Then there is a number c such that a < c < b and, f’(c) = 𝑓 𝑏 −𝑓(𝑎) 𝑏−𝑎 Or, f(b)−f(a)=f′(c)(b−a)
- 4. Graph Note that the Mean Value Theorem doesn’t tell us what c is. It only tells us that there is at least one number c that will satisfy the conclusion of the theorem.
- 5. Example 1 Determine all the numbers c which satisfy the conclusions of the Mean Value Theorem for the following function. f(x)=𝒙𝟑 + 𝟐𝒙𝟐 + 𝒙 on [−1,2] 𝑓′ 𝑐 = 𝑓 2 −𝑓 −1 2−(−1) 𝑓′ 𝑐 𝑓 𝑏 −𝑓 𝑎 𝑏−𝑎 3𝑐2 + 4𝑐 − 1 = 14 − 2 3 = 4 3𝑐2 + 4c − 5 = 0 𝑥 = −4 ± 42 − 4(3)(−5) 2(3)
- 6. Example 1 c = −4 + 76 6 ; c = −4 + 76 6 c = 0.7863 ; c = -2.1196 Here only one value (c = 0.7863) belongs to interval [a,b] so we will exclude the other value (since it isn’t in the interval), C = 0.7863
- 7. What is Rolle’s Theorem ? Rolle's theorem states that "If a function f is defined in the closed interval [a, b] in such a way that it satisfies the following condition: i) f is continuous on [a, b], ii) f is differentiable on (a, b), and iii) f (a) = f (b), then there exists at least one value of x, let us assume this value to be c, which lies between a and b i.e. (a < c < b ) in such a way that f‘(c) = 0."
- 8. Expression Rolle's theorem states that If • f(x) is continuous on [a, b] • f(x) is differentiable on (a, b) • f(a) = f(b) then there will exists at least one c belongs to (a,b) where, f’(c) = 0
- 9. Graph
- 10. Cases Case 1: the function is constant Case 2: the function is not constant
- 11. Case 1 In this case, every point satisfies Rolle's Theorem since the derivative is zero everywhere.
- 12. Case 2 Since the function isn't constant, it must change directions in order to start and end at the same y-value. It means at some point within the interval the function will either have a minimum, a maximum or both
- 13. Example 2 Verify Rolle’s theorem for the function y = x2 + 1, a = –1 and b = 1. f(-1) = (-1)2 + 1 = 1 + 1 = 2 f(1) = (1)2 + 1 = 1 + 1 = 2 Thus, f(– 1) = f(1) = 2 Hence, the function f(x) satisfies all conditions of Rolle's theorem.
- 14. Example 2 Now, f'(x) = 2x Rolle’s theorem states that there is a point c ∈ (– 2, 2) such that f′(c) = 0. 2c = 0 c = 0, where c = 0 ∈ (–1, 1)