The document discusses the Mean Value Theorem, which states that if a function f(x) is continuous on the closed interval [a,b] and differentiable on the open interval (a,b), then there exists some value c in (a,b) such that:
f(b) - f(a) = f'(c)(b - a)
In other words, there is at least one point where the slope of the tangent line equals the slope of the secant line between points a and b. The document provides examples and illustrations to demonstrate how to apply the Mean Value Theorem.
This ppt covers following topics of Unit - 2 of B.Sc. 2 Mathematics Rolle's Theorem , Lagrange's mean value theorem , Mean value theorem & its example .
The Mean Value Theorem states that for any given curve between two endpoints, there must be a point at which the slope of the tangent to the curve is same as the slope of the secant through its endpoints. Copy the link given below and paste it in new browser window to get more information on Mean Value Theorem www.askiitians.com/iit-jee-applications-of-derivatives/rolle-theoram-and-lagrange-mean-value-theorem/
This ppt covers following topics of Unit - 2 of B.Sc. 2 Mathematics Rolle's Theorem , Lagrange's mean value theorem , Mean value theorem & its example .
The Mean Value Theorem states that for any given curve between two endpoints, there must be a point at which the slope of the tangent to the curve is same as the slope of the secant through its endpoints. Copy the link given below and paste it in new browser window to get more information on Mean Value Theorem www.askiitians.com/iit-jee-applications-of-derivatives/rolle-theoram-and-lagrange-mean-value-theorem/
3 examples of PDE, for Laplace, Diffusion of Heat and Wave function. A brief definition of Fouriers Series. Slides created and compiled using LaTeX, beamer package.
The Mean Value Theorem is the Most Important Theorem in Calculus. It allows us to relate information about the derivative of a function to information about the function itself.
The Mean Value Theorem is the most important theorem in calculus. It is the first theorem which allows us to infer information about a function from information about its derivative. From the MVT we can derive tests for the monotonicity (increase or decrease) and concavity of a function.
3 examples of PDE, for Laplace, Diffusion of Heat and Wave function. A brief definition of Fouriers Series. Slides created and compiled using LaTeX, beamer package.
The Mean Value Theorem is the Most Important Theorem in Calculus. It allows us to relate information about the derivative of a function to information about the function itself.
The Mean Value Theorem is the most important theorem in calculus. It is the first theorem which allows us to infer information about a function from information about its derivative. From the MVT we can derive tests for the monotonicity (increase or decrease) and concavity of a function.
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Mean Value Theorem explained with examples.pptxvandijkvvd4
The Mean Value Theorem (MVT) is a crucial concept in calculus, connecting the average rate of change of a function to its instantaneous rate of change. It's a fundamental theorem that holds a significant place in calculus and has far-reaching implications across various mathematical fields. Exploring it through 3000 alphabets involves diving into its core principles, applications, and significance.
At its heart, the Mean Value Theorem asserts that if a function is continuous on a closed interval and differentiable on the open interval, there exists at least one point within that interval where the instantaneous rate of change (the derivative) equals the average rate of change of the function over that interval.
Geometrically, MVT can be visualized as a tangent line parallel to a secant line at a certain point within the function, signifying the equality between the average and instantaneous rates of change.
Understanding MVT involves grasping its conditions and implications. For a function
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f(x), the prerequisites for applying MVT are continuity and differentiability within the specified interval
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[a,b].
The theorem's application extends to various contexts in mathematics, science, and economics. It's utilized to prove the existence of solutions to equations, establish bounds for functions, and analyze behavior in optimization problems.
MVT plays a pivotal role in other fundamental theorems of calculus like the Fundamental Theorem of Calculus, aiding in the development of integral calculus and its applications in areas such as physics, engineering, and economics.
Beyond its practical applications, the Mean Value Theorem's elegance lies in its capacity to capture the essence of rates of change, providing a bridge between local and global behavior of functions.
Mathematicians and scientists rely on MVT to understand and model real-world phenomena, utilizing its principles to analyze trends, make predictions, and solve problems across diverse disciplines.
In essence, the Mean Value Theorem stands as a cornerstone of calculus, fostering a deeper comprehension of the relationship between a function and its derivatives while serving as a powerful tool in mathematical analysis and problem-solving.
The Mean Value Theorem (MVT) in calculus asserts that for a continuous and differentiable function on a closed interval, there exists at least one point within that interval where the derivative (instantaneous rate of change) of the function equals the average rate of change of the function over that interval. It's a fundamental concept connecting the behavior of functions locally and globally, pivotal in calculus, and extensively applied in various fields like physics, engineering, and economics. MVT's essence lies in relating the function's behavior at specific points to its overall behavior, aiding in problem-solving, equation-solving, and understanding rates of change in real-world scenarios.
MVT relates function's average to in
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I am Manuela B. I am a Calculus Assignment Expert at mathsassignmenthelp.com. I hold a Master's in Mathematics from, the University of Warwick Profession. I have been helping students with their assignments for the past 8 years. I solve assignments related to Calculus.
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Rolle's theorem:
Statement :
Let, F(x) be a real valued
function in interval [a, b] such
that,
1. F(x) is continuous in closed interval [a, b].
2. F(x) is differentiable in
open interval (a, b)
3. F(a) = F(b).
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2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
http://sandymillin.wordpress.com/iateflwebinar2024
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Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
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The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
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Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
1. 4.2
The Mean Value Theorem
Prepared By:
Sheetal Gupta
Asst. Professor
GDRCST, Bhilai
2. 2
The Mean Value Theorem
We will see that many of the results depend on one central
fact, which is called the Mean Value Theorem. But to arrive
at the Mean Value Theorem we first need the following
result.
Before giving the proof let’s take a look at the graphs of
some typical functions that satisfy the three hypotheses.
3. 3
The Mean Value Theorem
Figure 1 shows the graphs of four such functions.
Figure 1
(c)
(b)
(d)
(a)
4. 4
The Mean Value Theorem
In each case it appears that there is at least one point
(c, f(c)) on the graph where the tangent is horizontal and
therefore f′(c) = 0.
Thus Rolle’s Theorem is plausible.
5. 5
Example 2
Prove that the equation x3
+ x – 1 = 0 has exactly one real
root.
Solution:
First we use the Intermediate Value Theorem to show that
a root exists. Let f(x) = x3
+ x – 1. Then f(0) = –1 < 0 and
f(1) = 1 > 0.
Since f is a polynomial, it is continuous, so the Intermediate
Value Theorem states that there is a number c between 0
and 1 such that f(c) = 0.
Thus the given equation has a root.
6. 6
Example 2 – Solution
To show that the equation has no other real root, we use
Rolle’s Theorem and argue by contradiction.
Suppose that it had two roots a and b. Then f(a) = 0 = f(b)
and, since f is a polynomial, it is differentiable on (a, b) and
continuous on [a, b].
Thus, by Rolle’s Theorem, there is a number c between a
and b such that f′(c) = 0.
cont’d
7. 7
Example 2 – Solution
But
f′(x) = 3x2
+ 1 ≥ 1 for all x
(since x2
≥ 0) so f′(x) can never be 0. This gives a
contradiction.
Therefore the equation can’t have two real roots.
cont’d
8. 8
The Mean Value Theorem
Our main use of Rolle’s Theorem is in proving the following
important theorem, which was first stated by another
French mathematician, Joseph-Louis Lagrange.
9. 9
The Mean Value Theorem
Before proving this theorem, we can see that it is
reasonable by interpreting it geometrically. Figures 3 and 4
show the points A(a, f(a)) and B(b, f(b)) on the graphs of
two differentiable functions.
Figure 3 Figure 4
10. 10
The Mean Value Theorem
The slope of the secant line AB is
which is the same expression as on the right side of
Equation 1.
11. 11
The Mean Value Theorem
Since f′(c) is the slope of the tangent line at the point
(c, f(c)), the Mean Value Theorem, in the form given by
Equation 1, says that there is at least one point P(c, f(c)) on
the graph where the slope of the tangent line is the same
as the slope of the secant line AB.
In other words, there is a point P where the tangent line is
parallel to the secant line AB.
12. 12
Example 3
To illustrate the Mean Value Theorem with a specific
function, let’s consider
f(x) = x3
– x, a = 0, b = 2.
Since f is a polynomial, it is continuous and differentiable
for all x, so it is certainly continuous on [0, 2] and
differentiable on (0, 2).
Therefore, by the Mean Value Theorem, there is a number
c in (0, 2) such that
f(2) – f(0) = f′(c)(2 – 0)
13. 13
Example 3
Now f(2) = 6,
f(0) = 0, and
f′(x) = 3x2
– 1, so this equation becomes
6 = (3c2
– 1)2
= 6c2
– 2
which gives that is, c = But c must lie in
(0, 2), so
cont’d
14. 14
Example 3
Figure 6 illustrates this calculation:
The tangent line at this value of c is parallel to the secant
line OB.
Figure 6
cont’d
15. 15
Example 5
Suppose that f(0) = –3 and f′(x) ≤ 5 for all values of x.
How large can f(2) possibly be?
Solution:
We are given that f is differentiable (and therefore
continuous) everywhere.
In particular, we can apply the Mean Value Theorem on the
interval [0, 2]. There exists a number c such that
f(2) – f(0) = f′(c)(2 – 0)
16. 16
Example 5 – Solution
so f(2) = f(0) + 2f′(c) = –3 + 2f′(c)
We are given that f′(x) ≤ 5 for all x, so in particular we
know that f′(c) ≤ 5.
Multiplying both sides of this inequality by 2, we have
2f′(c) ≤ 10, so
f(2) = –3 + 2f′(c) ≤ –3 + 10 = 7
The largest possible value for f(2) is 7.
cont’d
17. 17
The Mean Value Theorem
The Mean Value Theorem can be used to establish some
of the basic facts of differential calculus.
One of these basic facts is the following theorem.
18. 18
The Mean Value Theorem
Note:
Care must be taken in applying Theorem 5. Let
The domain of f is D = {x | x ≠ 0} and f′(x) = 0 for all x in D.
But f is obviously not a constant function.
This does not contradict Theorem 5 because D is not an
interval. Notice that f is constant on the interval (0, ) and
also on the interval ( , 0).