Rolle's Theorem

ABSTRACT
This paper gives the details about the rolles theorem.. Indian mathematicianbhaskara (1114–1185) is credited with knowledge of Rolle's theorem.Although the theorem is named afterMICHEL ROLLE. Rolle's 1691 proof covered only the case of polynomial functions. His proof did not use the methods of differential calculus, which at that point in his life he considered to be fallacious. The theorem was first proved bycauchyin 1823 as a corollary of a proof of the mean value theorem.

KEYWORDS
·Rolle’s theorem
·Continuous/continuity
·Differentiability
·Velocity of object
·Sunrise and sunset

USE OF ROLLES THEOREM
Rolle's Theorem is about functions, and so a theorem about processes represented by functions, an affirmation among other things about the coordination of time and space. “Rolle's Theorem establishes a connection between continuity and differentiability. Continuity guarantees a maximum; differentiability delivers a number. Fermat's Theorem [which says that if f has a local extreme at c and if f'(c) exists, then f'(c) = 0] supplies the connection. The knowledge components required for the understanding of this theorem involve limits, continuity, and differentiability. The proof of the theorem is given using the Fermat’s Theorem and the Extreme Value Theorem, which says that any real valued continuous function on a closed interval attains its maximum and minimum values.

INTRODUCTION
Advanced mathematical concepts are characterized by complex interactions between intuitive and rigorous reasoning processes (Weber & Alcock, 2004). Learning calculus, which involves processes pertaining to advanced mathematical thinking, has been a subject of extensive research. One of the significant conclusions arising out of this research is that students typically develop routine techniques and manipulative skills rather than an understanding of theoretical concepts (Berry & Nymann, 2003; Davis & Vinner, 1986; Ervynck, 1981; Parameswaran, 2007; Robert, 1982; Sierpinska, 1987). The subject of calculus is rich in abstraction and calls for a high level of conceptual understanding, where many students have difficulties. Ferrini-Mundy and Graham (1991) argue that students’ understanding of central concepts of calculus is exceptionally primitive

Rolle's theoremorRolle's lemmaessentially states that any real-valued differential functionthat attains equal values at two distinct points must have at least one stationary pointsomewhere between them—that is, a point where the first derivative (the slope of the tangent line to the graph of the function) is zero.

f areal-valued functionfiscontinuouson a properclosed interval[a,b],differentiableon theopen interval(m,n), andf(m)=f(n), then there exists at least onecin the open interval (m,n) such that

f'(c)=0
f is continuous on [a, b] therefore assumes absolute max and min values on [a, b]. These can only occur at :

1. Points on [m,n] where f(x)=0 doesn’t exist.
2. The end points m&n.
3. Some internal point s where f(s) = 0
1. Void by hypothesis (f is continuous).
2. If either the end points m&n are a max or min then f is a constant function
and s can be taken anywhere in [m,n].
3. If a max or min occurs at some internal point s in [m,n] then
F’(s) = 0
and we have a point for the theorem

STEPS TO SOLVE
Step 1: Determine if the function is continuous.
Step 2: Figure out if the function is differentiable.
Step 3: Check that the derivative is continuous
Step 4: Put the given x-values into the given formula
Step 5: Set the first derivative formula

APPLICATION
Rolle's theorem can be used to show that a function has a horizontal tangent line inside an interval. If we can show that a function is continuous over an interval, differentiable over the same interval, and that the function has the same value at the endpoints of the interval, then we can use Rolle's theorem.

Since Rolle's theorem asserts the existence of a point where the derivative vanishes,One way to illustrate the theorem in terms of a practical example is to look at the calendar listing the precise time for sunset each day. One notices that around the precise date in the summer when sunset is the latest, the precise hour changes very little from day to day in the vicinity of the precise date. This is an illustration of Rolle's theorem because near a point where the derivative vanishes, the function changes very little.

if the average speed during a journey from A to B was say 50kms/hour then there had to be a time when your instantaneous speed was 50kms/hour as well.

The beauty of this theorem also reveals itself in its connection with real life. A ball, when thrown up, comes down and during the course of its movement, it changes its direction at some point to come down. Rolle’s Theorem thus can be used to explain that the velocity of the ball which is thrown upwards must become zero at some point

APPLICATION
Rolle's theorem can be used to show that a function has a horizontal tangent line inside an interval. If we can show that a function is continuous over an interval, differentiable over the same interval, and that the function has the same value at the endpoints of the interval, then we can use Rolle's theorem.

Since Rolle's theorem asserts the existence of a point where the derivative vanishes,One way to illustrate the theorem in terms of a practical example is to look at the calendar listing the precise time for sunset each day. One notices that around the precise date in the summer when sunset is the latest, the precise hour changes very little from day to day in the vicinity of the precise date. This is an illustration of Rolle's theorem because near a point where the derivative vanishes, the function changes very little.

if the average speed during a journey from A to B was say 50kms/hour then there had to be a time when your instantaneous speed was 50kms/hour as well.

The beauty of this theorem also reveals itself in its connection with real life. A ball, when thrown up, comes down and during the course of its movement, it changes its direction at some point to come down. Rolle’s Theorem thus can be used to explain that the velocity of the ball which is thrown upwards must become zero at some point.