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Suppose a function is continuous on $[a, b]$ and differentiable on $(a, b) .$ Additionally, suppose that $f(a)=f(b) .$ Sketch various possibilities for the graph of $\mathrm{f}$. Show from your graphs that there must be some point $c,$ where $a<c<b,$ such that the tangent line at $x=c$ is parallel to the $x$ -axis $\left(f^{\prime}(c)=0\right) .$ This result is known as Rolle 's Theorem.

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 2

The First Derivative Test

Derivatives

Campbell University

University of Nottingham

Boston College

Lectures

04:40

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

30:01

In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (the rate of change of the value of the function). If the derivative of a function at a chosen input value equals a constant value, the function is said to be a constant function. In this case the derivative itself is the constant of the function, and is called the constant of integration.

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Assume that $f$ and $g$ …

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If $f$ and $g$ are differe…

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Let $f$ and $g$ be the dif…

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Suppose that $f$ is contin…

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Rolle's Theorem Let $…

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Let $f(x)=x \arctan (1 / x…

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Let $f(x)=\left\{\begin{al…

it's suppose we have a function that's continuous and differentiable um on a to B. And then we'll have F A B equals F B. So we want to show that there must be a point C. Such that the tangent line is parallel to the X axis Or in other words at Prime FC is equal to zero and this is rule steer. Um Yeah, so let's consider a case of this, such as X squared. Um My distracts for example, and we'll consider I'm a negative X squared plus jacks. So for something like this man, and what role serum tells us is that um as long as we have a continuous differentiable function, which we do, um If we consider the fact that F. Of zero, which is zero, is equal to F. Of two, which is zero. And there has to be at some point in between see where F prime of C equals zero. And we see that's the case right here. The reason why this always holds is because we know to get from here to here, we either have to stay flat, Go like this or go like this. And in either case there's going to be a point where the slope is zero

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