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Suppose that a function satisfies the same conditions as in the previous exercise, except that $f(a) \neq f(b)$. One such function is illustrated in Figure 12 Show that if the figure is rotated so that the dotted line joining $A(a, f a)$ ) to $B(b, f b))$ is horizontal, then the rotated figure satisfies all the conditions of Rolle's Theorem. Therefore, there must be at least one point $c, a<c<b$ such that the tangent line at $x=c$ is parallel to the line joining $A$ to $B$. Thus, show that $f^{\prime}(c)=\frac{f(b)-f(a)}{b-a} .$ This result is known as the Mean Value Theorem, or The Law of the Mean.

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 2

The First Derivative Test

Derivatives

Harvey Mudd College

University of Michigan - Ann Arbor

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.

04:03

Using Rolle's Theorem…

01:16

Suppose a function is cont…

09:29

Suppose $f$ is differentia…

05:13

Determine whether the Mean…

04:52

(a) Let $f(x)=x^{2}$ and $…

01:10

02:57

Illustrate the Mean-Value …

Yeah, suppose that a function satisfies the same conditions of the previous exercise, but no F. Of A does not equal F. A. B. Um Now we see that There must be one point, see between A. And B. Such as the tangent line actually will see his parallel to the line joining A. And B. We want to show that act primacy is equal to five, Okay. And it's half of a. We'll be 99. What this is asking us to show is that the average rate of change is equal to the instantaneous rate of change at some point. And we know this to be true because This is the mean value theorem. And it explains why if a car averages 30 mph on a drive, it must be that there is at least one point. I wish the car was actually driving 30 mph, so it's gonna be our final answer.

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