Uploaded by TheIntegralCALC on 27.06.2011

Transcript:

Hi, everyone! Welcome back to integralcalc.com. Today we’re going to be doing a Rolle’s

theorem problem. And in this one, we’ve been asked to prove that the function f(x)

= x^2 – 2x satisfies the conditions of Rolle’s theorem on the range 0 to 2.

First of all, let’s talk about what it means to satisfy Rolle’s theorem. Rolle’s theorem

has three premises and then a conclusion so it says that if three things are true, then

a fourth thing must be true. So let’s go ahead and talk about what that is. Rolle’s

theorem tells that if a function is continuous over a certain range, in this case our range

is 0 to 2, if it is differentiable meaning that you can differentiate it on the entire range 0 to 2, it has a derivative

and three, if f(a) = 0 = f(b). If these three things are true, then it means that at some

point c on the range a to b, it means that the derivative at the point c is going to

be equal to zero. Let’s talk about what this means and we can pull up the graph of

this particular function. So what this means is that in order for this

function to satisfy Rolle’s theorem on the range zero to 2, the function is continuous

over the range 0 to 2. So we’ve got zero here and 2. So the function is continuous

over this entire range. And we can see pretty much just by looking at it that it is continuous.

Remember that continuity means that there’s no breaks in the graph, there’s no holes

in it. So this function is continuous on the range 0 to 2; that it’s differentiable which

in this case the function is because there’s no vertical asymptote and there’s no break

or point like an absolute value or something like that. It’s differentiable. This third

condition here that f(a) = 0 = f(b) means that at the endpoints 0 and 2, the function

is equal to zero;, meaning that it’s touching the x axis at that point; that it’s equal

to zero. So the endpoints of this range have to be right on the x axis. So that’s what

that third condition means. And what Rolle’s theorem is saying is if all three of those

things are true, then there’s got to be some point c within the range between the endpoints that the derivative

of the function is going to be equal to zero which means that the tangent line at that

point is going to be horizontal. So that’s all that that means. We’ve got c here. The

point c corresponds to this point here in the graph, 1. And if we draw the tangent line

through that point, we can see that the tangent line is horizontal.

So basically, just to recap, all Rolle’s theorem is saying is that if you’ve got

a range a to b, if the function is continuous and if it’s differentiable and if the endpoints

are on the x axis, then there’s going to be some point inside the range where the tangent

line of the function is horizontal or the slope of the tangent line is equal to zero.

This is what this is saying here; that the slope of the tangent line is equal to zero.

So in order to prove that this function satisfies this theorem, here’s what we’ll do. We

will take the derivative of f(x). So that’s how you prove it. We take the derivative of

f(x) and of course when we do that you get 2x – 2 and then you’re going to set that

equal to zero and solve for x. So we say 0 = 2x – 2. We add 2 to both sides to get

2 = 2x and then we divide both sides by 2 and we get 1 = x. So that’s the point c

in the range a to b where the tangent line is horizontal. We know that that’s the point

but we need to prove that it’s continuous. Remember that continuity is going to exist

unless there is a point in the graph of which it’s undefined. So if your function is a

fraction, it would be the point at which the denominator would be equal to zero that the

function would be discontinuous. If you’ve got a vertical asymptote, if you have a square

root sign and if there are negative values within the square root sign, then you might

have discontinuity in your function. In this case, none of those things exist for us so

we know that the function is continuous. We also know it’s differentiable and we know

that there are no holes, there are no vertical asymptotes; there’s no point in the graph

at which it breaks. You can take your pencil and draw in one continuous motion without

picking your pencil up. So we know that it’s also differentiable. And we know that f(a)

= 0 = f(b) and we can prove that by plugging in the endpoints of our range zero and 2.

So if we plug in 0 and 2 to the original function, we’d get f(0) = 0^2 – 2(0). So that is

equal to zero. And if we plug in 2, we’d get 4 – 4 which is also going to be equal

to zero. So we’ve now proved that both of the endpoints a and b are equal to zero. So

that condition is satisfied. So we know all three conditions are satisfied, which proves

the existence of a point c. And in order to find that point, we just take the derivative,

set it equal to zero and solve it for x and we know that there is a point x = 1 where

the tangent line is horizontal and again, if we look at our graph, we can see that that

point is right here at x = 1. The tangent line at that point of the graph is horizontal.

So that’s it. I hope this video helped you guys and I will see you in the next one. Bye!

theorem problem. And in this one, we’ve been asked to prove that the function f(x)

= x^2 – 2x satisfies the conditions of Rolle’s theorem on the range 0 to 2.

First of all, let’s talk about what it means to satisfy Rolle’s theorem. Rolle’s theorem

has three premises and then a conclusion so it says that if three things are true, then

a fourth thing must be true. So let’s go ahead and talk about what that is. Rolle’s

theorem tells that if a function is continuous over a certain range, in this case our range

is 0 to 2, if it is differentiable meaning that you can differentiate it on the entire range 0 to 2, it has a derivative

and three, if f(a) = 0 = f(b). If these three things are true, then it means that at some

point c on the range a to b, it means that the derivative at the point c is going to

be equal to zero. Let’s talk about what this means and we can pull up the graph of

this particular function. So what this means is that in order for this

function to satisfy Rolle’s theorem on the range zero to 2, the function is continuous

over the range 0 to 2. So we’ve got zero here and 2. So the function is continuous

over this entire range. And we can see pretty much just by looking at it that it is continuous.

Remember that continuity means that there’s no breaks in the graph, there’s no holes

in it. So this function is continuous on the range 0 to 2; that it’s differentiable which

in this case the function is because there’s no vertical asymptote and there’s no break

or point like an absolute value or something like that. It’s differentiable. This third

condition here that f(a) = 0 = f(b) means that at the endpoints 0 and 2, the function

is equal to zero;, meaning that it’s touching the x axis at that point; that it’s equal

to zero. So the endpoints of this range have to be right on the x axis. So that’s what

that third condition means. And what Rolle’s theorem is saying is if all three of those

things are true, then there’s got to be some point c within the range between the endpoints that the derivative

of the function is going to be equal to zero which means that the tangent line at that

point is going to be horizontal. So that’s all that that means. We’ve got c here. The

point c corresponds to this point here in the graph, 1. And if we draw the tangent line

through that point, we can see that the tangent line is horizontal.

So basically, just to recap, all Rolle’s theorem is saying is that if you’ve got

a range a to b, if the function is continuous and if it’s differentiable and if the endpoints

are on the x axis, then there’s going to be some point inside the range where the tangent

line of the function is horizontal or the slope of the tangent line is equal to zero.

This is what this is saying here; that the slope of the tangent line is equal to zero.

So in order to prove that this function satisfies this theorem, here’s what we’ll do. We

will take the derivative of f(x). So that’s how you prove it. We take the derivative of

f(x) and of course when we do that you get 2x – 2 and then you’re going to set that

equal to zero and solve for x. So we say 0 = 2x – 2. We add 2 to both sides to get

2 = 2x and then we divide both sides by 2 and we get 1 = x. So that’s the point c

in the range a to b where the tangent line is horizontal. We know that that’s the point

but we need to prove that it’s continuous. Remember that continuity is going to exist

unless there is a point in the graph of which it’s undefined. So if your function is a

fraction, it would be the point at which the denominator would be equal to zero that the

function would be discontinuous. If you’ve got a vertical asymptote, if you have a square

root sign and if there are negative values within the square root sign, then you might

have discontinuity in your function. In this case, none of those things exist for us so

we know that the function is continuous. We also know it’s differentiable and we know

that there are no holes, there are no vertical asymptotes; there’s no point in the graph

at which it breaks. You can take your pencil and draw in one continuous motion without

picking your pencil up. So we know that it’s also differentiable. And we know that f(a)

= 0 = f(b) and we can prove that by plugging in the endpoints of our range zero and 2.

So if we plug in 0 and 2 to the original function, we’d get f(0) = 0^2 – 2(0). So that is

equal to zero. And if we plug in 2, we’d get 4 – 4 which is also going to be equal

to zero. So we’ve now proved that both of the endpoints a and b are equal to zero. So

that condition is satisfied. So we know all three conditions are satisfied, which proves

the existence of a point c. And in order to find that point, we just take the derivative,

set it equal to zero and solve it for x and we know that there is a point x = 1 where

the tangent line is horizontal and again, if we look at our graph, we can see that that

point is right here at x = 1. The tangent line at that point of the graph is horizontal.

So that’s it. I hope this video helped you guys and I will see you in the next one. Bye!