jgens said:Well, what does the mean value theorem say?
Mstenbach said:Could the "real values x and y" have anything to do with the (a, b) interval?
jgens said:Well, what do you think?
l'Hôpital said:How about you let [tex] x = b[/tex] and [tex] y = a[/tex]?
You're close.Mstenbach said:Hmm, ok, thank you.
So I get now
|sinb - sina|
__________ <= 1
| b - a|
Which is similar to the theorem f(b)-f(a) / b-a ?
If it is, this basically says that the equation I came up with (which is the derivative) is equal to 1?
Can I work on this further by taking the derivative and getting cosb - cosa?Sorry if I'm coming off as a bit.. stubborn. This question is really confusing me.
l'Hôpital said:You're close.
It's not exactly the derivative. I mean, it is the derivative at point c between a and b.
So, we have the equation in the form of
[tex]
\frac{\sin b - \sin a}{b - a} = \cos c
[/tex]
As for your derivative idea, no. Derivative of sin a would be 0 because sin a is a constant. Instead, consider the following:
What do we know about [tex]\cos[/tex]'s range?
Mstenbach said:Aha! I see it now. The range of cos is between 1 and -1, so it cannot exceed 1... I think I see that now.
Just one question, how did you find "cosc"?
Thank you very much.
The Mean Value Theorem is a fundamental theorem in calculus that states that for a continuous and differentiable function on a closed interval, there exists at least one point where the slope of the tangent line is equal to the average rate of change of the function over that interval.
In proving |sinx-siny| ≤ |x-y|, the Mean Value Theorem is used to establish the existence of a point where the derivative of the function f(x) = sinx is equal to the average rate of change of the function over the interval [x,y]. This point can then be used to prove the inequality.
Proving |sinx-siny| ≤ |x-y| is important because it allows us to establish the relationship between the absolute value of the difference of two trigonometric functions and the absolute value of their argument. This relationship is useful in solving many problems in calculus and real analysis.
The key steps in proving |sinx-siny| ≤ |x-y| are:
1. Restate the problem in terms of the Mean Value Theorem.
2. Use the Mean Value Theorem to establish the existence of a point c where f'(c) = [f(b)-f(a)]/(b-a).
3. Simplify the equation using trigonometric identities.
4. Use the properties of absolute value to eliminate any negative signs.
5. Substitute the values of a and b to obtain the final inequality.
Yes, the Mean Value Theorem can be used to prove other trigonometric inequalities. It is a powerful tool in establishing relationships between the values of trigonometric functions and their arguments. However, it is important to note that the conditions of the Mean Value Theorem must be satisfied for it to be applied in proving any inequality.