Thread: Please help prove trig Iinequality with the mean value theorem

1. Please help prove trig Iinequality with the mean value theorem

Hello everyone Im new to the forum but looking foward to getting to know everyone. Im a first year calculus desperately needing help to solve this problem. I have to prove thie following inequality using the mean value theorem;

a-b <=sin (b)-sin (a)<=b-a

I appreciate any help you can provide. Thanks

2. Re: Please help prove trig Iinequality with the mean value theorem

Originally Posted by sicbagger
Hello everyone Im new to the forum but looking foward to getting to know everyone. Im a first year calculus desperately needing help to solve this problem. I have to prove thie following inequality using the mean value theorem;

$a-b\le\sin (b)-\sin (a)\le b-a$
You know that $\sin(x)$ is differentiable on every $[a,b]$.

You know that $\forall x,~-1\le \cos(x)\le 1$

Apply the mean value theorem to $\sin(x)$ on $[a,b]$

3. Re: Please help prove trig Iinequality with the mean value theorem

Thank you for the reply but this is my first semester of calc. I feel great that I understand your first sentence but your second sentence definitely lost me. Lol

4. Re: Please help prove trig Iinequality with the mean value theorem

Originally Posted by sicbagger
Thank you for the reply but this is my first semester of calc. I feel great that I understand your first sentence but your second sentence definitely lost me. Lol
Are you saying that you do not understand $\forall x,~-1\le \cos(x)\le 1~?$

If you are, then you do not understand enough Pre-Calculus to do this question.

5. Re: Please help prove trig Iinequality with the mean value theorem

I dont understand what the symbol in front of your x that looks like an upside down A is. I understand what -1 <=cos (x)<=1 means

6. Re: Please help prove trig Iinequality with the mean value theorem

Originally Posted by sicbagger
I dont understand what the symbol in front of your x that looks like an upside down A is. I understand what -1 <=cos (x)<=1 means
An upside-down A, $~\forall~$, reads "for all"; just as for a backwards E, $~\exists\;\; ,$ reads "there exists".

7. Re: Please help prove trig Iinequality with the mean value theorem

Ok awesome Ive never seen either of those before. I understand that cos lives between 1 and -1 but how does that help me on my inequality?

8. Re: Please help prove trig Iinequality with the mean value theorem

Originally Posted by sicbagger
Ok awesome Ive never seen either of those before. I understand that cos lives between 1 and -1 but how does that help me on my inequality?
The Mean Value Theorem: if $f$ is differentiable on $[a,b]$ then $\exists c\in (a,b)$ such that $f'(c)=\frac{f(b)-f(a)}{b-a}$.

Now just apply that theorem to the question.