High School Average angle made by a curve with the ##x-axis##

[Kumar8434]

The average angle made by a curve $f(x)$ between $x=a$ and $x=b$ is:
$$\alpha=\frac{\int_a^b\tan^{-1}{(f'(x))}}{b-a}$$
I don't think there should be any questions on that. Since $f'(x)$ is the value of $\tan{\theta}$ at every point, so $tan^{-1}{(f'(x))}$, should be the angle made by the curve at that point.

Now, I expected this to hold:
$$\tan^{-1}\left({\frac{f{(b})-f{(a)}}{b-a}}\right)=\alpha=\frac{\int_a^b\tan^{-1}{(f'(x))}}{b-a}$$
because, $\tan^{-1}\left({\frac{f{(b})-f{(a)}}{b-a}}\right)$ is also the 'average angle' made by the curve between $x=a$ and $x=b$.
It was true only approximately for $f(x)=\log{|\sec{x}|}$ when I checked for $a=0$ and $b=\frac{\pi}{4}$. It obviously holds for linear functions and I checked that it only approximately holds for quadratic functions. I don't know anything beyond high-school calculus, so couldn't check it for polynomials of degree greater than $2$.

I also tried root-mean-square instead of average angle but that expression too didn't hold accurately.

I took one step further and replaced $\tan^{-1}{x}$ with any function $g(x)$ and expected this to hold:
$$g\left({\frac{f{(b})-f{(a)}}{b-a}}\right)=k=\frac{\int_a^bg{(f'(x))}}{b-a}$$
But this one too only holds approximately for some $g(x)$ that I checked.
So, why don't these expressions hold as expected?

 

[jedishrfu]

arctan depends on x so you can't just pull it out of the integral over x like that.

In general, you can't pull any function dependent on x out of an integral over x.

If you changed the integral to a summation and ask yourself about this simpler example:

atan( f(x₁) + f(x₂) + f(x₃) )

- vs -

atan( f(x₁) ) + atan( f(x₂) ) + atan( f(x₃) )

 

[Dale]

So, why don't these expressions hold as expected?

Because your expectation was wrong.

 

[jedishrfu]

Have you tried using the chain rule to evaluate the righthand side to see if it matches the lefthand side:

$$\tan^{-1}(f(b)-f(a)) = \int_a^b\tan^{-1}(f'(x)) dx$$

to see if you can derive your conclusion. It may also explain why the results are different from what you expected.

https://en.wikipedia.org/wiki/Chain_rule

In particular heck out the first proof which uses limits to prove the rule.

 

[Mark44]

Thread closed. As @Dale pointed out in post #3, your expectation is wrong. In future posts, please make more of an attempt on proving your conjectures instead of asking why something doesn't work.