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

• B
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?

Last edited:

jedishrfu
Mentor
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.

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

- vs -

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

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

jedishrfu
jedishrfu
Mentor
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.

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Mark44
Mentor