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Omega_Prime
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Been doing some calculus review to knock the rust off for this coming fall semester and I got stuck...
From Stewart's book (Early Transcendentals: 6E): (7.8 pg517 #69)
Determine how large the number "a" has to be so that:
[itex]\int[/itex][itex]\stackrel{\infty}{a}[/itex][itex]\frac{1}{x^{2}+1}[/itex]dx <.001
None.
Ok, I can easily picture the graph and the area under it. I figured I'd integrate, use "a" for my lower bound and "t" for the upper bound, then by using the potential equation it's just a simple matter of solving for "a" while taking the limit of said equation as t goes to infinity.
I managed to get:
arctan (t) - arctan (a) < 1/1000 (I suck at "latex" but this should technically be the limit of those arctans as t -> infinity < .001)
Here is where I think I'm screwing up... I take the tangent of both sides:
tan [arctan t -arctan a] < tan (1/1000) - and I'm stuck, I know I can't just apply the tangent function independently to both parameters giving me t - a < tan (.001) is there some trig identity I'm not thinking of..?
Homework Statement
From Stewart's book (Early Transcendentals: 6E): (7.8 pg517 #69)
Determine how large the number "a" has to be so that:
[itex]\int[/itex][itex]\stackrel{\infty}{a}[/itex][itex]\frac{1}{x^{2}+1}[/itex]dx <.001
Homework Equations
None.
The Attempt at a Solution
Ok, I can easily picture the graph and the area under it. I figured I'd integrate, use "a" for my lower bound and "t" for the upper bound, then by using the potential equation it's just a simple matter of solving for "a" while taking the limit of said equation as t goes to infinity.
I managed to get:
arctan (t) - arctan (a) < 1/1000 (I suck at "latex" but this should technically be the limit of those arctans as t -> infinity < .001)
Here is where I think I'm screwing up... I take the tangent of both sides:
tan [arctan t -arctan a] < tan (1/1000) - and I'm stuck, I know I can't just apply the tangent function independently to both parameters giving me t - a < tan (.001) is there some trig identity I'm not thinking of..?