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klio [65]
3 years ago
11

Can you please tell me what’s wrong in this plot? I can’t see what.

Mathematics
1 answer:
Readme [11.4K]3 years ago
7 0

Answer:

The Upper Quartile would not be 59.  It would be 60.5.

Step-by-step explanation:

The Upper Quartile would not be 59.  It would be 60.5.

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slava [35]
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Taking x=7\tan\theta gives \mathrm dx=7\sec^2\theta\,\mathrm d\theta, so that the integral becomes

\displaystyle\int\frac{(7\tan\theta)^3}{\sqrt{(7\tan\theta)^2+49}}(7\sec^2\theta)\,\mathrm d\theta
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=\displaystyle7^3\int\frac{\tan^3\theta\sec^3\theta}{\sqrt{\tan^2\theta+1}}\,\mathrm d\theta
=\displaystyle7^3\int\frac{\tan^3\theta\sec^3\theta}{\sqrt{\sec^2\theta}}\,\mathrm d\theta
=\displaystyle7^3\int\frac{\tan^3\theta\sec^3\theta}{|\sec\theta|}\,\mathrm d\theta

When \sec\theta>0, we have

=\displaystyle7^3\int\frac{\tan^3\theta\sec^3\theta}{\sec\theta}\,\mathrm d\theta
=\displaystyle7^3\int\tan^3\theta\sec^2\theta\,\mathrm d\theta

and from here we can substitute u=\tan\theta to proceed from here.

Quick note: When we set x=7\tan\theta, we are implicitly enforcing -\dfrac\pi2 just so that the substitution can be undone later via \theta=\tan^{-1}\dfrac x7. But note that over this domain, we automatically guarantee that \sec\theta>0, so the absolute value bars can be dropped immediately.
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Step-by-step explanation:

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