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Question 6 for 06/08 HW

tyoung4

I had quite a bit of trouble on question 6 for last night’s homework… not really sure why but I tried it a bunch (8 to be exact) of different ways and couldn’t quite seem to crack it.

The question was:

\int_5^7 \left(\frac{d}{dt}\sqrt{5 + 3 t^4}\right)\, dt

I also wasn’t sure whether you were supposed to multiply by d/dt (the derivative) or if it was asking you to take the derivative first, inside of the integral and then integrate.

Thanks for the help!

mark

First off, d/dt is an operator and when we prepend it to an expression, we aren’t indicating multiplication we’re indicating that we should take the derivative. For example,

\frac{d}{dt} t^2 = 2t.

Now, to evaluate your integral, I guess you could take the derivative of \sqrt{5+3t^4} and then integrate, but the Fundamental Theorem of Calculus tells us you should land right back where you started. Thus, I believe the answer should be

\sqrt{5+3(7^4)} - \sqrt{5+3(5^4)}.

tyoung4

wow. I did this in much more complicated ways. Thanks!