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Numerade Educator



Problem 67 Hard Difficulty

A manufacturer of lightbulbs wants to produce bulbs that last about 700 hours but, of course, some bulbs burn out faster than others. Let $ F(t) $ be the fraction of the company's bulbs that burn out before $ t $ hours, so $ F(t) $ always lies between 0 and 1.
(a) Make a rough sketch of what you think the graph of $ F $ might look like.
(b) What is the meaning of the derivative $ r(t) = F^\prime (t) $?
(c) What is the value of $ \displaystyle \int_0^\infty r(t)\ dt $? Why?


b. $r(t)=F^{\prime}(t)$ is the rate at which the fraction $F(t)$ of burnt-out bulbs increases as t increasees. This could be interpreted as a fractional burnout rate.
c. 1


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Video Transcript

The problem is, a manufacturer of light bombs wants to produce bombs that last about seven hundred hours, that of cast. Some bums run out faster than others. But after Bebe is a fraction off the company's bombs that bring out forty hours. So after he always last between zero and one a midget Roth's catch of what you think the graph of half might look like for a rest notified After Errol, it's zero things no bombs bring out before zero hours and what he is, whether large we have one after you zero if they go to what and after, he should be increasing function. So, Graham, it looks like this. This is one from zero to one and two hours after you, I'd be what is many half that derivative after prom? That means the writ reed, at which the fraction after increases at tio increases I see what is a while you're off integral from zero to infinity. Tt Why better set by the magisterial more calculus to we have integral. Is there a tree? Infinity? Artie, this is the limit. A cost to your vanity. You have to be minus zero. Destiny they thought was the limit in costume. Bennett E A from part a half. This is a country. Why? Reason is our bombs. Well, bring out you weren't really