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Problem 68 Hard Difficulty

Let $ f(x) $ be the temperature at time $ t $ where you live and suppose that at time $ t = 3 $ you feel uncomfortably hot. How do you feel about the given data in each case?

(a) $ f'(3) = 2 $, $ f"(3) = 4 $
(b) $ f'(3) = 2 $, $ f"(3) = -4 $
(c) $ f'(3) = -2 $, $ f"(3) = 4 $
(d) $ f'(3) = -2 $, $ f"(3) = - 4 $

Answer

a) From the graphs above, we see that $f(x)$ is
concave upward on $(-1.8,0.3) \cup(1.5, \infty)$ and
concave downward on $(-\infty,-1.8) \cup(0.3,1.5)$
b) From the graphs above, we see that $f(x)$ is
concave upward on $(-1.8,0.3) \cup(1.5, \infty)$ and
concave downward on $(-\infty,-1.8) \cup(0.3,1.5)$
c) From the graphs above, we see that $f(x)$ is
concave upward on $(-1.8,0.3) \cup(1.5, \infty)$ and
concave downward on $(-\infty,-1.8) \cup(0.3,1.5)$
d) From the graphs above, we see that $f(x)$ is
concave upward on $(-1.8,0.3) \cup(1.5, \infty)$ and
concave downward on $(-\infty,-1.8) \cup(0.3,1.5)$

Discussion

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Video Transcript

is the temperature and x. Is the time now at f of three F of three is the temperature at three hours? You feel uncomfortably hot. So at three hour mark, uh the temperature f is Very high for you. You feel high temperature, you feel uncomfortably hot at F of three. No, if f prime of three is positive too, what does that mean? We'll look at f double prime of uh three being positive for in a second. But if F prime of three equals two, what does that mean? Well, if F prime, the derivative is positive, that means to function is increasing. So F prime of three equaling to a positive to that means here we are at the three hour mark already feeling uncomfortably hot, but F prime of three is positive, meaning the function is increasing, meaning the temperature is going to get even hotter uh after we move past three hours. So we're uncomfortably hot. Now we're going to get really uncomfortably hot. Now, f double prime at three is 4. Let's see if I can show this little bit. No, if F double prime is positive which it is here in part A. That means F prime is increasing. Now, if prime is the rate at which F is increasing, the temperature is increasing. So if that double prime is positive, like it is here, that means the rate at which F is increasing is going to increase. So not only is it getting hotter after the three hour mark, but the rate at which is going to get higher is going to keep increasing. So a is not a good situation for you. If you already are uncomfortable at the three hour mark. Second, I just want to try to shoot a function here. Yeah. Okay, now let's look at B. That prime at three is too remember At the three hour mark, f the temperature is uncomfortably hot for you. F. Prime looking at B. F. Prime at at the three hour mark is positive. If that crime is positive, that means the function F is increasing. So F is going to be increasing uh in part B because the first derivative is positive, is positive. So if you're already uncomfortably hot At three hours, uh if crime is positive, meaning the function is going to increase, meaning it's going to get hotter after the three hour market. So you're already uncomfortable, you're gonna get more uncomfortable. Now the second derivative is negative. The second derivative being negative means F prime is starting to decrease. So even though the first derivative is too, meaning the rate at which the temperature is going to increase with respect to time increasing is uh to uh the second derivative being negative. This means that uh the rate at which uh f prime uh is changing F crime is going to be decreasing. So not to confuse you. Okay, F F €3 uncomfortably hot, looking at part B. F prime at three is a positive to the fact crime is positive. That means your functions increasing. So F F three, the temperature at three hours you're already hot, the first derivative is positive. That means the function is going to increase. That means the temperature f is going to increase, its going to get hotter, you're going to be more uncomfortable. But the second derivative being negative uh means that even though the function the temperature is increasing, the rate at which uh the temperature is increasing is going to start slowing down. Uh That's what the second derivative being negative is. So in the short term you're still going to be getting hotter and more uncomfortable but it won't last long. Uh second derivative being negative, it means the rate at which your function is getting hotter is going to start slowing down. Let's look at sea here, we finally have a f prime of three being a negative number. If f prime is negative, your function is decreasing. So f of three. The temperature at the three hour mark, you feel uncomfortably hot but F prime uh at three is negative. Two. If if F prime is negative, that means your function is decreasing. So even though ff three, you're very hot, you're uncomfortably hot because f prime is negative. Your function is starting to decrease, meaning temperature is going to start coming down. So you might be uncomfortably hot at f of three. But with the F prime being negative, the temperature is coming down because f is decreasing when it's first derivative is negative. So, if f is decreasing temperatures coming down, you're gonna feel better. Uh F double prime is positive. Uh That means to rate um the f crime. Uh F prime itself is increasing. So let's look at that if prime is negative, meaning F is decreasing the temperature's coming down. But F double prime is positive. If F double prime is positive, that means f prime is increasing. So even though F prime is negative, which is good, uh F double prime being positive means your f prime function is going to increase. So it's not going to stay down at negative two. It might even, you know, start increasing, get close to zero and increase and start being positive begin, which means F would start increasing and getting hotter again. All right, let's look at deep. Uh one more time F of three to temperature to three hour mark, you feel uncomfortably hot. Part d f prime of three is negative. That's a good thing. If that crime is negative, that means your function is decreasing. So, even though you're uncomfortably hot at the three hour mark, uh F prime is negative, meaning f the temperature is decreasing. The temperature function, F is decreasing because it's derivative is negative. So you might be uncomfortably hot at the three hour mark. But f prime is negative, meaning the f function the temperature is decreasing. So, if you're uncomfortable at the three hour mark, at least the temperature now is going down and you should start feeling better. Uh F double prime is negative. If F double prime is negative, the f prime is decreasing, so f prime is already negative. F double prime uh Being negative means F prime is going to become uh is going to continue decrease, which means it's going to be more negative, meaning the rate at which your temperature is coming down uh is gonna, the temperature is going to come down faster and faster. So this is your best scenario. Uh you're uncomfortably hot At F of three. F prime is negative, meaning the temperature function f is decreasing. So the temperature's coming down, you're going to feel better, but double prime is negative, meaning F prime is going to decrease. So the more F prime decreases, the more negative it gets uh faster that your temperature is coming down. So d is your happiest.

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