Suppose that $f(t)$ is the temperature at time $t$ where you live and when at $t = 3$ you feel uncomfortably hot. What will you feel on the following conditions?
$
\begin{equation}
\begin{aligned}
\text{a.) } f'(3) &= 2, &&& f''(3) &= 4\\
\text{b.) } f'(3) &= 2, &&& f''(3) &= -4\\
\text{c.) } f'(3) &= -2, &&& f''(3) &= 4\\
\text{d.) } f'(3) &= -2, &&& f''(3) &= -4\\
\end{aligned}
\end{equation}
$
a.) Since $f'(3) > 0$, it means that the temperature is rising and if $f''(3) > 0$, it means that the temperature is not leveling off at any time. If I already feel hot, I might die on a heat stroke at some time.
b.) Since $f'(3) > 0$, it means that the temperature is rising. However, $f''(3) < 0$ it means that the temperature is about to level off. So I will feel hot for only a short period of time.
c.) Since $f'(3) < 0$, it means that the temperature is dropping and if $f''(3) > 0$, it means that the temperature is going to increase. So it looks like I will feel uncomfortable only for a short period of time.
d.) Since $f'(3) < 0$ and $f''(3) < 0$, the temperature is always decreasing. So I might suffer from Hypothermia at some time.
Monday, December 23, 2013
Single Variable Calculus, Chapter 4, 4.3, Section 4.3, Problem 50
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