평균변화율과 미분계수

(a) Using the central difference formula, we find that $$F'(10) \approx \frac{F(10.01) - F(9.99)}{0.02} = \frac{(75 + 110 e^{-0.05\cdot 10.01}) - (75 + 110 e^{-0.05\cdot 9.99})}{0.02} \approx -3.33592\,.$$ (b) Since $F$ is measured in degrees Fahrenheit and $t$ in minutes, we have $F'(10) \approx -3.33592$ degrees F per minute. This means that at the instant $t=10$, the temperature of the coffee is decreasing at an instantaneous rate of about $3.33592$ degrees per minute. Said differently, in the next minute, we'd expect the temperature of the coffee to drop about $3.33592$ degrees. (c) We expect the coffee's temperature to decrease quickly early on when the difference between its temperature and the surroundings is greatest, and as time goes on for the coffee's temperature to approach that of the surrounding room. Keeping in mind that $F$ is an always decreasing function so that $F'$ is always negative, we observe that $F'(10)$ should be less (more negative) than $F'(20)$. These observations can also be confirmed by plotting a graph of $F$. (d) We have noted that $F'$ is always negative, and that we expect $F'$ to get closer and closer to $0$ as time goes on (that is, $F'$ is increasing). As such, we expect $F'$ to look something like the graph of $y = -e^{-x}$. This is reasonable since $-e^{-x}$ is also an exponential function, is always negative, is always increasing, and approaches $0$ as $x$ increases.