이계도함수와 오목·볼록

(a) We will use central differences to estimate the requested derivatives. Thus, $$\begin{aligned}h'(4.5) &\approx \frac{h(5)-h(4)}{5-4} = \frac{69.8 - 55.5}{1} = 14.3 \\ h'(5) &\approx \frac{h(5.5)-h(4.5)}{5.5-4.4} = \frac{81.6 - 60.4}{1} = 21.2 \\ h'(5.5) &\approx \frac{h(6)-h(5)}{6-5} = \frac{93.7 - 69.8}{1} = 23.9\end{aligned}$$ The bungee jumper is rising most rapidly at $t = 5.5$, at a rate of about $23.9$ feet per second. (b) We again use a central difference, this time to estimate $h''(5)$. Observe that $$h''(5) \approx \frac{h'(5.5) - h'(4.5)}{5.5-4.5} = \frac{23.9 - 14.3}{1} = 9.6$$ (c) $h''(5)$ measures the acceleration of the bungee jumper at $t = 5$. Thus we know that the person's acceleration at $t = 5$ is about $9.6$ feet per second per second. This measures the instantaneous rate of change of the jumper's velocity. (d) The bungee jumper's position function is concave down on approximately the intervals $0 < t < 2$, $6 < t < 10$. This is easiest to see if we plot the data using a program like Excel. On these time intervals, the velocity of the jumper is decreasing: either the velocity is positive and getting less positive, or negative and getting more negative.

(a) When a function is increasing, its graph rises as $x$ increases, and when a function is concave up, that means the graph of the function looks bowl-shaped. A graph of an increasing, concave up function $f$ on the interval $(-3,0)$ is shown in the following figure. When a function is concave down, its graph looks like an upside down bowl. The function $f$ in the figure is increasing and concave down on the interval $(0,3)$ Note that $f$ is also increasing on the entire interval $(-3,3)$. (b) When a function is increasing, its graph rises as $x$increases, and when a function is concave down, that means the graph of the function looks like an upside down bowl. A graph of an increasing, concave down function $g$ on the interval $(-3,0)$ is shown in the figure below. When a function is concave up, its graph looks like a bowl. The function $g$ in the figure is increasing and concave up on the interval $(0,3)$. Note that $g$ is also increasing on the entire interval $(-3,3)$. (c) When a function is decreasing, its graph falls as $x$ increases, and when a function is concave up, that means the graph of the function looks bowl-shaped. A graph of a decreasing, concave up function $h$ on the interval $(-3,-1)$is shown in the figure below. To be neither concave up or concave down, a graph has to be linear, so the graph of $h$ in the following figure is a straight line on $(-1,1)$. When a function is concave down, its graph looks like an upside down bowl. The function $h$ in the figure is decreasing and concave down on the interval $(1,3)$. Note that $h$ is decreasing on the entire interval $(-3,3)$. (d) When a function is decreasing, its graph falls as $x$ increases, and when a function is concave down, that means the graph of the function looks like an upside down bowl. A graph of a decreasing, concave down function $p$ on the interval $(-3,0)$ is shown in the figure below. When a function is increasing, its graph rises as $x$ increases. The function $p$ in the figure is increasing and concave down on the interval $(0,3)$.