## Calculus: Early Transcendentals 8th Edition

(a) $f'(3)=2$ and $f''(3)=4$ First, $f'(3)=2\gt0$, so the graph of $f(t)$ is increasing, and the temperature is rising. (you feel a part of yourself dead already). $f''(3)=4\gt0$, the point where $t=3$ is inside a concave upward. Combining this fact with $f$ is increasing, now you know the temperature would be rising even faster and faster in the time ahead. Of course you can judge for yourself how you would feel. (b) $f'(3)=2$ and $f''(3)=-4$ First, $f'(3)=2\gt0$, so the graph of $f(t)$ is increasing, and the temperature is rising. $f''(3)=-4\lt0$, the point where $t=3$ is concave downward. Combining this fact with $f$ is increasing, you know the temperature would be rising, but with a decreasing rate compared with before. The feeling is saved for yourself. (c) $f'(3)=-2$ and $f''(3)=4$ First, $f'(3)=-2\lt0$, the graph of $f(t)$ is decreasing, so the temperature is decreasing. $f''(3)=4\gt0$, the point where $t=3$ concave upward. Combining this fact with $f$ is decreasing, you know the temperature would still decrease, but slower and slower in the time ahead. (d) $f'(3)=-2$ and $f''(3)=-4$ First, $f'(3)=-2\lt0$, the graph of $f(t)$ is decreasing, so the temperature is decreasing. $f''(3)=-4\lt0$, the point where $t=3$ is concave downward. Combining this fact with $f$ is decreasing, you know the temperature would decrease faster and faster from now on. Models of graphs of all 4 cases are depicted in the image below.