Chapter 2.8, Problem 13E

To determine

To find:The graph of $K\left(t\right)$ is concave upwardand concave downward.

Explanation of Solution

Given: $K\left(t\right)$ be a measure of the knowledge that you gain by studying for a test for $t$ hours.

$K\left(8\right)-K\left(7\right)$ = Knowledge gain during 1 hour of study after 7 hours.

  $K\left(3\right)-K\left(2\right)$ = Knowledge gain during 1 hour of study after 2 hours.

Ability to gain knowledge decreases as time increases because our brain becomes tired.

That means our ability to learn further will be less after 7 hours

Therefore

  $K\left(3\right)-K\left(2\right)$ $>K\left(8\right)-K\left(7\right)$

There is only finite amount of knowledge that can be gained

  $K^{\text{'}}\left(t\right)$ is decreasing function and approaches 0 as $t\to \infty$ .

Since $K^{\text{'}}\left(t\right)$ is decreasing, it can say $K^{\text{'}}\left(t\right)$ , hence graph is concave down