Question

Let \(K\left( t \right)\) be a measure of the knowledge you gain by studying for a test for \(t\) hours. Which do you think is larger, \(K\left( {\bf{8}} \right) - K\left( {\bf{7}} \right)\) or \(K\left( {\bf{3}} \right) - K\left( {\bf{2}} \right)\)? Is the graph of \(K\) concave upward or concave downward? Why?

Short answer

\(K\left( 3 \right) - K\left( 2 \right)\) is larger than \(K\left( 8 \right) - K\left( 7 \right)\) because students learn in the third hour of their study than the eight hours. The graph of \(K\) is concave downward because the rate of knowledge gain is large.

Step-by-step solution

Find which is larger \(K\left( {\bf{8}} \right) - K\left( {\bf{7}} \right)\) or \(K\left( {\bf{3}} \right) - K\left( {\bf{2}} \right)\)

As most of the students learn in the third hour of their study than the eight hour that is \(K\left( 3 \right) - K\left( 2 \right)\)then \(K\left( 3 \right) - K\left( 2 \right)\) is larger than \(K\left( 8 \right) - K\left( 7 \right)\).

Step 2: Find whether the graph of \(K\) concave upward or concave downward

As \(K\left( t \right)\) be a measure of the knowledge gain by studying for a test for \(t\) hours.

The rate of knowledge gain is large and then starts to taper off, so \(K'\left( t \right)\) decreases and the graph of \(K\) is concave downward.