Question

The president announces that the national deficit is increasing, but at a decreasing rate. Interpret this statement in terms of a function and its first and second derivatives.

Short answer

As the deficit is increasing at the time of the speech, \[D'\left( t \right) > 0\] and the second derivative of deficit, \[D''\left( t \right)\] decreasing since the rate is increasing.

Step-by-step solution

Define the first and second derivative

The first derivative examines the rate at which the function changes while the second derivative examines the rate at which the first derivative changes.

Step 2: Find the statement in terms of a function and its first and second derivatives

Assume the size of the national deficit as \[D\left( t \right)\] which is a function of \[t\]. As the deficit is increasing at the time of the speech, \[D'\left( t \right) > 0\] and the second derivative of deficit, \[D''\left( t \right)\] decreasing since the rate is increasing.