Question

(a) at what time was the circulation increasing the fastest? (round your answer to the nearest integer.) (b) express using functional - notation the circulation of the magazine 18 months after it was started. (c) over what time interval is the graph of c concave up? the graph of c is concave up from t = (smaller value) to t = (larger value) years. (d) find that value. (round your answer to two decimal places.) (thousand magazines) explain what “concave up” means in practical terms. o the circulation was increasing more and more slowly. o the circulation was increasing more and more quickly. o the circulation was decreasing more and more slowly. o the circulation was decreasing more and more quickly.

Explanation:

Step1: Analyze part (a)

We need to find where the circulation is increasing the fastest. This occurs where the slope of the curve is the steepest. By visual inspection of the graph, we look for the point with the greatest positive - slope.

Step2: Analyze part (b)

To express the circulation at 18 months using functional notation, if we assume the function is \(C(t)\) where \(t\) is the time in months, then at \(t = 18\), we write \(C(18)\).

Step3: Analyze part (c)

A graph is concave up when the second - derivative is positive. Visually, it means the graph is curving upwards like a cup. We look for the interval on the \(t\) - axis where the graph has this shape.

Step4: Analyze part (d)

We are asked to find a value from the graph. We read the value of \(C\) from the graph at the appropriate \(t\) value and round to two decimal places.

Answer:

(a) [Value of \(t\) (rounded to nearest integer) where circulation is increasing fastest]
(b) \(C(18)\)
(c) [Smaller value of \(t\), Larger value of \(t\) for the interval where graph is concave up]
(d) [Rounded value to two decimal places]

(Note: Without the actual graph values, we cannot provide numerical answers for (a), (c) and (d). But the above is the general way to solve such problems.)