The following curve is a model of a certain country\'s total prison population a

Question

The following curve is a model of a certain country's total prison population as a function of time in years (years since 1995).

(a) Which is correct?

Over the period [5, 10] the instantaneous rate of change of N is increasing.

Over the period [5, 10] the instantaneous rate of change of N is decreasing.    

(b) Which is correct?

The instantaneous rate of change of prison population at t = 4 was less than the average rate of change over the interval [0, 10].

The instantaneous rate of change of prison population at t = 4 was greater than the average rate of change over the interval [0, 10].    

The instantaneous rate of change of prison population at t = 4 was approximately equal to the average rate of change over the interval [0, 10].

(c) Which is correct?

Over the period [0, 10] the instantaneous rate of change of N is increasing, then decreasing.

Over the period [0, 10] the instantaneous rate of change of N is decreasing, then increasing.    

Over the period [0, 10] the instantaneous rate of change of N is always increasing.

Over the period [0, 10] the instantaneous rate of change of N is always decreasing.

(d) According to the model, this country's total prison population was increasing fastest around what year?


(e) Roughly estimate the instantaneous rate of change of N at

t = 4 by using a balanced difference quotient with h = 1.5.

(Round your answer to three decimal places.)
( ) million prisoners per year

Interpret the result.

In 1999, this country's total number of prisoners was (increasing/decreasing) increasing decreasing at a rate of approximately( ) prisoners per year.

Explanation / Answer

(A) Instantaneous rate of change of population is nothing but slope of tangent line, or derivative at that point. On the period [5,10] we can see that slope of tangent is decreasing. Therefore, second option is correct!

(B) In this part we are comparing the slope of tangent at x=4 and slope of line joining the end points of the curve. We can see that slope of tangent at x=4 is slightly larger than the slope of tangent joining end points of the curve. Therefore, second option is right.

(C) Since slope of tangent line increases first and then starts to decrease, therefore, first option is right!

(D) Country's population was increasing fastest when slope of tangent is largest. It happens around the year 2000.

(E) Instantaneous Rate of change =(1.55-1.5)/(5.5-4) = 0.05/1.5 = 1/30 = 0.033

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