AP Calculus BC 4.3: Applied Rates

Practice AP Calculus BC questions on using derivative signs and values to explain whether an applied quantity is increasing or decreasing.

[Maximum number: 2]

From 5 A.M. to 10 A.M., the rate at which vehicles arrive at a certain toll plaza is given by
$A(t)=450 \sqrt{\sin (0.62 t)}$ , where t is the number of hours after 5 A.M. and A(t) is measured in vehicles per hour. Traffic is flowing smoothly at 5 A.M. with no vehicles waiting in line.

Is the rate at which vehicles arrive at the toll plaza at 6 A.M. ( t=1 ) increasing or decreasing? Give a reason for your answer.

$A^{\prime}(1)=148.947272$

Considers $A^{\prime}(1)$

1 point

Because $A^{\prime}(1)>0$ , the rate at which the vehicles arrive at the toll
plaza is increasing.

Answer with reason

1 point

Scoring notes:

- The response need not present the value of $A^{\prime}(1)$ . The second line of the model solution earns both

points.

- An incorrect value assigned to $A^{\prime}(1)$ earns the first point (but will not earn the second point).

- Without a reference to t=1, the first point is earned by any of the following:

○ 148.947 accurate to the number of decimals presented, with zero up to three decimal places

(i.e., 149, 148, 148.9, 148.95, or 148.94)

○ $A^{\prime}(t)=148.947$ by itself

- To be eligible for the second point, the first point must be earned.

- To earn the second point, there must be a reference to t=1.

- Degree mode: $A^{\prime}(1)=23.404311$

Total for part (c)

2 points

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Covered syllabus points

4.3 Rates of Change in Applied Contexts Other Than Motion question 1