AP Calculus AB - Understanding Free Response Questions

1. AP Calculus AB -Understanding Free Response Questions By Katherine Lopez

2. The Question Intro: 2010 #3 There are 700 people in line for a popular amusement-park ride when the ride begins operation in the morning. Once it begins operation, the ride accepts passengers until the park closes 8 hours later. While there is a line, people move onto the ride at a rate of 800 people per hour. The graph above shows the rate, r(t), at which people arrive at the ride throughout the day. Time t is measured in hours from the time the ride begins operation.

3. Information You Are Given This is a first derivative graph, which shows you a rate. This is not the number of people on line, but the rate at which people move onto the ride per hour.

4. Other Given Information • 700 is the initial condition because at 0 hours, there are 700 people • The interval is • People moving off the line= people moving onto the ride

5. Part A How many people arrive at the ride between t=0 and t=3? Show the computations that lead to your answer. First: You must understand what the question is asking you. The question is asking you for the number of people arriving at the ride. You must keep in mind that the graph you are given only tells you the rate at which people arrive. You can use the given information to solve this question. Method: When you are given a first derivative graph and you can solve for the number of people between a specific integral by using differentiation, also known as integration Amount of people who arrived from hour 0 to hour 3

6. To solve for you must use the Trapezoidal Rule. People

7. Part B Is the number of people waiting in line to get on the ride increasing or decreasing between t = 2 and t = 3 ? Justify your answer. Tips: for this problem you should once again remember that the graph given to you shows the rate at which people arrive at the ride Method: To solve this problem you must simply look at the graph and understand what information it contains

8. These green vertical lines show the intervals between t=2 and t=3 The blue horizontal line shows the given rate at which people arrive at the ride (800 people per hour) Answer: Between time t=2 and t=3 the number of people waiting on line is increasing. You know this because the rate of people arriving is greater than the number at which people move onto the ride.

9. Part C At what time t is the line for the ride the longest? How many people are in line at that time? Justify your answers. Tips: Begin by explaining the graph so you will have a better understanding of what math calculations you must do. Method: This question involves explanation of different elements in the given graph and then the use of integration to solve for the number of people.

10. The longest wait on line must be at t=3 r(t) > 800, or the rate at which people move onto the ride between 0 t3 r(t) < 800 between 3t8 so the rate at which people move onto the line is shorter R(t) = 800 This graph is the first derivative. Here y=800 serves as the line where the graph shifts from positive to negative. In most problems the x axis serves as this line. t=3 is the relative maximum. The number of people on line at t=3 can be determined by the function: 3900-2400= 1500 people Initial Condition Total number of people who move onto the ride (off the line) in three hours Total number of people who get on the line in 3 hours

11. Part D Write, but do not solve, an equation involving an integral expression of r whose solution gives the earliest time t at which there is no longer a line for the ride. Tip: Remember that for there to no longer be a line, the number of people on line must be zero 0= 800t is the number of people who have boarded the ride by hour t. As they board the ride, they get off the line The entire expression is equal to zero because the number of people on line is zero This integral represents the total number of people who have arrived at the line by hour t.