AP CALCULUS AB 2022 Exam Full Solution FRQ#1(a,b)

Weily Lin
17 May 202205:36
EducationalLearning
32 Likes 10 Comments

TLDRThe video script discusses a problem related to the 2022 AP Calculus AB exam, specifically focusing on a free-response question about the rate at which vehicles arrive at a toll plaza. The rate function given is a(t) = 450sqrt(sin(0.62t)), with t representing hours after 5 a.m. The task is to find the total number of vehicles from 6 a.m. to 10 a.m., which is solved by integrating the rate function from t = 1 to t = 5. Additionally, the script explores finding the average value of the rate function over the given interval, which is calculated by dividing the total area under the curve by the interval length. The total number of vehicles is found to be 1502.148, and the average rate of vehicles per hour is determined to be 375.537.

Takeaways
  • 📈 The rate at which vehicles arrive at a toll plaza is given by a function a(t) = 450sqrt(sin(0.62t)), representing vehicles per hour.
  • ⏰ The time frame considered is from 6 a.m. to 10 a.m., which is 5 hours after 5 a.m.
  • 🚗 The integral expression for the total number of vehicles is the area under the curve of a(t) from t = 1 to t = 5.
  • 🔢 The accumulation of vehicles is found by integrating the rate function over the given interval.
  • 📐 The average value of the rate is calculated by dividing the total area under the curve by the interval length.
  • 🕒 The interval length is 4 hours, from 6 a.m. to 10 a.m. (5 hours after 5 a.m.).
  • 📊 The average value of the rate function is also the average slope of the accumulation function over the interval.
  • 🧮 The total number of cars that arrive at the toll plaza from 6 a.m. to 10 a.m. is calculated to be 1502.148.
  • 🚨 The average rate of vehicles per hour is found by dividing the total number of cars by the total time, resulting in 375.537 cars per hour.
  • 🎯 The average value formula is applied to find the average value of a(t) over the interval from 6 a.m. to 10 a.m.
  • 🛣️ Traffic is flowing smoothly at 5 a.m. with no vehicles waiting in line, indicating the starting condition for the rate equation.
Q & A

  • What is the subject of the video series?

    -The video series is for the 2022 AP Calculus AB exam, focusing on the free response section, specifically question number one.

  • What is the rate equation given for the vehicles arriving at the toll plaza?

    -The rate equation given is a(t) = 450 sqrt(sin(0.62t)), where t is the number of hours after 5 a.m., and the rate is measured in vehicles per hour.

  • What does the term 'rate equation' imply in the context of the problem?

    -A rate equation implies that it describes the rate at which something occurs over time, in this case, the rate at which vehicles arrive at a toll plaza.

  • What is the time interval considered for the total number of vehicles arriving at the toll plaza?

    -The time interval considered is from 6 a.m. to 10 a.m., which corresponds to t = 1 to t = 5 hours after 5 a.m.

  • How is the total number of vehicles that arrive at the toll plaza from 6 a.m. to 10 a.m. represented mathematically?

    -The total number of vehicles is represented by the integral of the rate function a(t) from t = 1 to t = 5, which is the area under the curve of a(t) over this interval.

  • What is the formula for finding the average value of a rate function over an interval?

    -The average value of the rate function is found by taking the integral of the rate function over the interval and dividing it by the length of the interval, which is (b - a).

  • How is the average rate of change related to the total number of vehicles and the time interval?

    -The average rate of change is the total number of vehicles (which is the area under the curve) divided by the total time interval, representing the average number of vehicles per hour over the given period.

  • What is the total number of vehicles that arrive at the toll plaza from 6 a.m. to 10 a.m. according to the video?

    -The total number of vehicles that arrive at the toll plaza from 6 a.m. to 10 a.m. is 1502.148.

  • What is the average value of the rate function a(t) over the interval from 6 a.m. to 10 a.m.?

    -The average value of the rate function a(t) over the interval from 6 a.m. to 10 a.m. is 375.537 vehicles per hour.

  • How does the video calculate the area under the curve for the integral?

    -The video calculates the area under the curve by integrating the function a(t) from t = 1 to t = 5, which involves using a graphing calculator to find the numerical value of the integral.

  • What is the significance of the square root function in the rate equation?

    -The square root function in the rate equation a(t) = 450 sqrt(sin(0.62t)) contributes to the fluctuation of the rate at which vehicles arrive, simulating a more realistic traffic pattern that varies over time.

  • How does the sine function in the rate equation affect the rate of vehicle arrival?

    -The sine function in the rate equation modulates the rate of vehicle arrival, creating a periodic pattern that simulates the ebb and flow of traffic over the course of the hours considered.

Outlines
00:00
📈 Calculus AP Exam: Vehicle Arrival Rate and Total Count

This paragraph discusses a question from the 2022 AP Calculus AB exam, specifically focusing on graphing calculator questions. The scenario involves calculating the rate at which vehicles arrive at a toll plaza from 5 AM to 10 AM, given by a rate function a(t) = 450√(sine(0.62t)). The paragraph emphasizes that this is a rate equation, with 't' representing hours after 5 AM and the rate given in vehicles per hour. The traffic is described as flowing smoothly at 5 AM with no vehicles waiting in line. The task in part A is to write, but not evaluate, an integral expression that gives the total number of vehicles arriving from 6 AM to 10 AM (from t=1 to t=5). The paragraph explains that the integral represents the accumulation of the rate function over the given interval, which corresponds to the area under the curve of the function a(t). Part B involves finding the average value of the rate at which vehicles arrive, which is done by dividing the total area under the rate function by the length of the interval (5-1=4 hours). The total number of cars is calculated to be 1502.148, and the average rate is found to be 375.537 cars per hour.

05:01
📊 Average Arrival Rate Calculation for Toll Plaza

The second paragraph continues the discussion from the first, focusing on calculating the average rate at which vehicles arrive at the toll plaza from 6 AM to 10 AM. It explains that the average value of the rate function is found by dividing the total area under the rate function by the interval length, which in this case is 4 hours (from t=1 to t=5). The paragraph outlines the process of finding the average value using the average value formula, which involves integrating the rate function over the interval and then dividing by the interval length. The result of this calculation is presented as 375.537 cars per hour, which represents the average rate of vehicle arrivals during the specified time period.

Mindmap
Keywords
💡AP Calculus AB Exam
The AP Calculus AB Exam is a standardized test administered by the College Board that assesses students' understanding of calculus concepts. In the video, it is the context for the problem-solving exercise, which revolves around a question from the free response section of the exam.
💡Free Response Section
The Free Response Section is a part of the AP Calculus AB Exam where students must write out and solve complex problems, demonstrating their understanding and application of calculus concepts. The video focuses on question number one from this section, emphasizing the need for both conceptual understanding and problem-solving skills.
💡Graphing Calculator
A graphing calculator is a specialized device that can plot graphs, solve complex mathematical functions, and perform various other mathematical operations. In the context of the video, it is an essential tool for visualizing and solving the calculus problem involving the rate of vehicle arrival at a toll plaza.
💡Rate Equation
A rate equation is a mathematical expression that describes the rate at which a quantity changes over time. In the video, the rate equation 'a(t) = 450√sin(0.62t)' is used to model the rate at which vehicles arrive at a toll plaza, with 't' representing the time in hours after 5 a.m.
💡Integral Expression
An integral expression in calculus represents the accumulation of a quantity over a given interval. In the video, the integral expression is used to find the total number of vehicles that arrive at the toll plaza from 6 a.m. to 10 a.m., which is a key part of solving the problem.
💡Total Number of Vehicles
The total number of vehicles is the sum of all vehicles that arrive at the toll plaza over a specified period. The video involves calculating this total by integrating the rate function over the interval from 6 a.m. to 10 a.m., which is a fundamental concept in the application of calculus to real-world problems.
💡Average Value
The average value of a function over an interval is a measure of the central tendency of the function's values over that interval. In the video, the average value of the rate function is calculated to find the average rate at which vehicles arrive at the toll plaza from 6 a.m. to 10 a.m.
💡Area Under the Curve
The area under the curve of a graph is a visual representation of the integral of a function over a certain interval. In the context of the video, finding the area under the curve of the rate function from 6 a.m. to 10 a.m. is essential for determining the total number of vehicles that arrive at the toll plaza.
💡Accumulation Function
An accumulation function in calculus is used to represent the total amount of a quantity that has accumulated over time. In the video, the accumulation function is related to the total number of cars that have arrived at the toll plaza, which is found by integrating the rate function.
💡Displacement
In the context of calculus, displacement refers to the total change in position of a quantity, which can be represented by the area under the curve of its rate function. In the video, the displacement is analogous to the total number of cars that have arrived at the toll plaza, calculated by the integral of the rate function.
💡Average Rate of Change
The average rate of change is the average of the rates of change of a function over an interval. It is calculated by dividing the difference in the function's values at the endpoints of the interval by the length of the interval. In the video, the average rate of change is used to find the average value of the rate function, which represents the average number of vehicles arriving per hour.
Highlights

The series of videos is for the 2022 AP Calculus AB exam

Focus is on the free response section, question number one

The question involves graphing calculator questions

Vehicle arrival rate at a toll plaza from 5am to 10am is given by a rate equation

The rate function is a(t) = 450 * sqrt(sin(0.62t))

t represents the number of hours after 5am, with rate in vehicles per hour

Traffic is flowing smoothly at 5am with no vehicles waiting in line

Part A asks to write an integral expression for total vehicles from 6am to 10am (t=1 to t=5)

The accumulation is the integral of the rate function from a to b

The total number of vehicles is the area under the curve from t=1 to t=5

Part B asks to find the average value of the rate vehicles arrive from 6am to 10am

The average value of a rate function can be found using the area under the curve divided by the interval length

Alternatively, the average value can be interpreted as the average slope of the accumulation function

The total number of cars is the area under the curve, and the interval length is the total time

The average rate of change is the total number of cars divided by the total time

Using the average value formula, the average value of a(t) is calculated as 375.537 cars per hour

The area under the curve from t=1 to t=5 is 1502.148, representing the total number of cars

The interval length is 4 hours (5 hours - 1 hour)

The average value of the rate function is obtained by dividing the total number of cars by the interval length

Transcripts

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