2018 AP Calculus AB Free Response #1
TLDRIn the video, Alan from Bottle Stem Coaching delves into the 2018 free response questions from the AP Calculus AP exam. He tackles a problem involving the rate at which people enter and exit an escalator line, modeled by a function of time. Alan uses calculus to calculate the total number of people who enter the line within a 300-second period, finding it to be 127 people. He then determines the number of people in line at the 300-second mark, which is 80. Alan also explores when the line would be empty post 300 seconds, concluding it would be at 114.3 seconds. Further, he investigates the minimum number of people in line during the interval, using calculus to find the critical points and eventually determining the minimum to be 4 people at 33.01 seconds. The video provides a step-by-step walkthrough of the mathematical process, complete with calculations and corrections, offering viewers a clear understanding of the problem-solving approach in AP Calculus.
Takeaways
- 📚 Alan is discussing the 2018 free response questions from the AP Calculus exam.
- 🚶♂️ The problem involves modeling the rate at which people enter and exit an escalator line.
- ⏱️ The rate of people entering the line is given by a function of time, with T measured in seconds.
- 🔢 The integral from 0 to 300 of the rate function minus the exit rate gives the total number of people in line after 300 seconds.
- 🧮 Alan initially calculates an incorrect integral result but then corrects it to 127 people.
- 🚫 After 300 seconds, 80 people are in line, considering the exit rate of 0.7 persons per second.
- 🕒 To find when the line is empty after time T > 300, Alan calculates T = 80 / 0.7 seconds.
- 📉 Alan seeks to determine the minimum number of people in line by setting the derivative of the rate function equal to zero.
- 📈 He identifies critical points and uses a table to evaluate the minimum at the endpoints and the critical points.
- 🔍 Alan plots the rate function minus the exit rate to visualize where the minimum occurs.
- 🕒 The minimum number of people in line occurs at T = 33.01 seconds, with approximately 3.8 people.
- 🔄 Alan corrects an oversight regarding the calculation for time T > 300 seconds, adjusting the time to 404.3 seconds.
Q & A
What is the subject of the video Alan is discussing?
-Alan is discussing the 2018 free response questions from the AP Calculus AP exam.
What is the rate at which people enter the escalator line in the given model?
-The rate at which people enter the escalator line is modeled by the function R(T), which is T/100^3 * (1 - T/300)^7 people per second.
How many people are initially in line at time T equals zero?
-There are 20 people initially in line at time T equals zero.
What is the integral Alan calculates to find the number of people who enter the line during a certain time period?
-Alan calculates the integral from 0 to 300 of the function T/100^3 * (1 - T/300)^7 to find the number of people who enter the line during the time period from 0 to 300 seconds.
What is the result of the integral Alan calculates?
-The result of the integral is initially stated as 271, but Alan corrects himself and states the correct result should be 127 people.
How many people are in line at time T equals 300?
-At time T equals 300, there are 80 people in line after accounting for the people entering and exiting the line.
What is the first time after 300 seconds when there are no people in line for the escalator?
-The first time after 300 seconds when there are no people in line is at approximately 114.3 seconds, calculated by dividing the remaining 80 people by the exit rate of 0.7 persons per second.
What method does Alan use to find the time when the number of people in line is at a minimum?
-Alan uses calculus, specifically by finding the first derivative of the function representing the number of people in line and setting it to zero to find the critical points, which are potential minimums.
What are the critical points Alan identifies for the function representing the number of people in line?
-Alan identifies two critical points at T = 166.57 and T = 33.01, where the function R(T) - 0.7 equals zero.
How does Alan determine the absolute minimum number of people in line?
-Alan determines the absolute minimum by evaluating the function at the critical points and the endpoints of the interval (time zero and 300 seconds), then comparing the values to find the true minimum.
What is the minimum number of people in line according to Alan's calculations?
-The minimum number of people in line, according to Alan's calculations, is 4 people at T = 33.01 seconds.
What is the mistake Alan makes in part C of the video?
-In part C, Alan forgets to add 300 to the time after 300 seconds when calculating the time after which there are no people in line, leading to an incorrect answer of 4.3 seconds instead of the correct 304.3 seconds.
Outlines
📚 AP Calculus Exam Free Response Analysis
In this segment, Alan discusses the 2018 free response questions from the AP Calculus exam. He begins by setting up an integral to calculate the number of people entering an escalator line over a period of time. The rate at which people enter is modeled by a function, R(T), which is then integrated from T=0 to T=300 seconds. Alan uses a calculator to find the integral's value, correcting a mistake to arrive at 127 people. He then calculates the number of people in line at T=300 seconds, considering the exit rate of 0.7 persons per second. The final count is 80 people. To find when there are no people in line after T=300 seconds, Alan determines the time it takes for the remaining 80 people to leave the line. The problem also explores finding the minimum number of people in line during the interval, which involves setting the derivative of the function equal to zero and solving for T. Alan plots the function and identifies critical points to determine the minimum, which occurs at approximately T=33.01 seconds with about 4 people in line.
📊 Deriving the Minimum People in Line
This paragraph focuses on finding the time T when the number of people in line is at its minimum. Alan starts by plotting the rate function R(T) minus the exit rate of 0.7 to visualize the critical points where the function could achieve a minimum. He identifies two zeros at T=166.57 and T=33.01. To find the minimum, Alan explains that he needs to check the slopes around these points, indicating a change from negative to positive slope signifies a relative minimum. After calculating the number of people at T=0, T=33.01, and T=300, Alan concludes that the minimum occurs at T=33.01 with 4 people in line. However, he acknowledges a mistake in his previous calculation, correcting the time after 300 seconds to 404.3 seconds. The paragraph ends with Alan noting that the minimum number of people in line, as concluded from the relative minimum, is indeed correct.
🔄 Conclusion and Future Engagement
In the final paragraph, Alan wraps up the video by summarizing the findings and encouraging viewers to engage with future content. He acknowledges the corrected time of 404.3 seconds for the scenario after 300 seconds and confirms the minimum number of people in line. Alan invites viewers to comment, like, or subscribe for more content and mentions that he offers free homework help on platforms like Twitch and Discord. He teases the next video, prompting viewers to stay tuned for further educational insights.
Mindmap
Keywords
💡AP Calculus
💡Free Response Questions
💡Escalator Model
💡Integral
💡Rate
💡Derivative
💡Fundamental Theorem of Calculus
💡Critical Points
💡Relative Minima and Maxima
💡Endpoints
💡Twitch and Discord
Highlights
Alan is going to look at the 2018 free response questions from the AP Calculus exam.
People enter an escalator line at a rate modeled by the function T/100^3 * (1 - T/300)^7 people per second.
People exit the escalator line at a constant rate of 0.7 persons per second.
There are 20 people in line at time T=0.
To find how many people enter the line from T=0 to T=300, Alan does the integral ∫(T/100^3 * (1 - T/300)^7) dT from 0 to 300.
The integral evaluates to 270 people.
At T=300, there are 80 people in line after starting with 20 and adding 270, but 0.7*300 people leave.
To find when the line is empty after T=300, Alan solves 80 / 0.7 = 114.3 seconds.
To find when the line has the minimum number of people for T ≤ 300, Alan sets up the integral ∫(R(T) - 0.7) dT + 20.
Taking the derivative and setting it to 0, Alan finds critical points at T = 166.57 and T = 33.01.
By analyzing the sign of the derivative, Alan determines T = 33.01 is a local minimum and T = 166.57 is a local maximum.
Alan calculates the number of people at the endpoints T=0 (20 people), T=33.01 (3.8 people), and T=300 (80 people).
The minimum number of people is at T = 33.01 with 4 people.
Alan double checks the solution and finds a mistake in part c, the time after 300 seconds.
The corrected time after 300 seconds is 404.3 seconds.
The minimum number of people in line is 4 at T=33.01, not 5 as Alan initially stated.
Alan offers free homework help on Twitch and Discord.
Transcripts
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