2016 AP Calculus AB Free Response #1
TLDRIn this video, Alan from Bothell STEM, a coach, dives into AP Calculus by examining the 2016 exam's free response questions. Alan begins by addressing a challenge with his graphing calculator but proceeds to estimate the derivative of a function representing the rate of water being pumped into a tank. He uses a secant line method to approximate the derivative at a specific time. Alan then employs the left Riemann sum to estimate the total water removed from the tank over eight hours, noting it as an overestimate due to the decreasing rate function. He also calculates the total amount of water in the tank at the end of eight hours by integrating the rate function and subtracting the removed water. Finally, Alan explores if there's a time when the water input rate equals the output rate, using the intermediate value theorem to confirm such a time exists. Despite a minor error in sign interpretation, Alan provides a comprehensive walkthrough of calculus problems, offering help on Twitch or Discord for further learning.
Takeaways
- 📈 The video discusses AP Calculus, specifically the 2016 exam's free response questions.
- 💡 Alan, the speaker, is looking for a graphing calculator to solve the problems but attempts to estimate derivatives without one.
- ⏱️ The problem involves water being pumped into and out of a tank, with rates given by different functions of time T.
- 📊 Alan uses a table of selected values for the function R(T) to estimate the derivative at T=2 using a secant line method.
- 🧮 He calculates the units of the derivative as liters per hour per hour, or liters per hour squared.
- 🔍 Alan uses the left Riemann sum to estimate the total amount of water removed from the tank over eight hours.
- 📉 He notes that using the left Riemann sum results in an overestimate when the rate R(T) is decreasing.
- 🚰 To estimate the total amount of water in the tank at the end of eight hours, Alan considers both the water pumped in and out.
- ⌛️ Alan attempts to integrate the rate function for water being pumped in from 0 to 8 hours but acknowledges the need for a graphing calculator.
- 🔢 He uses the graphing calculator to find the integral and combines this with the initial volume and the estimated water removed.
- 🔄 Alan explores whether there is a time T when the rate of water being pumped in equals the rate of water being removed, using the intermediate value theorem.
- 🤔 He identifies a mistake in his calculation of the derivative's sign and corrects it, emphasizing the importance of accuracy.
- 📢 Alan offers free homework help on Twitch or Discord for those with questions in math and physics.
Q & A
What is the topic of the video Alan is discussing?
-Alan is discussing AP Calculus, specifically looking at the 2016 exam's free response questions.
What tool does Alan mention he needs to solve the problems but has not yet resolved the issue of?
-Alan mentions that he needs a graphing calculator to solve the problems but has not yet resolved his graphing calculator issue.
What is the rate at which water is pumped into the tank modeled by?
-The rate at which water is pumped into the tank is modeled by a function of time, denoted as W(T), measured in liters per hour.
How does Alan estimate the derivative of R at T=2?
-Alan estimates the derivative of R at T=2 by using a secant line slope between T=1 and T=3, which is calculated as (R(3) - R(1)) / (3 - 1).
What is the unit of measure for the derivative of R?
-The unit of measure for the derivative of R is liters per hour per hour, or liters per hour squared.
How does Alan estimate the total amount of water removed from the tank during eight hours using the left Riemann sum?
-Alan uses the left Riemann sum with four subintervals indicated by the table (0 to 1, 1 to 3, 3 to 6, and 6 to 8) and multiplies the height of each rectangle (rate of water removal at the left endpoint of each interval) by the width of each rectangle (the duration of each interval) to estimate the total amount of water removed.
Why is the left Riemann sum considered an overestimate when the rate function R(T) is decreasing?
-The left Riemann sum is considered an overestimate when the rate function R(T) is decreasing because the rectangles used in the sum will be taller than the curve of the function for most of the interval, leading to a larger total area than the actual curve would enclose.
How does Alan calculate the total amount of water in the tank at the end of eight hours?
-Alan calculates the total amount of water in the tank at the end of eight hours by starting with the initial volume (50,000 liters), subtracting the estimated volume of water removed, and adding the volume of water pumped in, which is the integral of the rate function W(T) from 0 to 8 hours.
What does Alan use to find the integral of the rate function W(T) from 0 to 8?
-Alan uses a graphing calculator to find the integral of the rate function W(T) from 0 to 8, as it involves a complex integral that cannot be solved by hand easily.
What theorem does Alan refer to when discussing if there is a time when the rate at which water is pumped into the tank equals the rate at which it is removed?
-Alan refers to the Intermediate Value Theorem when discussing if there is a time when the rate at which water is pumped into the tank equals the rate at which it is removed.
What mistake does Alan make during the video?
-Alan makes a mistake by not realizing that the derivative he calculated was negative, which led to an incorrect estimate in one of the problems.
What additional help does Alan offer at the end of the video?
-Alan offers free homework help on Twitch or Discord for anyone who has homework questions or wants to learn about different parts of math and physics.
Outlines
📊 Calculus Exam Problem: Water Tank Dynamics
In this segment, Alan from Bothell StemCoach tackles an AP Calculus exam question focused on the rate at which water is pumped into and removed from a tank. The problem involves estimating the derivative of a function to determine the rate of change at a specific time, using a secant line approach. Alan also discusses the use of a left Riemann sum to estimate the total water removed from the tank over eight hours, noting that this method typically overestimates the actual amount due to the decreasing nature of the function. Finally, he addresses how to calculate the total amount of water in the tank after eight hours by considering both the water pumped in and the water pumped out, integrating the rate function from zero to eight hours.
🧮 Estimating Water Volume and Finding Equilibrium Time
Alan continues the AP Calculus problem by estimating the volume of water in the tank at the end of eight hours. He explains that the initial volume is 50,000 liters and accounts for both the water pumped out and the water pumped in over time. To find the volume pumped in, Alan uses a graphing calculator to integrate a given rate function from zero to eight hours. After calculating the integral, he combines this with the initial volume and the estimated water removed to find the final volume in the tank. In the final part of the problem, Alan investigates if there's a time when the rate of water being pumped into the tank equals the rate of water being removed. Using the intermediate value theorem and examining the table of values, he concludes that such a time exists where the rates are equal.
🎓 Conclusion and Additional Learning Resources
Alan concludes the video by acknowledging a mistake in his calculations, where he inadvertently provided a negative estimate instead of a positive one. He corrects the error and reiterates the use of the intermediate value theorem to ensure that there is a time when the rates of water flow into and out of the tank are equal. Alan also invites viewers to seek further help with their homework questions on Twitch or Discord, offering his assistance in various areas of math and physics. He encourages interaction by asking viewers to leave comments, like, or subscribe to his content.
Mindmap
Keywords
💡Graphing Calculator
💡Derivative
💡Secant Line
💡Riemann Sum
💡Integral
💡Intermediate Value Theorem
💡Water Tank
💡Liters per Hour
💡Free Response Questions
💡Twitch
💡Discord
Highlights
Alan from Bothell STEM is coaching AP Calculus, focusing on the 2016 exam.
The first free response question involves graphing calculator usage, which Alan is working to resolve.
A tank's water is pumped in and out at rates modeled by different functions of time T, with R(T) being differentiable and decreasing.
At time T=0, the tank contains 50,000 liters of water.
Alan estimates the derivative of R at T=2 without a graphing calculator using a secant line slope.
The units of the derivative are liters per hour per hour, or liters per hour squared.
A left Riemann sum with four subintervals is used to estimate the total water removed over eight hours.
The left Riemann sum is an overestimate when the rate function R(T) is decreasing.
Alan uses the answer from Part B to estimate the total water in the tank at the end of eight hours.
The integral of the rate water is pumped in from T=0 to T=8 is calculated to find the total water pumped in.
Alan uses a graphing calculator to solve the integral of a function from 0 to 8 hours.
The final volume of water in the tank is calculated by starting volume minus pumped out plus pumped in.
Part D involves finding if there's a time T when the water pumped in equals the water removed.
Alan uses the intermediate value theorem to confirm that there exists a time T when the rates are equal.
Alan acknowledges a mistake in his calculation, where he had a negative sign error.
Alan offers free homework help on Twitch or Discord for any math or physics questions.
The video concludes with an invitation for viewers to engage with Alan on his platforms for further assistance.
Transcripts
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