2017 AP Calculus AB Free Response #1
TLDRIn this engaging video, Alan, a Bothell stem coach, takes viewers on a journey through AP Calculus by revisiting the 2017 AP exam's free response questions. The video begins with a problem involving a tank with a height of 10 feet, where Alan uses a Riemann sum with three subintervals to approximate the tank's volume, resulting in an overestimation due to the decreasing nature of the function. Alan then demonstrates how to find the exact volume using an integral, which is approximately 101.325 cubic feet. The video concludes with a problem on calculating the rate of change of water volume in the tank, which Alan solves using the fundamental theorem of calculus, resulting in a rate of 1.694 cubic feet per minute. Throughout the video, Alan's candid approach and willingness to correct his own mistakes make the complex concepts of calculus more accessible and relatable to the audience.
Takeaways
- 📚 The video discusses AP Calculus problems from the 2017 AP exam, focusing on free response questions.
- 💡 The first problem involves calculating the volume of a tank with a given height and varying cross-sectional area.
- 📏 The cross-sectional area of the tank at height H is represented by a function a(H), which decreases as H increases.
- 🔢 A table of selected values for a(H) is provided, and a Riemann sum with three subintervals is used to approximate the tank's volume.
- 📐 The subintervals chosen are 0 to 2, 2 to 5, and 5 to 10 feet, based on the data from the table.
- 🧮 The volume is calculated as the sum of the areas of the rectangles formed by the left Riemann sum method.
- 📏 The units of the volume are cubic feet, derived from the area in square feet multiplied by the height in feet.
- ⏯️ The video mentions that the approximation might overestimate the actual volume due to the nature of the Riemann sum method.
- 🧮 An integral calculation is performed to find the exact volume of the tank using the function a(H) over the interval from 0 to 10 feet.
- 🚰 In part C, the rate at which water is pumped into the tank is given, and the video calculates the rate of change of the water's volume with respect to time.
- 🔁 The fundamental theorem of calculus is applied to find the derivative of the volume with respect to time, using the chain rule when necessary.
- 📉 The final answer for the rate of volume change is provided, along with a check on the units to ensure correctness.
- 📝 The video concludes by comparing the calculated answers with the official AP Calculus 2017 free response solution and scoring guidelines.
Q & A
What is the main topic of the video?
-The main topic of the video is AP Calculus, specifically discussing and solving free response questions from the 2017 AP exam.
What is the shape of the tank discussed in the video?
-The tank has a height of 10 feet and a horizontal cross section with an area given by a function of height H, which is measured in square feet.
How is the volume of the tank approximated in the video?
-The volume of the tank is approximated using a Riemann left sum with three subintervals based on the given data in a table.
What are the three subintervals used for the Riemann sum?
-The three subintervals used are from 0 to 2, 2 to 5, and 5 to 10 feet.
What is the result of the volume approximation using the Riemann left sum?
-The approximated volume of the tank is 176.3 cubic feet.
Why does the Riemann left sum overestimate the volume of the tank?
-The Riemann left sum overestimates the volume because the function a(H) decreases as H increases, and the left sum does not account for the decreasing area as accurately as a midpoint or trapezoidal rule might.
What is the integral used to find the exact volume of the tank?
-The integral used is the definite integral from 0 to 10 of the function f(H) times dH, where f(H) represents the area of the horizontal cross section at height H.
What is the result of the integral used to find the exact volume?
-The exact volume calculated from the integral is 101.325 cubic feet.
How is the rate of change of the volume of water in the tank found?
-The rate of change of the volume of water is found by differentiating the volume function with respect to time and applying the chain rule, considering H as a function of time.
What is the rate at which the volume of water is changing when the height is 5 feet?
-The rate at which the volume of water is changing when the height is 5 feet is 1.694 cubic feet per minute.
What is the unit of measure for the volume of the tank?
-The unit of measure for the volume of the tank is cubic feet.
What is the final step in the video regarding the AP Calculus exam?
-The final step is to compare the solutions and calculations done in the video with the official AP Calculus 2017 free response solution scoring guidelines.
Outlines
📚 AP Calculus Exam Review: Tank Volume Calculation
In this segment, Alan, a Bothell stem coach, is reviewing AP Calculus by solving a 2017 AP exam question. The question involves calculating the volume of a tank with a height of 10 feet and a varying horizontal cross-sectional area given by a function a(H). Alan uses a Riemann left sum with three sub-intervals to approximate the volume, which he calculates as 176.3 cubic feet. He also discusses the concept of overestimation due to the decreasing nature of the function a(H). Alan then proceeds to find the volume of the tank more precisely using an integral from 0 to 10, resulting in an approximation of 101.325 cubic feet. Finally, he addresses a rate of change problem, calculating the rate at which the volume of water in the tank is changing when the water height is increasing at a rate of 0.26 feet per minute, obtaining a rate of 1.694 cubic feet per minute. Alan acknowledges a minor mistake in his initial setup but corrects it to provide the final answer.
🔢 AP Calculus Exam Review: Correcting and Finalizing Calculations
Alan continues his AP Calculus review by addressing a mistake in his initial calculation setup. He corrects his approach to find the rate at which the volume of water in the tank is changing with respect to time. Using the fundamental theorem of calculus and applying the chain rule, he recalculates the rate, obtaining a corrected value of 1.694 cubic feet per minute. Alan emphasizes the importance of units, ensuring that the rate is expressed in cubic feet per minute. He compares his work with the official AP Calculus 2017 free response solution scoring guidelines, confirming that his final answers align with the expected results. The video concludes with an invitation for viewers to comment, like, or subscribe for more content, and Alan expresses his willingness to create further instructional content based on viewer interest.
Mindmap
Keywords
💡AP Calculus
💡Riemann Sum
💡Volume Approximation
💡Units of Measure
💡Integral
💡Fundamental Theorem of Calculus
💡Chain Rule
💡Rate of Change
💡Graphing Calculator
💡Free Response Question
💡Modeling
Highlights
Alan, a Bothell stem coach, is teaching AP Calculus.
The class is reviewing the 2017 AP exam's free response questions.
A tank with a height of 10 feet is used as an example to illustrate the problem.
The area of the tank's horizontal cross-section at height H is given by a function a(H).
The function a(H) is continuous and decreases as H increases.
Selected values of a(H) are provided in a table for the students to use.
A Riemann left sum with three subintervals is used to approximate the tank's volume.
The subintervals chosen are 0 to 2, 2 to 5, and 5 to 10 feet.
The volume is calculated as 176.3 cubic feet using the left Riemann sum method.
The approximation is an overestimate because the area function a(H) decreases with increasing H.
The integral method is used to find the exact volume of the tank.
The integral calculation results in an approximate volume of 101.325 cubic feet.
Water is pumped into the tank at a rate of 0.26 feet per minute when the height is 5 feet.
The rate at which the volume of water changes with respect to time is calculated using the model.
The derivative of the volume with respect to time is found using the fundamental theorem of calculus.
The rate of volume change is determined to be 1.694 cubic feet per minute at the height of 5 feet.
Alan discusses the importance of using a calculator for more precise integral calculations.
Alan invites viewers to comment, like, or subscribe for more videos and to ask questions.
Transcripts
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