2015 AP Calculus AB Free Response #3
TLDRIn this educational video, Alan from Bothell STEM, a coach, guides viewers through AP Calculus 2015 free response question number three, focusing on the non-calculator portion. He uses a table to estimate the derivative of a jogger's velocity at 16 minutes, explaining the units and their significance. Alan then interprets the definite integral of the absolute value of velocity over 40 minutes as the total distance traveled. He proceeds to approximate this value using a right Riemann sum. The video also covers calculating Bob's acceleration and average velocity using given velocity functions, with Alan correcting his arithmetic mistake during the process. Alan concludes by offering free homework help on Twitch or Discord for further learning and assistance.
Takeaways
- 📚 Alan is teaching AP Calculus, focusing on the 2015 free response questions, specifically question number three without a calculator.
- ⏱ The problem involves a jogger named Joanna, who moves along a straight path, with her velocity given as a function of time in minutes.
- 🔍 Alan uses a table of selected values to estimate the derivative at a specific time, employing the secant line method.
- 📊 The units of the derivative are meters per minute squared, which represents acceleration.
- 🏃 The definite integral of the absolute value of velocity (speed) over 40 minutes represents the total distance traveled by Joanna.
- 🚴 Bob, another character, is riding a bicycle along the same path, with his velocity modeled by a different function.
- 📐 Bob's acceleration is found by taking the derivative of his velocity function, specifically evaluated at time T equals five minutes.
- 🧮 Alan approximates the value of a definite integral using a right Riemann sum with four subintervals, accounting for the absolute value of velocities.
- 🤔 Alan acknowledges a mistake in his arithmetic during the calculation of the Riemann sum but corrects it to find the total distance.
- 🎯 The average velocity of Bob is calculated by integrating his velocity function over a specific interval and dividing by the length of that interval.
- 🤓 Alan offers free homework help on platforms like Twitch or Discord for those interested in learning math and physics.
- 👍 The video concludes with a reminder that mistakes happen but it's important to learn from them, and an invitation to join Alan's online study sessions.
Q & A
What is the subject of the video Alan is discussing?
-Alan is discussing AP Calculus, specifically the 2015 free response questions, focusing on question number three which is the non-calculator portion.
What is the context for the problem Alan is solving?
-The context is a problem involving a jogger named Joanna who jogs along a straight path, with her velocity given by a differentiable function V(T), where T is measured in minutes and V is measured in meters per minute.
How does Alan estimate the derivative of V at T=16?
-Alan uses the secant line method, calculating it as V(20) - V(12) over the interval from 12 to 20, which results in (240 - 200) / (20 - 12), simplifying to 5 meters per minute squared.
What does the definite integral of the absolute value of V(T) from 0 to 40 represent in the context of the problem?
-The definite integral of the absolute value of V(T) from 0 to 40 represents the total distance traveled by Joanna over the 40-minute period, as the absolute value of velocity corresponds to speed.
How does Alan approximate the value of the definite integral using a right Riemann sum with four subintervals?
-Alan uses the absolute values of the velocities at the endpoints of each subinterval and multiplies them by the width of the subintervals, summing these products to approximate the integral.
What is the correct method to find Bob's acceleration, according to the video?
-Bob's acceleration is found by taking the derivative of his velocity function, which is given by a specific formula involving T cubed and T squared terms.
At what time does Alan calculate Bob's acceleration?
-Alan calculates Bob's acceleration at time T equals five minutes.
What is the formula for calculating the average velocity during an interval in the context of the video?
-The average velocity is calculated by integrating the velocity function V(T) over the interval and then dividing by the length of the interval.
What is the correct unit for the average velocity calculated in the video?
-The correct unit for the average velocity is meters per minute.
What does Alan offer to help with homework and learning about math and physics?
-Alan offers free homework help on platforms like Twitch or Discord for those who have questions or want to learn about different parts of math and physics.
What is the main mistake Alan acknowledges during the video?
-Alan acknowledges a mistake in his arithmetic when calculating the total distance using the right Riemann sum, particularly in adding the values to get the final distance.
What is the final correct average velocity that Alan calculates for Bob?
-The final correct average velocity that Alan calculates for Bob is 350 meters per minute.
Outlines
📚 AP Calculus 2015 Free Response Question 3
In this video, Alan from Bothell STEM, Coach is discussing AP Calculus 2015 Free Response Question 3, which is the non-calculator portion. He begins by estimating the derivative of a differentiable function V(T) at T=16 using the secant line method, which yields a value of 5 meters per minute squared. Alan then explains the meaning of the definite integral of the absolute value of V(T) from 0 to 40 in the context of the problem, which represents the distance traveled by Joanna over 40 minutes. He proceeds to approximate the value of this integral using a right Riemann sum with four subintervals, resulting in a total distance of 10,000 meters. The video also covers the calculation of Bob's acceleration, modeled by a function B(T), which is found to be 15 meters per minute squared at T=5. Lastly, Alan calculates Bob's average velocity during a specific interval, which is determined to be 350 meters per minute. Despite a minor arithmetic error, Alan provides a comprehensive walkthrough of the calculus problem, offering viewers a clear understanding of the concepts involved.
🚴♂️ Bob's Bicycle Velocity and Acceleration Calculation
The second paragraph of the video script focuses on Bob's bicycle velocity and acceleration. Alan explains that acceleration is the derivative of velocity and calculates Bob's acceleration at time T equals five, resulting in 15 meters per minute squared. For the final part of the video, Alan addresses the task of finding Bob's average velocity during a given interval. He integrates the velocity function from zero to ten and divides by the interval to find the average velocity, which is correctly identified as 350 meters per minute. Alan acknowledges a mistake in his arithmetic during the calculation but corrects it to provide the accurate result. The video concludes with an invitation for viewers to join Alan on Twitch or Discord for free homework help in math and physics.
Mindmap
Keywords
💡AP Calculus
💡Free Response Questions
💡Derivative
💡Definite Integral
💡Secant Line
💡Velocity
💡Acceleration
💡Riemann Sum
💡Absolute Value
💡Distance
💡Average Velocity
Highlights
Alan is discussing AP Calculus 2015 free response question number three, focusing on the non-calculator portion.
The problem involves a jogger named Joanna, who jogs along a straight path, with her velocity given by a differentiable function V(T).
A table of selected values of V(T) is provided, where T is in minutes and V is in meters per minute.
The secant line method is used to estimate the derivative of V at T=16, resulting in an approximate value of 5 meters per minute squared.
The units of the derivative are explained as meters per minute squared, representing acceleration.
The definite integral of the absolute value of V(T) from 0 to 40 is interpreted as the total distance traveled by Joanna over 40 minutes.
A right Riemann sum with four subintervals is used to approximate the value of the integral of the absolute value of V(T).
The absolute value of velocity is considered as speed for the calculation, which affects the Riemann sum.
The arithmetic error in the calculation of the Riemann sum is acknowledged, with a corrected total distance of 10,000 meters.
Bob's velocity is modeled by a different function B(T), and the problem asks to find his acceleration at T=5.
Bob's acceleration is calculated to be 15 meters per minute squared using the derivative of his velocity function.
The average velocity of Bob is determined over the interval from 0 to 10, using an integral and division by the interval length.
The average velocity calculation involves integrating a polynomial function and simplification, resulting in 350 meters per minute.
Alan offers free homework help on Twitch or Discord for those with questions in math and physics.
The video concludes with an invitation to join Alan on his platforms for further learning and interaction.
Alan emphasizes the commonality of making mistakes in arithmetic and the importance of careful calculation.
The scoring guide is briefly mentioned, indicating the correct answers for Bob's acceleration and average velocity.
Transcripts
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