2017 AP Calculus AB Free Response #4

Allen Tsao The STEM Coach
12 Sept 201810:57
EducationalLearning
32 Likes 10 Comments

TLDRIn this video, Alan from Bothell STEM continues the AP Calculus 2017 free response question series, focusing on question number four. The problem involves a boiled potato cooling down in a kitchen, with its internal temperature modeled by a function H(T) that satisfies a differential equation. Alan explains how to find the equation for the line tangent to the graph of H at T=0, using the initial condition of the potato's temperature being 91°C. He then approximates the internal temperature at T=3 minutes using the tangent line equation and discusses whether this approximation is an underestimate or an overestimate by examining the second derivative. Finally, Alan solves the differential equation to find the expression for G(T), the internal temperature of the potato at time T, and calculates the temperature at T=3 minutes. The video concludes with a brief review of the scoring guidelines and a prompt for viewer engagement.

Takeaways
  • 📚 The video discusses AP Calculus 2017 free response question number four, focusing on the non-calculator portion.
  • 🥔 The problem involves a boiled potato with an initial internal temperature of 91°C cooling in a kitchen.
  • ⏱ At time T=0, the potato is taken from the pot, and its internal temperature remains above 27°C for all times T > 0.
  • 📈 The internal temperature of the potato at time T is modeled by a function H(T), which satisfies a given differential equation.
  • 🔍 The task is to write an equation for the line tangent to the graph of H at T=0 and use it to approximate the internal temperature at T=3 minutes.
  • 🧮 The slope of the tangent line is found using the derivative of H(T), which is negative 16 at T=0.
  • 📐 The equation of the tangent line is derived as H = -16t + 91, representing the linear approximation of the potato's temperature at T=0.
  • 🔢 By substituting T=3 into the tangent line equation, the internal temperature is approximated to be 43°C at T=3 minutes.
  • ⛰ The second derivative of H(T) is used to determine if the approximation is an underestimate or overestimate at T=3, revealing that the function is concave up.
  • 📉 Because the second derivative is positive, the tangent line underestimates the actual temperature, as the curve flattens out more than the linear approximation.
  • 🔧 The model for the internal temperature of the potato at time T is represented by a function G(T), which satisfies a different differential equation.
  • 🧬 G(T) is solved using separation of variables, leading to an expression involving a cubed term and an integration constant.
  • 🌡️ The final expression for G(T) is used to find the internal temperature of the potato at T=3, which is calculated to be 54°C.
Q & A
  • What is the subject of the video Alan is discussing?

    -Alan is discussing AP Calculus 2017 free response questions, specifically focusing on question number four.

  • What is the initial condition given for the internal temperature of the potato?

    -The initial condition given is that the internal temperature of the potato is 91 degrees Celsius at time T equals zero.

  • What is the differential equation that the function H must satisfy?

    -The function H must satisfy a differential equation, which is not explicitly provided in the transcript but is related to the cooling of the potato.

  • What is the purpose of finding the equation for the line tangent to the graph of H at T equals zero?

    -The purpose is to approximate the internal temperature of the potato at a later time using the slope of the tangent line, which is derived from the derivative of the function H at T equals zero.

  • What is the approximated internal temperature of the potato at time T equals three, using the tangent line equation?

    -The approximated internal temperature of the potato at time T equals three is 43 degrees Celsius.

  • How does the second derivative of the function H affect the accuracy of the approximation at T equals three?

    -The second derivative of the function H is positive, indicating that the function is concave up. This suggests that the approximation using the tangent line is an underestimate because the actual temperature values are higher than the line suggests.

  • What is the differential equation model that Alan is trying to solve for the function G?

    -The differential equation model for the function G is not explicitly provided in the transcript, but it is related to the internal temperature of the potato over time.

  • How does Alan solve the differential equation for G?

    -Alan uses the method of separation of variables to solve the differential equation for G, integrating both sides and applying initial conditions to find the constant.

  • What is the final expression for the function G of T?

    -The final expression for G of T is G(T) = -1/3 * T + 4^(1/3) + 27.

  • What is the internal temperature of the potato at time T equals three, according to the model for G?

    -The internal temperature of the potato at time T equals three, according to the model for G, is 54 degrees Celsius.

  • What is the significance of the second derivative test in this context?

    -The second derivative test is used to determine the concavity of the function, which in this case indicates whether the tangent line approximation is an overestimate or an underestimate of the actual temperature at T equals three.

  • What feedback does Alan encourage from his viewers?

    -Alan encourages viewers to leave comments, likes, or subscribe if they have any feedback or enjoyed the video.

Outlines
00:00
📚 AP Calculus 2017 Question 4 - Tangent Line and Linear Approximation

In this paragraph, Alan from Bothell Stem introduces the AP Calculus 2017 free response question number four. The problem involves a boiled potato with an initial internal temperature of 91 degrees Celsius cooling down in a kitchen. The temperature of the potato at any time T is greater than 27 degrees Celsius and can be modeled by a function H(T) that satisfies a given differential equation. Alan explains the process of writing an equation for the line tangent to the graph of H at T=0, which involves finding a point and the slope. The slope is derived from the differential equation, leading to a linear approximation of the potato's internal temperature at T=3. The paragraph concludes with an approximation of the internal temperature at T=3 and a discussion on whether the approximation is an underestimate or overestimate using the second derivative.

05:02
🔢 Differential Equation Solution and Potato Temperature Estimation

Alan continues by addressing the second part of the problem, which involves finding an expression for the function G(T) that models the internal temperature of the potato based on a given differential equation. He explains the method of separation of variables to solve the differential equation and finds an expression for G(T). Using the initial condition that G(0) is 91 degrees Celsius, Alan calculates the constant C and provides the final form of G(T). He then uses this model to find the internal temperature of the potato at T=3, which is approximately 54 degrees Celsius. Alan also discusses the implications of the concavity of the function and its impact on the accuracy of the linear approximation.

10:05
📝 Conclusion and Engagement Invitation

In the final paragraph, Alan wraps up the video by summarizing the key points of the problem and the solution process. He apologizes for not explicitly answering a part of the question and reiterates the steps to find the internal temperature at T=3 using the model. Alan encourages viewers to engage with the content by leaving comments, liking, or subscribing if they found the video helpful. He also invites feedback and expresses his gratitude for watching before signing off, promising to see the audience in the next video for another free response question.

Mindmap
Keywords
💡AP Calculus
AP Calculus is a high school mathematics course that covers the study of calculus, which is a branch of mathematics focused on limits, functions, derivatives, integrals, and series. In the video, Alan is discussing AP Calculus 2017 free response questions, indicating that the content is related to advanced problems in this subject area.
💡Free Response Questions
Free response questions are a type of question found on certain standardized tests, including the AP exams, where students must provide a detailed answer rather than select from multiple-choice options. Alan is focusing on these types of questions, specifically question number four, to demonstrate problem-solving skills.
💡Differential Equation
A differential equation is a mathematical equation that involves a function and its derivatives. In the context of the video, Alan is using differential equations to model the cooling process of a potato, which is a key concept in the calculus problems he is discussing.
💡Tangent Line
A tangent line is a line that touches a curve at a single point without crossing it. In calculus, finding the equation of the tangent line to a curve at a given point involves using the derivative of the function at that point. Alan uses the concept of a tangent line to approximate the internal temperature of a potato at a specific time.
💡Derivative
The derivative is a fundamental concept in calculus that represents the rate at which a function is changing at a certain point. In the video, Alan uses derivatives to find the slope of the tangent line and to analyze the concavity of the function modeling the potato's temperature.
💡Second Derivative
The second derivative is the derivative of the first derivative of a function. It provides information about the concavity of the function and can indicate if the function is concave up or down. Alan uses the second derivative to determine whether the linear approximation of the potato's temperature is an underestimate or an overestimate.
💡Concave Up
A graph is said to be concave up if, when viewed from above, the graph curves upward like the top of a bowl. In the video, Alan discusses the implications of a function being concave up, which suggests that the linear approximation of the potato's temperature at a given time might be an underestimate.
💡Separation of Variables
Separation of variables is a method used to solve differential equations by rearranging the equation so that all instances of one variable are on one side and the other variable on the opposite side. Alan uses this technique to solve the differential equation that models the cooling of the potato.
💡Integration
Integration is the process of finding the integral, or the area under the curve of a function, and is the reverse process of differentiation. In the video, Alan uses integration to solve for the function G of T that models the internal temperature of the potato over time.
💡Initial Condition
An initial condition is a specified value or condition that applies at the starting point of a problem, which helps to determine the constants in a solution. In the context of the video, Alan uses the initial condition that the internal temperature of the potato at time zero is 91 degrees Celsius to find the constant in the function G of T.
💡Linear Approximation
Linear approximation is a method of approximating the value of a function using a linear function, typically the tangent line at a point of interest. Alan uses linear approximation to estimate the internal temperature of the potato at a future time based on the current rate of cooling.
Highlights

Alan is discussing AP Calculus 2017 free response question number four focusing on the non-calculator portion.

The problem involves a boiled potato with an initial temperature of 91 degrees Celsius cooling down in a kitchen.

The potato's internal temperature is always above 27 degrees Celsius.

The temperature of the potato at time T is modeled by a function H satisfying a given differential equation.

The task is to write an equation for the line tangent to the graph of H at T equals zero.

The tangent line equation is derived using the point-slope form, with the point being T equals zero and 91 degrees Celsius.

The slope of the tangent line is determined by differentiating the function H, resulting in a slope of -16 at T equals zero.

The linear approximation equation for the tangent line is H = -16t + 91.

The internal temperature at time T equals three is approximated to be 43 degrees Celsius using the tangent line equation.

The second derivative of the function is used to determine if the approximation is an underestimate or overestimate.

Since the second derivative is positive, the function is concave up, indicating an underestimate for the temperature at T equals three.

A new model for the internal temperature of the potato is introduced, with the function G satisfying a different differential equation.

The differential equation is solved using separation of variables, leading to an expression for G of T.

The initial condition G(0) = 91 is used to find the constant in the expression for G of T.

The final expression for G of T is G = -1/3T + 4^3 + 27, with the constant determined to be 4.

The internal temperature of the potato at time T equals three is calculated to be 54 degrees Celsius using the model G.

The scoring guidelines for the AP Calculus exam are briefly mentioned, indicating the importance of accurate approximation and model representation.

Alan encourages viewers to comment, like, or subscribe for more educational content on AP Calculus.

Transcripts
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