2017 AP Calculus AB Free Response #6
TLDRIn the video, Alan from Bothell STEM tackles the last AP Calculus 2017 free response question, which is question number six. The problem involves functions F, G, and H, where F is a trigonometric function, G is a differentiable function with given values and derivatives, and H is a function represented by a graph with five line segments. Alan calculates the slope of the tangent line to the graph of F at x equals pi, finds the derivative of K, which is a function of H composed with F, and determines M'(2) for a function M defined by a product of G and H. He also discusses the existence of a value C in a given interval where G'(C) equals a specific slope using the Mean Value Theorem. Throughout the video, Alan emphasizes the importance of understanding the unit circle and the rules of differentiation. Despite a few computational errors and oversights, he provides a detailed explanation of the steps involved in solving the problem, offering valuable insights into the process.
Takeaways
- ๐ Alan is working through the last AP Calculus 2017 free response question, number six, which involves calculus concepts.
- ๐ข The function F(x) is defined as cosine(2x) + e^(sine(x)), which is a combination of exponential and trigonometric functions.
- ๐ The task involves finding the slope of the tangent line to the graph of function f at x equals pi, which requires knowledge of derivatives and the unit circle.
- ๐ Alan identifies a computational error in his work, emphasizing the importance of accuracy in mathematical calculations.
- ๐ The script covers the application of the chain rule and product rule in differentiation, which are fundamental calculus concepts.
- ๐ Alan discusses the use of the Mean Value Theorem to find a value C in a given interval where the derivative equals a specific value.
- ๐ค The video script highlights the necessity of understanding the continuity of differentiable functions, even if it's not explicitly stated in the problem.
- ๐งฎ Alan demonstrates the process of finding derivatives of composite functions and emphasizes the importance of correctly applying the rules of differentiation.
- ๐ The script includes a table of values for a differentiable function G and its derivative, which is used to solve related calculus problems.
- ๐ Alan finds the slope of the tangent line to the graph of f at x = pi by evaluating the derivative of f at that point.
- ๐ Alan admits to making a mistake with the sign of the cosine function at pi, which affected the subsequent calculations.
- ๐ The video concludes with a reminder that minor mistakes can still allow for a good performance on exams, as long as the process is well-understood.
Q & A
What is the function F defined by in the transcript?
-The function F is defined by f(x) = cosine(2x) + e^(sine(x)).
What is the significance of knowing the unit circle for the value of cosine(2ฯ) in the script?
-Knowing the unit circle is crucial because it helps to quickly determine that cosine(2ฯ) equals 1, as 2ฯ corresponds to a full rotation on the unit circle where the cosine value repeats.
What is the point that the tangent line to the graph of f at x equals ฯ passes through?
-The point that the tangent line passes through is (ฯ, 2), as f(ฯ) equals cosine(2ฯ) + e^(sine(ฯ)), which simplifies to 2 due to the properties of cosine and sine at ฯ.
How is the slope of the tangent line to the graph of f at x equals ฯ calculated?
-The slope is calculated by taking the derivative of f, which is f'(x) = -sine(2x) * 2 + e^(sine(x)). Then, plugging in x = ฯ, the slope is found to be 1 because sine(2ฯ) is 0 and e^(sine(ฯ)) is e^0, which is 1.
What is the function K defined by in the transcript?
-The function K is defined by K(x) = H(f(x)), where H and f are given functions.
How is K'(x), the derivative of K, found using the chain rule?
-K'(x) is found by applying the chain rule, which states that K'(x) = H'(f(x)) * f'(x), where H' and f' are the derivatives of H and f, respectively.
What is the value of K'(ฯ) in the transcript?
-K'(ฯ) is calculated to be H'(2) * f'(ฯ), which after substituting the known values, is determined to be -1/3.
What is the function M defined by in the transcript?
-The function M is defined by M(x) = G(-2x) * H(x), where G and H are given functions.
How is M'(2), the derivative of M at x=2, calculated?
-M'(2) is calculated using the product rule, which results in M'(2) = G'(-4) * H'(2) + H(2) * G'(-4) * (-2), after substituting the given values it simplifies to -4/3 - 9/3.
What theorem is used to determine if there exists a number C in the closed interval [-5, 3] such that G'(C) = -4?
-The Mean Value Theorem is used to determine the existence of such a number C, stating that there must be a point in the interval where the derivative equals the average rate of change over that interval.
What mistake was made in the calculation of the slope of the tangent line to the graph of f at x equals ฯ?
-The mistake was that cosine(ฯ) was incorrectly stated as 1 instead of -1, which affected the final calculation of the slope.
What is the importance of the Mean Value Theorem in solving the problem for part C of the script?
-The Mean Value Theorem is important as it guarantees the existence of at least one point C in the given interval where the derivative of G equals the average rate of change over that interval, which in this case is -4.
Outlines
๐ Calculus AP Exam Question 6: Tangent Line and Derivatives
In this paragraph, Alan from Bothell Stem, Coach tackles the last free response question from the 2017 AP Calculus exam. The question involves a function F(x) = cos(2x) + e^(sin(x)), a differentiable function G(x) with given values and derivative, and a function H(x) represented by a graph consisting of five line segments. Alan explains the process of finding the slope of the tangent line to the graph of F at x = ฯ. He calculates the point that the line passes through and the slope using the derivative F'(x). Afterward, he discusses the function K(x) = H(f(x)) and its derivative K'(x), applying the chain rule. Finally, he addresses the function M(x) = G(-2x) * H(x) and calculates M'(2) using the product rule. Alan also explores the existence of a value C in the interval [-5, 3] where G'(C) = -4, using the Mean Value Theorem.
๐ค Errors and Corrections in Calculus AP Exam Analysis
Alan reviews his work on the AP Calculus exam question and realizes he made a mistake with the unit circle values, particularly with the cosine of ฯ, which should be -1 instead of 1. He corrects the slope of the tangent line accordingly. He also acknowledges a sign error in calculating K'(ฯ) and M'(2), providing the correct values. Alan discusses the implications of G being differentiable, which implies continuity, although he did not explicitly state it. He admits to the computational errors but emphasizes that the steps he outlined are crucial for solving such problems. Alan concludes by reflecting on his performance, noting that minor mistakes are common but do not necessarily detract from doing well on the exam.
๐ Wrapping Up the 2017 AP Calculus Exam Discussion
In the final paragraph, Alan summarizes his experience with the 2017 AP Calculus exam, acknowledging that despite a few minor errors, he believes he performed well. He emphasizes the importance of understanding the process and steps for solving calculus problems, even if perfection is not achieved. Alan invites viewers to comment, like, or subscribe for more content and hints at reviewing a previous year's exam in the next video. He signs off, indicating he will cover the 2016 AP Calculus exam in his next session.
Mindmap
Keywords
๐กAP Calculus
๐กFree Response Question
๐กFunction
๐กDerivative
๐กTangent Line
๐กChain Rule
๐กProduct Rule
๐กMean Value Theorem
๐กUnit Circle
๐กSlope
๐กDifferentiable
Highlights
Alan is solving the last AP Calculus 2017 free response question, number six.
Function F is defined by f(x) = cosine(2x) + e^(sine(x)).
Function G is given with values and its derivative for selected x values in a table.
Function H is represented by a graph consisting of five line segments.
The slope of the tangent line to the graph of f at x=ฯ is required.
At x=ฯ, f equals 2 as cosine(2ฯ) is 1 and e^(sine(ฯ)) is 1.
The derivative F' is calculated using the chain rule and the derivative of e^x.
The slope of the tangent line at x=ฯ is found to be 1 after plugging in the values.
The function K is defined by K(x) = H(f(x)) and its derivative K'(x) is found using the chain rule.
K'(ฯ) is calculated to be -1/3 using the values of H' and F' at the given points.
Function M is defined by M(x) = G(-2x) * H(x) and M'(2) is calculated using the product rule.
M'(2) is determined to be -3 after substituting the given values.
The question asks if there exists a number C in the interval [-5, 3] such that G'(C) is -4.
The mean value theorem is applied to find that there exists a C with G'(C) = -4.
Alan made a computational mistake by incorrectly stating the value of cosine(ฯ) as 1 instead of -1.
The correct slope of the tangent line is actually -1 due to the error in cosine(ฯ).
Alan acknowledges the mistake and explains the importance of knowing the unit circle well.
The final answer for part A should be y - 2 = -1(x - ฯ), which is y = x - ฯ - 2.
Alan concludes the video by reflecting on the process and encouraging viewers to learn from mistakes.
Transcripts
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