2017 AP Calculus AB Free Response #6

Allen Tsao The STEM Coach
14 Sept 201811:24
EducationalLearning
32 Likes 10 Comments

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
  • ๐Ÿ“š Alan is working through the last AP Calculus 2017 free response question, number six, which involves calculus concepts.
  • ๐Ÿ”ข The function F(x) is defined as cosine(2x) + e^(sine(x)), which is a combination of exponential and trigonometric functions.
  • ๐Ÿ“ˆ 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.
  • ๐Ÿ‘‰ Alan identifies a computational error in his work, emphasizing the importance of accuracy in mathematical calculations.
  • ๐ŸŽ“ The script covers the application of the chain rule and product rule in differentiation, which are fundamental calculus concepts.
  • ๐Ÿ“Š 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.
  • ๐Ÿค” The video script highlights the necessity of understanding the continuity of differentiable functions, even if it's not explicitly stated in the problem.
  • ๐Ÿงฎ Alan demonstrates the process of finding derivatives of composite functions and emphasizes the importance of correctly applying the rules of differentiation.
  • ๐Ÿ“‰ The script includes a table of values for a differentiable function G and its derivative, which is used to solve related calculus problems.
  • ๐Ÿ“ 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.
  • ๐Ÿ˜“ Alan admits to making a mistake with the sign of the cosine function at pi, which affected the subsequent calculations.
  • ๐Ÿ“ 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
00:00
๐Ÿ“š 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.

05:01
๐Ÿค” 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.

10:02
๐Ÿ 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
๐Ÿ’กAP Calculus
AP Calculus is a high school mathematics course offered by the College Board in the United States. It is designed to be equivalent to a college-level calculus course and is widely recognized for its rigorous curriculum. In the video, Alan is solving a free response question from the 2017 AP Calculus exam, which is a significant part of the educational theme of the video.
๐Ÿ’กFree Response Question
A free response question is a type of question on the AP Calculus exam that requires students to write out their solutions to problems, as opposed to multiple-choice questions. These questions test the students' ability to apply mathematical concepts and perform calculations. Alan is working through question number six, which is a free response question, demonstrating the depth and application of calculus concepts.
๐Ÿ’กFunction
In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. In the video, Alan defines a function F(x) using trigonometric and exponential expressions, which is central to solving the calculus problem at hand.
๐Ÿ’กDerivative
The derivative of a function measures the rate at which the function value changes with respect to a change in its variable. It is a fundamental concept in calculus and is used to find the slope of tangent lines and to analyze the behavior of functions. Alan calculates the derivative of function F(x) to determine the slope of the tangent line at a specific point.
๐Ÿ’กTangent Line
A tangent line to a curve at a given point is a straight line that 'just touches' the curve at that point. The slope of the tangent line at a particular point on a curve is equal to the derivative of the function at that point. In the video, Alan finds the equation of the tangent line to the graph of function F at x equals pi, which is a key step in solving the problem.
๐Ÿ’กChain Rule
The chain rule is a fundamental theorem in calculus for finding the derivatives of compositions of functions. It is used when the derivative of a function involves another function whose derivative must also be considered. Alan applies the chain rule when differentiating the function F(x) and when finding the derivative of the composite function K(x).
๐Ÿ’กProduct Rule
The product rule is a formula used to find the derivatives of products of two or more functions. It states that the derivative of a product of two functions is the derivative of the first function times the second function plus the first function times the derivative of the second function. Alan uses the product rule to find the derivative of the function M(x), which is a product of two other functions.
๐Ÿ’กMean Value Theorem
The mean value theorem is a statement in calculus that formalizes the idea that if a function satisfies certain conditions, there is at least one point in its domain where the derivative of the function equals the average rate of change of the function over an interval. Alan uses the mean value theorem to find a value of c in a given interval where the derivative of function G equals a specific value.
๐Ÿ’กUnit Circle
The unit circle is a circle with a radius of one, centered at the origin of a coordinate system. It is used in trigonometry to define the trigonometric functions for all angles. Alan refers to the unit circle when evaluating the function F at x equals pi, as the values of cosine and sine at pi are crucial to finding the function's value at that point.
๐Ÿ’กSlope
Slope, in the context of a line, is a measure of the steepness of an incline or the angle at which the line is inclined with respect to the horizontal. In calculus, the slope of the tangent line to a curve at a given point is equal to the derivative of the curve at that point. Alan calculates the slope of the tangent line to the graph of function F at x equals pi, which is essential for determining the equation of the tangent line.
๐Ÿ’กDifferentiable
A function is said to be differentiable at a point if it has a derivative at that point. Differentiability is a key concept in calculus, as it allows for the study of rates of change and the application of various differentiation rules. Alan mentions that function G is differentiable, which implies it is continuous and allows for the computation of its derivative.
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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