2016 AP Calculus AB Free Response #6

Allen Tsao The STEM Coach
14 Sept 201806:06
EducationalLearning
32 Likes 10 Comments

TLDRIn this video, Alan from Bothell STEM, a math coach, concludes the 2016 AP Calculus Free Response section by tackling question number six. He begins by determining the value of K at x=3, which is necessary to find the equation of the tangent line to the graph of K. After calculating K(3) as 6, he finds the point (3,4) and then uses the chain rule to find the slope of the tangent line, which is 10. The equation of the tangent line is thus derived as y - 4 = 10(x - 3), simplifying to y = 10x - 3 + 4. Alan then moves on to find H'(1) using the quotient rule, which involves calculating derivatives and simplifying the expression. Lastly, he evaluates the integral from 1 to 3 of the second derivative of 2x, using a u-substitution to ensure the signs are correct and simplifying the integral to find its value. Alan emphasizes the importance of showing all work and using methodical approaches like u-substitution to avoid losing track of signs. He offers free homework help on Twitch or Discord and invites viewers to join for further learning and discussion.

Takeaways
  • 📚 The video is a wrap-up of the 2016 AP Calculus free response section focusing on question number six.
  • 🔢 To find the equation of the tangent line to the graph of K at x=3, Alan first calculates K(3) which is F(G(3)) = 6.
  • 📈 Alan then finds the point (3, F(6)) which is (3, 4) to use in the equation of the tangent line.
  • 🧮 Using the chain rule, he calculates the derivative of K at x=3, resulting in a slope of 10.
  • 💡 The final equation of the tangent line is y - 4 = 10(x - 3), which simplifies to y = 10x - 3 + 4.
  • 📝 For H(x) = G(x)/x, Alan applies the quotient rule to find H'(1), using values from the table provided.
  • 🧐 Alan calculates H'(1) as (F(1) * G'(1) - G(1) * F'(1)) / (F(1)^2), which simplifies to -3/2.
  • 🧬 Alan uses a u-substitution method to evaluate the integral from 1 to 3 of the second derivative, 2x dx.
  • 🔄 The bounds of integration change to from 2 to 6 after the substitution, and the integral is solved to be 7/2.
  • 🤓 Alan emphasizes the importance of showing all work and using methodical approaches like u-substitution to avoid sign errors.
  • 🎓 The video concludes with Alan offering free homework help on Twitch or Discord for those interested in math and physics.
  • 📅 Alan hints at starting on the 2015 AP Calculus exam in the next video, which will be released the following week.
Q & A

  • What is the first step in finding the equation of the tangent line to the graph of K at x equals 3?

    -The first step is to determine the value of K at x equals 3, which gives the point of tangency on the graph.

  • What is the point of tangency on the graph of K when x equals 3?

    -The point of tangency is (3, 4), as found by evaluating F of G of 3, which equals 6, and then evaluating F of 6, which equals 4.

  • What mathematical rule is used to find the derivative of K at x equals 3?

    -The chain rule is used to find the derivative of K at x equals 3, as it involves the derivative of the outer function times the derivative of the inner function.

  • What is the slope of the tangent line to the graph of K at x equals 3?

    -The slope of the tangent line is 10, which is found by applying the chain rule to the functions F and G at the given point.

  • What is the equation of the tangent line to the graph of K at x equals 3?

    -The equation of the tangent line is y - 4 = 10(x - 3), which simplifies to y = 10x - 3 + 4, or y = 10x + 1.

  • How does Alan find H prime of 1?

    -Alan uses the quotient rule to find H prime of 1, which involves the derivative of the numerator minus the product of the numerator and the derivative of the denominator, all divided by the square of the denominator.

  • What is the final result of evaluating H prime of 1?

    -The final result of evaluating H prime of 1 is -3/2, after simplifying the expression obtained from the quotient rule.

  • What substitution method does Alan use to evaluate the integral from one to three of the second derivative of 2x?

    -Alan uses the u-substitution method to evaluate the integral, which helps to ensure the correct signs and simplifies the integration process.

  • What are the new bounds of integration after applying u-substitution to the integral?

    -The new bounds of integration after u-substitution are from 2 to 6, as u of 3 equals 6 and u of 1 equals 2.

  • What is the final result of the integral evaluation from one to three of the second derivative of 2x?

    -The final result of the integral evaluation is 7.5, after performing the u-substitution and evaluating the antiderivative.

  • What additional help does Alan offer to those who have homework questions or want to learn about math and physics?

    -Alan offers free homework help on Twitch or Discord, where individuals can ask questions or learn about different parts of math and physics.

  • What does Alan plan to do in his next video?

    -Alan plans to start on the 2015 AP Calculus exam in his next video.

Outlines
00:00
📚 Calculus Problem Walkthrough

In this paragraph, Alan from Bothell StemCoach is concluding a 2016 AP Calculus exam review by addressing question number six. The problem involves functions F and G with continuous second derivatives and requires finding the equation of the tangent line to the graph of K at x equals 3. Alan demonstrates the process of finding the value of K at x equals 3, which is F(G(3)) resulting in 6, and then finding the slope of the tangent line using the chain rule for derivatives. The final equation of the tangent line is derived as y = 10x - 3 + 4. Additionally, Alan calculates H'(1) using the quotient rule and evaluates an integral involving the second derivative of 2x. The paragraph concludes with a correct reiteration of the derived equations and a summary of the steps taken.

05:03
🎓 Finalizing the AP Calculus Exam and Offering Further Help

Alan wraps up his review of the 2016 AP Calculus exam and confirms the correctness of his solutions. He emphasizes the importance of showing all work and using methods like u-substitution to keep track of signs during integration. He also mentions his preference for a methodical approach to calculus problems. The paragraph ends with Alan inviting viewers to engage with him for free homework help on Twitch or Discord, where he offers assistance with math and physics, and encourages viewers to join for a more interactive learning experience.

Mindmap
Keywords
💡AP Calculus
AP Calculus is an advanced high school mathematics course that covers topics such as limits, derivatives, integrals, and infinite series. In the video, Alan is discussing the AP Calculus free response section, which is a part of the exam that requires students to demonstrate their understanding of calculus concepts through written responses.
💡Free Response Section
The free response section of an AP exam requires students to answer questions that involve short answers, essays, or mathematical explanations. In the video, Alan is wrapping up the discussion on the 2016 AP Calculus free response section, specifically focusing on problem number six.
💡Continuous Second Derivatives
Continuous second derivatives refer to the mathematical property where a function's first and second derivatives are both continuous over an interval. This concept is important in calculus as it often indicates the smoothness of a function's graph and is a requirement for applying certain theorems. In the video, Alan mentions that functions F and G have continuous second derivatives, which is a precondition for the problem-solving approach he is demonstrating.
💡Tangent Line
A tangent line is a straight line that touches a curve at a single point without crossing it. It is used to approximate the function's value near the tangent point. In the video, Alan is looking to write an equation for the tangent line to the graph of the function K at x equals 3, which is a common problem in calculus to understand the function's behavior at a specific point.
💡Chain Rule
The chain rule is a fundamental theorem in calculus for finding the derivative of a composite function. It states that the derivative of a function composed of two functions is the product of the derivative of the outer function and the derivative of the inner function. Alan uses the chain rule to find the derivative of K of X, which is F of G of X, by multiplying the derivatives of F and G.
💡Quotient Rule
The quotient rule is a method for finding the derivative of a function that is the quotient of two other functions. It is used when differentiating expressions of the form f(x)/g(x). In the video, Alan applies the quotient rule to find H prime of X, which is G of X over X, by using the formula for the derivative of a quotient.
💡U-Substitution
U-substitution is a technique used in calculus to evaluate integrals, especially those that are difficult to solve by direct integration. It involves a change of variables to transform the integral into a simpler form. Alan uses U-substitution to evaluate the integral from one to three of the second derivative of 2x with respect to x, which simplifies the integration process and ensures the correct signs are maintained.
💡Derivative
The derivative of a function at a point measures the rate at which the function's value changes with respect to a change in its variable. It is a fundamental concept in calculus and is used to analyze the behavior of functions. In the video, Alan calculates the derivatives of functions F and G to determine the slope of the tangent line and to evaluate integrals.
💡Integral
An integral is a mathematical concept that represents the area under a curve, the accumulated value of a function over an interval, or the antiderivative of a function. In the video, Alan evaluates an integral to find the accumulated change over a given interval, which is a common application of integrals in calculus.
💡Table of Values
A table of values is a list that shows the input-output pairs of a function for selected input values. It is a useful tool for understanding the behavior of a function without graphing it. In the video, Alan refers to a table that gives the values of functions F and G at selected values of X, which helps him to find the necessary values to solve the calculus problems.
💡Antiderivative
An antiderivative, also known as an indefinite integral or primitive, is a function whose derivative is equal to the original function. Finding antiderivatives is a key part of integration in calculus. In the video, Alan finds the antiderivative of the function to evaluate the integral using U-substitution.
💡Twitch
Twitch is a popular live streaming platform often used for video games but also for various other content, including educational streams. In the video, Alan mentions offering free homework help on Twitch, indicating that he uses this platform to provide live educational support for subjects like math and physics.
Highlights

Alan from Bothell STEM is wrapping up the 2016 AP Calculus free response section.

The focus is on problem number six involving functions F and G with continuous second derivatives.

The table provides values of functions at selected X values.

The task is to write the equation of the tangent line to the graph of K at x equals 3.

K of 3 is calculated to find the point of tangency, resulting in F of G of 3 equals 6.

The point of tangency is determined to be (3, 4).

The slope of the tangent line is found using the chain rule, resulting in a slope of 10.

The equation of the tangent line is derived as y - 4 = 10(x - 3), simplifying to y = 10x - 3 + 4.

H of X is defined as G of X over X, and H prime of 1 is calculated using the quotient rule.

H prime of X is simplified to the square of the bottom function divided by the top function.

The final expression for H prime of 1 is calculated as -3/2 after simplification.

An integral from one to three of the second derivative 2x is evaluated using a u-substitution.

The bounds of integration are adjusted to reflect the substitution, changing from 1 to 3 to 2 to 6.

The antiderivative of the integral is found, resulting in a final answer of 7.5 after simplification.

Alan confirms the correctness of the derived equations and integral solutions.

Alan discusses his preference for using u-substitution in integral calculus for better sign management.

The video concludes with an invitation for viewers to join Alan on Twitch or Discord for free homework help.

Alan expresses hope to see viewers in the next video and on his streaming platforms.

Transcripts

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