2007 AP Calculus AB Free Response #4

Allen Tsao The STEM Coach
10 Dec 201807:44
EducationalLearning
32 Likes 10 Comments

TLDRIn this educational video, Alan from Bothell Stem Coach dives into solving the AP Calculus 2007 free response question number four, which involves a particle moving along the x-axis with a given position function. The focus is on finding the time 'T' when the particle is furthest to the left, which corresponds to the minimum position value. Alan explains the process of finding the minimum by setting the first derivative of the position function to zero and solving for 'T'. He identifies two potential solutions, ฯ€/4 and 5ฯ€/4, and uses the first derivative test to confirm that 5ฯ€/4 is indeed when the particle is at its minimum position. The video then proceeds to solve for the constant 'a' in a given differential equation by calculating the first and second derivatives and plugging them into the equation. Alan finds that 'a' must equal 1/2 for the equation to hold true for all 'T'. The video is a hands-on walkthrough of calculus concepts, emphasizing the importance of understanding unit circles and the first derivative test, and it concludes with an invitation for viewers to engage with the content and seek further help through offered platforms.

Takeaways
  • ๐Ÿ“š The video discusses solving AP Calculus 2007 free response question 4, which involves a particle moving along the x-axis with a given position function.
  • ๐Ÿ•’ The focus is on the non-calculator portion of the exam, where the problem requires finding the time T when the particle is furthest to the left.
  • ๐Ÿ“‰ To determine the leftmost position, the video explains looking for the minimum value of the position function X(T), which is when X is most negative.
  • ๐Ÿ”ข The process involves finding the first derivative X'(T) and setting it equal to zero to find critical points.
  • ๐Ÿ”„ By using the product rule, the derivative is expressed as a function of cosine and sine, which is then factored to isolate cosine T - sine T.
  • ๐Ÿงฎ It is noted that the exponential function e^(-T) is never zero, so the equation T - sine T = 0 is solved to find the critical points T = ฯ€/4 and 5ฯ€/4.
  • ๐Ÿ” The first derivative test is used to determine if these points are minimums by checking the sign of the derivative at points around the critical points.
  • ๐Ÿ“Œ The video emphasizes the importance of understanding the unit circle and the relationship between cosine and sine for different angles.
  • ๐Ÿ”— The second derivative X''(T) is calculated using the product rule, which simplifies to -2e^(-T)cosine T.
  • ๐Ÿ”‘ To satisfy the given differential equation aX''(T) + X'(T) + X(T) = 0 for all T, the video concludes that a must be equal to 1/2.
  • ๐Ÿ“ The presenter mentions the importance of checking endpoints and the absolute minimum, which was not fully addressed in the solution.
  • ๐Ÿ‘ The video concludes with an invitation for viewers to engage by leaving comments, likes, or subscribing, and mentions offering free homework help on Twitch and Discord.
Q & A
  • What is the main topic of the video?

    -The main topic of the video is working on the AP Calculus 2007 free response questions, specifically focusing on the non-calculator portion and finding the time at which a particle is furthest to the left given its position function.

  • What is the position function of the particle given in the video?

    -The position function of the particle is not explicitly provided in the transcript, but it is mentioned that it is given for the time interval 0 <= T <= 2ฯ€.

  • What does 'X' represent in the context of the video?

    -'X' represents the position of the particle along the x-axis at a given time 'T'.

  • How does the video determine the time when the particle is furthest to the left?

    -The video determines this by finding the minimum value of the position function, which corresponds to the particle being furthest to the left since a negative X value indicates a position to the left of the origin.

  • What mathematical concept is used to find the minimum value of the position function?

    -The concept of derivatives is used to find the minimum value. Specifically, the first derivative of the position function is set to zero to find critical points, which are potential locations for minimums or maximums.

  • What is the significance of setting the derivative equal to zero?

    -Setting the derivative equal to zero finds the critical points where the slope of the tangent to the curve is horizontal, which are potential locations for minima, maxima, or points of inflection.

  • How does the video use the first derivative test to determine the minimum?

    -The first derivative test examines the sign of the first derivative to the left and right of a critical point. If the sign changes from negative to positive, it indicates a minimum.

  • What is the role of the exponential function e^-T in the calculations?

    -The exponential function e^-T is a common factor in both the first and second derivatives of the position function, and it is used to simplify the equations and find the values of 'T' that satisfy the given conditions.

  • What is the equation that the constant 'a' must satisfy according to the video?

    -The constant 'a' must satisfy the equation aX''(T) + X'(T) + X(T) = 0, where X''(T), X'(T), and X(T) are the first and second derivatives and the position function of the particle, respectively.

  • How does the video find the value of the constant 'a'?

    -The video finds the value of 'a' by substituting the expressions for the first and second derivatives and the position function into the equation and then simplifying to find a value for 'a' that makes the equation true for all 'T'.

  • What is the final value of 'a' that the video concludes?

    -The final value of 'a' that the video concludes is a = 1/2.

  • What additional step did the video mention should have been done but was not?

    -The video mentioned that the endpoints at 'T = 0' and 'T = 2ฯ€' should have been checked to ensure the absolute minimum, which was not done in the video.

  • What resources does the video offer for further help?

    -The video offers free homework help on platforms like Twitch and Discord for those who need additional assistance.

Outlines
00:00
๐Ÿ“š AP Calculus 2007 Free Response Question Analysis

In this paragraph, Alan from Bothell Stem Coach begins by introducing the topic of AP Calculus 2007 free response questions, specifically focusing on the non-calculator portion. The analysis is centered around question number four, which involves a particle moving along the x-axis with a given position function of time T. The goal is to find the time T when the particle is furthest to the left, which corresponds to the most negative value of the position function. Alan uses calculus to find the minimum of the position function by taking the derivative and setting it equal to zero. He applies the product rule and factors out terms to solve for T, eventually concluding that the particle is furthest to the left at T equal to ฯ€/4 and 5ฯ€/4. The paragraph also includes a discussion on the first derivative test to confirm the minimum and a brief mention of finding the constant 'a' that satisfies a given differential equation involving the first and second derivatives of the position function.

05:03
๐Ÿงฎ Solving a Differential Equation for Constant 'a'

The second paragraph continues the mathematical discussion by addressing a differential equation involving the second derivative (X double prime of T), first derivative (X prime of T), and the original function (X of T). Alan computes the second derivative using the product rule and then substitutes the expressions for the first and second derivatives, as well as the original function, into the given equation. By factoring out terms and simplifying, he finds that for the equation to hold true for all values of T, the constant 'a' must be equal to 1/2. The paragraph concludes with a reminder to check endpoints, which Alan admits he overlooked, and a summary of the findings. The video ends with a prompt for viewers to engage by leaving comments, likes, or subscribing, and an invitation to receive free homework help on Twitch and Discord.

Mindmap
Keywords
๐Ÿ’กAP Calculus
AP Calculus refers to Advanced Placement Calculus, a set of standardized high school courses in the United States that are designed to be equivalent to first semester college calculus courses. In the video, the focus is on solving the 2007 AP Calculus exam's free response questions, particularly the non-calculator portion, emphasizing the application of calculus concepts without computational aids.
๐Ÿ’กFree Response
In the context of AP exams, 'Free Response' refers to questions that require the test taker to generate their own answers rather than selecting from provided options, like in multiple-choice questions. This portion of the exam tests a student's ability to apply mathematical theories and techniques to solve problems. In the video, the presenter is explaining how to approach solving these types of problems on the AP Calculus exam.
๐Ÿ’กDerivative
The derivative in calculus is a measure of how a function changes as its input changes. It's foundational for understanding rates of change and motion. In the video, the presenter calculates the derivative of the position function, X'(t), to determine when the particle is at its furthest left position by finding when this derivative equals zero.
๐Ÿ’กMinimum
A 'minimum' refers to the lowest point on a curve, which is of particular interest when determining extreme values of functions in calculus. In the transcript, the presenter is searching for the minimum of the function X(t) to find when the particle is furthest to the left, as this corresponds to the most negative value of X.
๐Ÿ’กProduct Rule
The product rule is a calculus formula used to find the derivative of products of functions. It states that d(uv)/dt = u'v + uv'. This rule is applied in the video when calculating the derivative of the position function, which involves the product of two functions involving e^{-t} and trigonometric functions.
๐Ÿ’กUnit Circle
The unit circle is a fundamental concept in trigonometry, describing a circle with radius one centered at the origin of the coordinate system. It's crucial for understanding the relationships between trigonometric functions. In the video, knowledge of the unit circle is used to determine when cos(t) = sin(t), which are critical points in solving the problem.
๐Ÿ’กFirst Derivative Test
The first derivative test is a method used in calculus to determine whether a function has a local minimum or maximum at certain critical points. This test involves checking the sign of the derivative before and after the critical point. The video applies this test to determine where the particle's position function X(t) has its minimum value.
๐Ÿ’กCritical Points
Critical points occur where the derivative of a function is zero or undefined, indicating potential local maxima, minima, or points of inflection. In the video, the presenter finds critical points for X(t) where the derivative X'(t) equals zero, which are crucial for determining the position's minimum value.
๐Ÿ’กSecond Derivative
The second derivative of a function, indicated as f''(x), represents the rate at which the function's rate of change is changing and is used to determine the concavity of the function. In the video, the second derivative is used in an equation to solve for the constant a that makes the function satisfy a given differential equation.
๐Ÿ’กExponential Function
Exponential functions are mathematical functions of the form f(x) = a^x, where a is a constant, representing continuous growth or decay. They are prominently featured in calculus for solving differential equations. In the transcript, e^{-t}, an exponential function, plays a key role in the position and velocity equations of the particle.
Highlights

Alan from Bothell stem coach is working on AP Calculus 2007 free response questions.

Focusing on the non-calculator portion of the exam, specifically question number four.

The problem involves finding the time when a particle is furthest to the left on the x-axis.

The position of the particle at time T is given by a specific function for 0 < T โ‰ค 2ฯ€.

The most negative value of X corresponds to the particle being furthest to the left.

To find the minimum value of X, Alan takes the derivative of the position function and sets it to zero.

Using the product rule, the derivative involves e^(-T), cosine(T), and sine(T).

The equation cos(T) - sin(T) = 0 is derived from setting the derivative equal to zero.

The solution to the equation involves looking at the unit circle where x and y values are the same.

Two possible times for the particle to be furthest left are identified: T = ฯ€/4 and T = 5ฯ€/4.

The first derivative test is used to confirm which of these times corresponds to the minimum.

Values are chosen around the suspected minimum times to check the sign of the derivative.

A change in the sign of the derivative from negative to positive indicates a minimum at T = 5ฯ€/4.

The constant 'a' is found by solving the equation aX''(T) + X'(T) + X(T) = 0.

The second derivative X''(T) is computed using the product rule and factorization.

After plugging the derivatives into the equation, factorization leads to e^(-T)cos(T) - 2a + 1 = 0.

For the equation to hold true for all T, a is determined to be equal to 1/2.

Alan emphasizes the importance of checking endpoints for absolute minimums, which was overlooked.

The video concludes with an offer for free homework help on twitch and discord.

Transcripts
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