2019 AP Calculus AB Free Response Question #6
TLDRThe video transcript discusses a challenging problem from the 2019 AP Calculus AB exam. It involves three differentiable functions, F, G, and H, with given conditions that G(2) and H(2) equal 4 and a line tangent to both G and H at x=2 with a slope of 2/3. The problem is broken down into parts: Part A asks for H'(2), which is straightforwardly found to be 2/3. Part B introduces a new function, a(x) = 3x^3 * H(x), and requires finding its derivative, a'(x), and evaluating it at x=2. This involves the product rule and substitution. Part C deals with a limit involving H(x) and requires using L'Hôpital's rule to find the values of F(2) and F'(2). Finally, Part D examines the continuity of a function K(x) that is bounded by G(x) and H(x) on the interval [1, 3], concluding that K must be continuous at x=2 based on the properties of G and H and the application of the sandwich theorem.
Takeaways
- 📚 The functions F, G, and H are all twice differentiable, which implies they are also continuous.
- 🔑 G(2) and H(2) are both equal to 4, and there's a line tangent to both G and H at the point (2, 4) with a slope of 2/3.
- 🔍 Part A asks to find H'(2), which is simply the slope of the tangent line, hence H'(2) = 2/3.
- 📝 Part B introduces a new function a(x) = 3x^3 * H(x) and requires finding its derivative, a'(x), using the product rule.
- 🧮 To find a'(2), substitute x with 2 in the expression for a'(x), and use the given values for H(2) and H'(2).
- 🟢 Part C involves using l'Hôpital's rule to find the limit of H(x) as x approaches 2, which requires differentiating the numerator and denominator separately.
- 🔬 It's determined that F(2) must be 1 due to the equation involving F(x) in the problem statement.
- 📉 F'(2) is found by applying l'Hôpital's rule to the expression resulting from the limit of H(x) as x approaches 2.
- 🤔 The continuity of H and G is used to argue that K(x) must also be continuous at x = 2, based on the given inequality G(x) ≤ K(x) ≤ H(x) for x in the interval [1, 3].
- 👌 The Sandwich Theorem is applied to show that K must equal 4 at x = 2, given that G(2) = H(2) = 4.
- 📌 The three conditions for continuity are checked: the limit as x approaches 2 exists, K(2) is defined, and these two values are equal, confirming the continuity of K at x = 2.
Q & A
What does it mean for a function to be differentiable?
-If a function is differentiable, it implies that it is continuous.
What is the significance of the fact that the line is tangent to both the graphs of G and H at x = 2?
-The slope of the tangent line corresponds to H prime of 2, as it represents the derivative of H at x = 2.
How is the product rule applied in Part B to find the derivative of the new function 'a'?
-The product rule is used by taking the derivative of the first function (3x^3) times the original second function (H of X), and adding to that the original first function times the derivative of the second function.
What is the value of a prime of 2, and how is it calculated?
-a prime of 2 is calculated by substituting 2 into the derivative expression of 'a'. It is obtained as -160.
Explain how L'Hopital's rule is used in Part C.
-L'Hopital's rule is applied to evaluate the limit as X approaches 2 for H of X. It is used when the limit yields 0/0 or ∞/∞, indicating an indeterminate form.
What is the value of F prime of 2, and how is it determined?
-F prime of 2 is determined as -1/3. It is obtained by applying L'Hopital's rule to the derivative expression of F, then substituting 2 into the resulting expression.
How does the continuity of functions G and H help in determining the continuity of function K at x = 2?
-The continuity of G and H ensures that the limit as X approaches 2 for G and H equals their respective function values at x = 2. This information is crucial in demonstrating the continuity of K at x = 2.
What is the Sandwich Theorem, and how is it applied in the context of this problem?
-The Sandwich Theorem states that if two functions, G and H, sandwich another function, K, between them on an interval, and the limits of G and H approach the same value at a point, then the limit of K at that point must also equal that value. In this problem, it's used to justify that the limit and function value of K at x = 2 are both 4, given the continuity of G and H.
What are the three criteria that need to be checked to determine the continuity of K at x = 2?
-The three criteria are: 1) The limit as X approaches 2 for K must exist, 2) K must be defined at x = 2, and 3) The limit and function value of K at x = 2 must be equal.
How is the continuity of functions G and H related to the continuity of K at x = 2?
-The continuity of G and H ensures that the limit and function value of K at x = 2 are both 4, as per the Sandwich Theorem. This information establishes the continuity of K at x = 2.
Outlines
📚 Understanding Differentiability and Tangent Lines
The first paragraph introduces a problem from the 2019 AP Calculus AB exam, focusing on the differentiability of three functions F, G, and H. It emphasizes the importance of differentiability, which implies continuity, a concept used later in the problem. The functions are given with specific values at x=2 and a line tangent to both G and H at that point. The paragraph outlines the tasks of Part A, which is to find H'(2), and Part B, which involves creating a new function a(x) and deriving it. The solution process includes using the product rule for derivatives and substituting given values to find a'(2). The paragraph concludes with the setup for Part C, which involves evaluating a limit using l'Hôpital's rule and understanding the implications of the differentiability of H.
🔍 Applying L'Hôpital's Rule and Finding Function Values
The second paragraph delves into the application of l'Hôpital's rule to find the limit of H(x) as x approaches 2. It discusses the necessity of having an indeterminate form (0/0) for the rule to apply and how the continuity of H helps in determining the function value at x=2 without direct calculation. The paragraph also explores finding F(2) and F'(2) by analyzing the given equation and using the chain rule. It concludes with the evaluation of the limit as 4, which is derived from the continuity of H and the provided function values, leading to the calculation of F'(2) as -1/3.
🏗️ Continuity and the Sandwich Theorem
The third paragraph addresses the continuity of a function K(x) that is bounded between G(x) and H(x) on the interval from 1 to 3. It uses the differentiability and continuity of G and H to argue that there are no discontinuities, holes, or asymptotes in their graphs within the interval. The paragraph applies the sandwich theorem to show that K(x) must also be continuous at x=2, given that the limits as x approaches 2 for G(x) and H(x) both equal 4. By substituting x=2 into the inequalities, it is shown that K(2) must also equal 4 to maintain the inequality, thus confirming the continuity of K at x=2.
Mindmap
Keywords
💡Differentiable
💡Tangent Line
💡Product Rule
💡L'Hôpital's Rule
💡Continuity
💡Derivative
💡Limit
💡Sandwich Theorem
💡Chain Rule
💡Exponent
💡Inequality
Highlights
Problem 6 from the 2019 AP Calculus AB exam is discussed in detail
Functions F, G, and H are all twice differentiable, which implies they are continuous
The line y-4 = (2/3)(x-2) is tangent to the graphs of G and H at x=2
Part A asks to find H'(2), which is the slope of the tangent line, 2/3
Part B introduces a new function a(x) = 3x^3 * H(x) and asks to find a'(x) and a'(2)
The product rule is used to find the derivative a'(x)
By substituting x=2 and using H(2)=4 and H'(2)=2/3, a'(2) is calculated to be 160
Part C deals with the equation H(x) = x^2 - 4 / (x^3 - f(x)^3 + 1)^(1/3)
L'Hopital's rule is used to find the limit of H(x) as x approaches 2
The continuity of H implies the limit as x approaches 2 is H(2) = 4
F(2) is determined to be 1 by setting the denominator of the fraction equal to zero
F'(2) is found by applying L'Hopital's rule and using the continuity of H
Part D discusses the inequality G(x) ≤ K(x) ≤ H(x) for x in [1,3]
The continuity of G and H implies K must also be continuous at x=2
The limit as x approaches 2 on K(x) is shown to exist and equal K(2)=4 using the Sandwich Theorem
The three conditions for continuity (limit exists, function is defined, limit equals function value) are verified for K at x=2
The solution process involves concepts of differentiability, continuity, derivatives, limits, and L'Hopital's rule
Transcripts
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