2011 Calculus AB free response #6a | AP Calculus AB | Khan Academy

Khan Academy
12 Sept 201104:22
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TLDRThe video script discusses the continuity of a piecewise function, f(x), at x=0. It explains that for a function to be continuous, the limit from the left and the right as x approaches 0 must equal the function's value at 0. By evaluating the function at x=0 and calculating the limits from both sides, the script demonstrates that f(x) is indeed continuous at x=0, as all values are found to be equal to 1.

Takeaways
  • 📌 The function f(x) is defined with two distinct cases based on the value of x.
  • 🔍 For x ≤ 0, f(x) = 1 - 2sin(x), and for x > 0, f(x) = e^(-4x).
  • 🌟 The continuity of f(x) at x = 0 is the focus of the problem.
  • 🤔 Continuity requires that the limit from the left equals the limit from the right, and both equal the function's value at that point.
  • 📈 The sine function is used in the definition of f(x) for x ≤ 0.
  • 📊 The limit of f(x) as x approaches 0 from the left is evaluated using the properties of the sine function.
  • 📂 The limit from the left is found to be 1, as sin(0) = 0 and e^(-4*0) = 1.
  • 🧠 For x > 0, the function f(x) utilizes the exponential function e^(-4x).
  • 📈 The limit of f(x) as x approaches 0 from the right is also evaluated, resulting in the same value of 1.
  • 🔗 The value of f(0) is calculated to be 1, using the definition of f(x) for x ≤ 0.
  • 🎯 All calculated values (f(0), limit from left, and limit from right) are equal to 1, confirming the continuity of f(x) at x = 0.
Q & A

  • What is the definition of function f(x) for x ≤ 0?

    -For x less than or equal to 0, the function f(x) is defined as 1 minus 2 times the sine of x.

  • How is function f(x) defined for x > 0?

    -For x greater than 0, f(x) is defined as e to the power of negative 4x.

  • What does it mean for a function to be continuous at a certain point?

    -A function is continuous at a certain point if the limit of the function as x approaches that point from the left is equal to the function's value at that point and also equal to the limit as x approaches from the right.

  • What is the value of f(0) according to the given definition?

    -The value of f(0) is 1, since sine of 0 is 0, and thus 1 minus 2 times 0 equals 1.

  • How do we determine the limit of f(x) as x approaches 0 from the left?

    -The limit of f(x) as x approaches 0 from the left is the same as the value of f(0) because the function's definition for x ≤ 0 applies, which we've determined to be 1.

  • What is the limit of e to the negative 4x as x approaches 0 from the right?

    -The limit of e to the negative 4x as x approaches 0 from the right is 1, since any number to the power of 0 is 1.

  • How does the continuity of the sine function and the exponential function play a role in this problem?

    -Both the sine function and the exponential function are continuous, which ensures that the limits as x approaches 0 from the left and from the right exist and can be equated to the function's value at 0.

  • What would happen if the function's value at a point was not equal to the limits approaching that point from the left and right?

    -If the function's value at a point was not equal to the limits approaching that point, the function would not be continuous at that point, potentially creating a gap in the graph.

  • How can we verify the continuity of f(x) at x = 0 using the given information?

    -By showing that the value of f(0), the limit of f(x) as x approaches 0 from the left, and the limit as x approaches 0 from the right are all equal to 1, we can verify that f(x) is continuous at x = 0.

  • What is the significance of a function being continuous over an interval?

    -A function being continuous over an interval means that there are no gaps, jumps, or breaks in the graph of the function within that interval, which is important for many mathematical and practical applications.

  • What other properties or characteristics of functions might be relevant when studying continuity?

    -Other relevant properties when studying continuity include differentiability, integrability, and the existence of limits at every point within the interval, all of which contribute to a thorough understanding of the function's behavior.

Outlines
00:00
📚 Continuity of Function f at x=0

This paragraph discusses the continuity of a function, f, at x=0. The function is defined with two cases: when x ≤ 0, f(x) = 1 - 2sin(x), and when x > 0, f(x) = e^(-4x). The main objective is to demonstrate that f is continuous at x=0. This requires showing that the left-hand limit and the right-hand limit of f(x) as x approaches 0 are equal to the value of f(0). The explanation includes a visual representation of the function and its behavior around x=0, emphasizing the importance of the function's value and the limits matching for continuity. The sine function's continuity is utilized to evaluate the left-hand limit, which is found to be 1. Similarly, the right-hand limit is evaluated using the exponential function's properties, also resulting in 1. Since both limits and the function's value at x=0 are equal, it is concluded that the function f is continuous at x=0.

Mindmap
Keywords
💡Continuous Function
A continuous function is one where, for every point within its domain, the limit as the function approaches the point is equal to the function's value at that point. This concept is central to the video's discussion on demonstrating the continuity of a piecewise function at a specific point (x = 0). The video elaborates on this by breaking down the condition into three parts: the function's behavior approaching from the left, from the right, and its value exactly at the point of interest, showing that all three must be equal for continuity.
💡Piecewise Function
A piecewise function is defined by different expressions for different parts of its domain. In the video, the function f(x) is a piecewise function with two cases: one expression (1 - 2 sin(x)) when x is less than or equal to 0, and another (e^(-4x)) when x is greater than 0. The video focuses on proving that this piecewise function is continuous at x = 0.
💡Limit
A limit refers to the value that a function approaches as the input approaches some value. The video uses the concept of limits to analyze the behavior of the piecewise function as x approaches 0 from both the left and the right. By showing that these limits equal the function's value at x = 0, the video demonstrates the function's continuity at this point.
💡Sine Function
The sine function, sin(x), is a periodic function that's important in trigonometry. The video uses the sine function as part of the expression defining the piecewise function for x ≤ 0. It's mentioned to illustrate that sin(0) = 0, contributing to the calculation of the function's value and the limit from the left as x approaches 0.
💡Exponential Function
The exponential function, denoted as e^(x), is a function where the base e (Euler's number) is raised to the power of x. The video discusses an exponential function, e^(-4x), for x > 0 part of the piecewise function. It explains how evaluating this function as x approaches 0 from the right contributes to showing the function's continuity.
💡Approaching from the Left
This concept refers to examining the behavior of a function as the input value gets increasingly closer to a certain point from values less than the point. In the video, the limit of f(x) as x approaches 0 from the left is considered to show that the piecewise function's left-hand limit at x = 0 is consistent with the function's value at x = 0.
💡Approaching from the Right
This concept refers to the examination of a function's behavior as the input value approaches a specific point from values greater than the point. The video looks at the limit of f(x) as x approaches 0 from the right, demonstrating that the piecewise function's right-hand limit at x = 0 aligns with the function's value at x = 0, thus supporting continuity.
💡Function Value at a Point
The value of a function at a particular point refers to the result of plugging that point into the function. The video explains that f(0) equals 1 by substituting 0 into the appropriate part of the piecewise function. This calculation is crucial for showing that the function's value at x = 0 matches the limits as x approaches 0 from both sides.
💡Gap
A gap in a graph of a function represents a discontinuity, where the function does not have a defined value at a certain point, or the limit does not equal the function's value at that point. The video uses this concept to explain what it means for a function not to be continuous, contrasting this with the continuous case being demonstrated.
💡Continuous at x = 0
This specific phrase refers to a function being continuous at the point x = 0, meaning there is no interruption, jump, or gap in its graph at this point. The entire video is dedicated to proving that the piecewise function in question is continuous at x = 0 by showing that the function's limits from the left and right and its value at x = 0 are all equal.
Highlights

The function f(x) is defined with two cases based on the value of x.

For x ≤ 0, f(x) = 1 - 2sin(x).

For x > 0, f(x) = e^(-4x).

The task is to demonstrate the continuity of f(x) at x = 0.

Continuity at x = 0 requires the left and right limits to equal the function's value at that point.

The limit from the left for f(x) when x approaches 0 is 1 - 2sin(0), which equals 1.

The sine function is continuous, which helps in determining the limit from the left.

The limit from the right for f(x) as x approaches 0 is e^(-4*0), which equals 1.

The exponential function used in the second case is also continuous.

The value of f(0) is calculated as 1 - 2sin(0), resulting in 1.

The function's value at x = 0 matches the limits from both the left and the right.

The continuity of f(x) at x = 0 is confirmed by the equality of the function's value and its limits.

A gap in the function's value and its limits would indicate discontinuity.

The concept of limits is crucial for determining continuity.

The transcript provides a step-by-step analysis of the function's continuity at a specific point.

The mathematical concepts of continuity, limits, and the properties of sine and exponential functions are central to the discussion.

Transcripts

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