|x| is Continuous but not Differentiable | Calculus 1 Exercises
TLDRThis video script explores the properties of the absolute value function, highlighting its continuity across its domain despite being non-differentiable at x=0. The presenter explains the function's definition, demonstrates its continuity at all points except zero, and shows why it fails to be differentiable at that point due to differing one-sided slopes. The script provides an intuitive understanding of the concepts of continuity and differentiability through the lens of a simple yet illustrative mathematical example.
Takeaways
- 📚 The study of derivatives and differentiable functions begins with the absolute value function due to its simplicity and common familiarity.
- 🔍 The absolute value function is continuous on its entire domain but is not differentiable at x = 0, making it an interesting example for analysis.
- 📈 The graph of the absolute value function has no holes or jumps, which aligns with the intuitive understanding of continuity.
- 📝 The absolute value function is defined piecewise, with x when x ≥ 0 and -x when x < 0, which helps in understanding its behavior at different points.
- 👉 The continuity of the absolute value function is confirmed by showing that the left and right limits at x = 0 are equal to the function's value at that point, which is zero.
- 📉 The absolute value function is differentiable at most points, with a derivative of -1 when x < 0 and +1 when x > 0.
- 🚫 The function is non-differentiable at x = 0 because the slopes from the left (-1) and right (+1) are not equal, indicating a corner where the derivative does not exist.
- 🤔 The non-differentiability at x = 0 is intuitively understood from the graph, where the function's slope changes from negative to positive as x crosses zero.
- 💡 A function is differentiable at x = c if the two-sided limit of (f(x) - f(c)) / (x - c) exists, which is the definition of the derivative.
- 📊 To show that the absolute value function is not differentiable at x = 0, one must demonstrate that the one-sided limits from the left and right are different, thus the two-sided limit does not exist.
- ❓ The script invites viewers to ask questions or request further explanations in the comments, emphasizing the interactive nature of the learning process.
Q & A
What is the absolute value function and why is it interesting in the context of derivatives?
-The absolute value function is a piecewise function defined as |x| = x when x ≥ 0 and |x| = -x when x < 0. It's interesting for studying derivatives because it is continuous on its entire domain but not differentiable at x = 0, providing a clear example of a function that meets the criteria for continuity but fails to be differentiable at a specific point.
How is the absolute value function defined for x values less than zero?
-For x values less than zero, the absolute value function is defined as the negation of x, which is -x. This is because the absolute value of a negative number is its positive counterpart.
Why is the absolute value function continuous for all x values except at x = 0?
-The absolute value function is continuous for all x values because it behaves like a simple polynomial (negative x when x < 0 and x when x > 0), which are inherently continuous. The only point of interest is x = 0, where the function changes its behavior from negative x to x, but it is still continuous at this point as the left and right limits both approach zero.
What is the definition of continuity for a function at a given point x?
-A function is continuous at a given point x if the limit of the function as it approaches x from both the left and the right is equal to the value of the function at x.
How do you show that the absolute value function is continuous at x = 0?
-To show that the absolute value function is continuous at x = 0, one must demonstrate that the limit of the absolute value function as x approaches 0 from the left is equal to the limit from the right, and both are equal to the function's value at x = 0, which is 0.
What is the derivative of the absolute value function for x values less than zero?
-The derivative of the absolute value function for x values less than zero is -1, as the function behaves like the simple polynomial -x in this domain.
What is the derivative of the absolute value function for x values greater than zero?
-The derivative of the absolute value function for x values greater than zero is 1, as the function behaves like the simple polynomial x in this domain.
Why is the absolute value function not differentiable at x = 0?
-The absolute value function is not differentiable at x = 0 because the one-sided limits from the left and right do not match. The left-hand limit gives a slope of -1, while the right-hand limit gives a slope of 1, indicating a corner where the function changes direction abruptly.
What does it mean for a function to be differentiable at a point x=c?
-A function is differentiable at a point x=c if the limit of (f(x) - f(c)) / (x - c) as x approaches c exists. This limit is also the definition of the derivative of the function at x=c.
How do you determine if the derivative of the absolute value function at x = 0 does not exist?
-To determine if the derivative at x = 0 does not exist, you evaluate the one-sided limits from the left and right. If these limits are not equal, it indicates that the two-sided limit, which defines the derivative, does not exist.
What is the significance of the absolute value function having a corner at x = 0?
-The significance of the absolute value function having a corner at x = 0 is that it demonstrates a point where the function is continuous but not differentiable. This is an important concept in calculus, as it shows that continuity and differentiability are not equivalent properties.
Outlines
📚 Continuity and Non-Differentiability of the Absolute Value Function
This paragraph delves into the properties of the absolute value function, emphasizing its continuity across its entire domain despite being non-differentiable at x=0. The explanation begins with a visual representation of the function's graph, confirming its lack of holes or jumps, indicative of continuity. The piecewise definition of the absolute value function is provided, highlighting its behavior for x < 0 and x > 0. The continuity is further justified by demonstrating that the left and right limits of the function as x approaches 0 are equal to the function's value at x=0, which is zero. The paragraph concludes by setting the stage for the next section, which will address the function's non-differentiability at x=0.
🔍 The Non-Differentiability of the Absolute Value Function at x=0
Building upon the established continuity, this paragraph explores why the absolute value function is not differentiable at x=0. The explanation is anchored in the function's graphical representation, which shows a clear 'corner' at x=0, where the slopes from the left and right are unequal. The formal definition of differentiability is recalled, stating that a function is differentiable at x=c if the two-sided limit of the function's difference quotient exists. The one-sided limits from the left and right are calculated, revealing a discrepancy: the left-hand limit yields a slope of -1, while the right-hand limit results in a slope of +1. This difference in slopes confirms that the two-sided limit, and thus the derivative, does not exist at x=0. The paragraph concludes by reinforcing the concept of a 'corner' in calculus, where the function's behavior changes abruptly, making it non-differentiable at that point.
Mindmap
Keywords
💡Derivatives
💡Differentiable Functions
💡Absolute Value Function
💡Continuity
💡Non-Differentiability
💡Piecewise Function
💡Limit
💡Slope
💡Polynomial
💡Corner
Highlights
The absolute value function serves as an interesting example in the study of derivatives and differentiable functions.
It is continuous on its entire domain but not differentiable at least at one point.
The graph of the absolute value function has no holes or jumps, indicating continuity.
The absolute value function is defined piecewise, with different expressions for x ≥ 0 and x < 0.
Every polynomial, including the negative x used for x < 0, is continuous.
The absolute value function behaves the same way from both the left and right of x > 0, confirming its continuity.
Special attention is given to the point x = 0, where the function's behavior changes.
The absolute value function is shown to be continuous at x = 0 by evaluating one-sided limits.
The limit from the left and right of x = 0 for the absolute value function equals zero, matching its value at x = 0.
The function is differentiable at most points, with derivatives of -1 for x < 0 and +1 for x > 0.
The absolute value function fails to be differentiable at x = 0, forming a corner where the slopes from the left and right differ.
Differentiability at x = c requires the existence of a two-sided limit of the function's slopes.
The one-sided limit from the left of x = 0 gives a slope of -1, while from the right it is +1.
The non-existence of the two-sided limit at x = 0 confirms the absolute value function is not differentiable at this point.
The concept of a corner in a function's graph is introduced as a point where the left and right derivatives do not match.
The explanation demonstrates the application of continuity and differentiability definitions to a practical example.
An invitation for questions or video requests is extended to the audience.
Transcripts
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