- #1
gillgill
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For what values of x is │x^2-4│ not differentiable?
Is there a way to solve it without looking at the graph?
Is there a way to solve it without looking at the graph?
Differentiability is a property of a function that indicates whether the function has a well-defined derivative at a given point. This means that the graph of the function is smooth and has a unique tangent line at that point.
The absolute value is used in this question because the function │x^2-4│ contains a term with an exponent, which could result in a negative value. Using the absolute value ensures that the function remains positive and allows us to analyze its differentiability more easily.
Continuity refers to the property of a function where there are no abrupt changes or breaks in the graph. A function can be continuous without being differentiable, but a function cannot be differentiable without being continuous.
Yes, we can determine these values by finding the points where the derivative of the function does not exist, or where the function has a sharp corner or cusp. These points are also known as points of non-differentiability.
To prove that a function is not differentiable, we can use the definition of differentiability and show that the limit of the difference quotient does not exist at a certain point. We can also look for points of non-differentiability, as mentioned in the previous question, and show that the function is not smooth at those points.