Discontinuous function at second derivative

In summary, the first and second derivatives of the function are both continuous at t = 1, but the first derivative has a cusp, causing the second derivative to not exist at t = 1.
  • #1
Applejacks
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Homework Statement



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Homework Equations





The Attempt at a Solution



I was wondering why the second derivative at t=1 does not exist but exists at the first derivative. What I did was draw the graph of the function, then the first derivative and lastly the second derivative. I found that at t=1, the function and its first derivative agreed on both the left hand and right hand sides. However, at t=1 for the second derivative, the top part of the function was 2 and the bottom was 0. Is this why it's not continuous for t=1 at the second derivative?
 
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  • #2
The function and its first derivative are both continuous at t = 1, but the first derivative has a cusp at t = 1, so the second derivative does not exist at t = 1 .
 

1. What is a discontinuous function at second derivative?

A discontinuous function at second derivative is a function whose second derivative does not exist at one or more points. This means that the rate of change of the function changes abruptly at those points, resulting in a discontinuity in the graph of the function.

2. How can I identify a discontinuous function at second derivative?

A discontinuous function at second derivative can be identified by looking for abrupt changes in the curvature of the function's graph. These changes will manifest as sharp corners, breaks, or gaps in the curve.

3. What causes a function to be discontinuous at its second derivative?

A function can be discontinuous at its second derivative due to a variety of reasons, including a sharp change in the slope of the function, a change in the type of function (e.g. from a polynomial to a trigonometric function), or a change in the domain of the function.

4. How is a discontinuous function at second derivative different from a continuous function?

A continuous function has a continuous second derivative, meaning that the rate of change of the function is smooth and continuous at all points. In contrast, a discontinuous function at second derivative has abrupt changes in the rate of change, resulting in a discontinuity in the graph of the function.

5. Can a function be discontinuous at its second derivative but still be differentiable?

Yes, a function can be discontinuous at its second derivative and still be differentiable. This means that while the second derivative may not exist at certain points, the first derivative (slope) of the function still exists and is continuous. However, if a function is discontinuous at its second derivative, it cannot be twice differentiable, meaning its third derivative does not exist at those points of discontinuity.

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