redolrn

2022-09-06

Is second derivative of a function related to curve smoothness?

If there exist a first derivative of a function at any point then the funtion is continuous at that point.

What if the second derivative of that function is also exist at that point ? Does this mean that function is smooth at that point ? Means instead of taking sharp bend the function is having curved shape ?

If there exist a first derivative of a function at any point then the funtion is continuous at that point.

What if the second derivative of that function is also exist at that point ? Does this mean that function is smooth at that point ? Means instead of taking sharp bend the function is having curved shape ?

Zara Pratt

Beginner2022-09-07Added 12 answers

If you found the first derivative then it is enough to say that the function is continuous. If you have also found that second derivative exists then it gives you more information about your function about what rate the funtions slope is increasing or decreasing at that point.

Also if the function were sharp at a particular point then first derivative itself wouldnt have existed at that point

Also if the function were sharp at a particular point then first derivative itself wouldnt have existed at that point

planhetkk

Beginner2022-09-08Added 2 answers

it's easier to speak about discontinuity than continuity.

If a function has a discontinuity (i.e. is not continuous), then it usually means that there's "leap" within the characteristic price, so its graph has a gap in it. This isn't always one hundred% real, although. There are bizarre features that oscillate infinitely fast in a small area, and these are also not non-stop. however permit's ignore ordinary features like that, for now.

If there may be a discontinuity in the first spinoff of a characteristic, it method that its graph has a pointy corner -- a place in which there may be an abrupt exchange in course.

If there may be a discontinuity in second derivative, it approach there may be an abrupt change in curvature (or radius of curvature). a few people can see those discontinuities, and some people can't. everybody with a history in photos or design will virtually see them.

things get a lot extra exciting while you make surfaces out of your curves, and you study reflections in those surfaces, as you would while searching at a car, for example. reflection "magnifies" discontinuities. A discontinuity in curvature will be certainly seen in a reflective surface, even to the untrained eye. Even a discontinuity in third derivative may be seen to a designer. it truly is why individuals who design automobiles don't use cubic splines (that you are studying, feels like), due to the fact those have best continuity of first and 2d derivatives.

If a function has a discontinuity (i.e. is not continuous), then it usually means that there's "leap" within the characteristic price, so its graph has a gap in it. This isn't always one hundred% real, although. There are bizarre features that oscillate infinitely fast in a small area, and these are also not non-stop. however permit's ignore ordinary features like that, for now.

If there may be a discontinuity in the first spinoff of a characteristic, it method that its graph has a pointy corner -- a place in which there may be an abrupt exchange in course.

If there may be a discontinuity in second derivative, it approach there may be an abrupt change in curvature (or radius of curvature). a few people can see those discontinuities, and some people can't. everybody with a history in photos or design will virtually see them.

things get a lot extra exciting while you make surfaces out of your curves, and you study reflections in those surfaces, as you would while searching at a car, for example. reflection "magnifies" discontinuities. A discontinuity in curvature will be certainly seen in a reflective surface, even to the untrained eye. Even a discontinuity in third derivative may be seen to a designer. it truly is why individuals who design automobiles don't use cubic splines (that you are studying, feels like), due to the fact those have best continuity of first and 2d derivatives.

Read carefully and choose only one option

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