Proving Curvature at Point (a,f(a)): A Guide

In summary, the conversation discusses how to prove the curvature at a point (a, f(a)) assuming that f(x) is smooth at that point. The suggested approach is to consider a circle and use the formula f"(a)/(1+f'(a)^2)^(3/2)). The concept of curvature is also discussed and the use of a definition is recommended. It is also suggested to refer to Wikipedia for more information on curvature.
  • #1
typhoonss821
14
1
hallow everyone
i am a tenth-grade student in Taiwan.What i want to know is that how to proove the curvature at point (a,(f(a))(assume f(x) is smooth at this point) is
f"(a)/(1+f'(a)^2)^(3/2))
i've thought this way:consider a circle first
未命名.JPG

in this circle the curvature at point P is lim arcPR/A as R approaches P ,curvature at point Q supposesd to be lim arcQR/(A-B) as R approaches Q
it might be d(arcQR)/d(A-B),equaling to d(arcQR)/d(A-(A-B)),because (A-B) is constant,it also equals to arcPR/A
So if we draw an ossculating cirsle at (a,(f(a)) ,we can reply the conclusion to deal with the problem,is this saying right??
 
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  • #2

1. How do you prove curvature at a specific point?

To prove curvature at a specific point, you will need to use the second derivative test. This involves taking the second derivative of the function at the given point and evaluating it. If the second derivative is positive, the function is concave up and there is a positive curvature at that point. If the second derivative is negative, the function is concave down and there is a negative curvature at that point.

2. What does it mean to have positive curvature at a point?

Having positive curvature at a point means that the function is concave up at that point. This means that the graph of the function is curving upwards, like a bowl. In other words, the function is increasing at an increasing rate at that point.

3. Can a function have both positive and negative curvature at the same point?

No, a function cannot have both positive and negative curvature at the same point. The curvature at a point is determined by the second derivative of the function at that point. If the second derivative is positive, the curvature is positive and if the second derivative is negative, the curvature is negative. It is not possible for the second derivative to be both positive and negative at the same point.

4. How does curvature affect the shape of a graph?

Curvature affects the shape of a graph by determining whether the graph is curving upwards or downwards at a specific point. If the curvature is positive, the graph is curving upwards and has a concave shape. If the curvature is negative, the graph is curving downwards and has a convex shape. The magnitude of the curvature also affects the steepness of the curve at that point.

5. Are there any other methods for proving curvature at a point besides the second derivative test?

Yes, there are other methods for proving curvature at a point, such as using the curvature formula or the radius of curvature formula. However, the second derivative test is the most commonly used method and is often the easiest and most straightforward way to determine curvature at a point.

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