What values determine the concavity of a parametric curve?

In summary, to determine the concavity of a parametric curve, we can calculate the second derivative of the curve and analyze its sign. If the second derivative is positive, the curve is concave upward, and if it is negative, the curve is concave downward. This can be determined by solving inequalities involving the second derivative and identifying the values of t that satisfy them.
  • #1
sapiental
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0
Hello,

My textbook says that to determine concavity we calculate the second derivative of the curve. This is a problem from my book,

x = t^2 and y = t^3 - 3t

the second derivative of this is (3(t^2+1))/(4t^3)

I know all the steps to get to this point.. However, the book says that the curve is concave upward when t > 0 and concave downward when t < 0.

Can somebody please explain to me what values this last statement refers to. Is there a general theorem/procedure that I can apply to any second derivative of a parametric curve to determine the concavity?

Thanks a lot in advance.
 
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  • #2
If [tex] f''(x) > 0 [/tex] then the curve is concave upward and vice versa.Looking at [tex] \frac{d^{2}y}{dx^{2}} = \frac{3(t^{2}+1)}{4t^{3}} [/tex]

it is greater than 0 when both numerator and denominator are positive (cant do negative over negative because the numerator will always be positive).

So solve the following inequalities: [tex] 3t^{2} + 3 > 0 [/tex] and [tex] 4t^{3} > 0 [/tex]. Clearly, all values of t work for the first inequality, but only positive values of t work for the second inequality.
 

1. What is parametric curve concavity?

Parametric curve concavity refers to the direction in which a curve is bending at any given point. It is determined by the curvature of the curve, which is the rate at which the tangent line to the curve changes.

2. How is parametric curve concavity calculated?

The concavity of a parametric curve can be calculated by first finding the second derivative of the curve. The second derivative will be a measure of the curvature at any given point. If the second derivative is positive, the curve is concave up, and if it is negative, the curve is concave down.

3. What is the significance of parametric curve concavity?

The concavity of a curve is important because it can provide information about the behavior of the curve. For example, a concave up curve indicates a function that is increasing at an increasing rate, while a concave down curve indicates a function that is decreasing at an increasing rate.

4. How does parametric curve concavity relate to optimization problems?

In optimization problems, the concavity of a curve can determine whether a point is a maximum or minimum point. A concave up curve will have a minimum point, while a concave down curve will have a maximum point.

5. Can parametric curve concavity change at different points along the curve?

Yes, the concavity of a parametric curve can change at different points along the curve. This is because the curvature and second derivative of the curve can vary at different points, resulting in changes in concavity.

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