Second derivative in parametric equations

Vickson for the help in explaining and solving the problem. In summary, to find the second derivative of a parametric function, we need to use the chain and Leibniz rules, and set y' = dy/dx = y'(t) before computing d^2y/dx^2.
  • #1
Karol
1,380
22

Homework Statement


Snap2.jpg

Only the second part

Homework Equations


Second derivative:
$$\frac{d^2y}{dx^2}=\frac{d}{dx}\frac{dy}{dx}$$

The Attempt at a Solution


$$dx=(1-2t)\,dt,~~dy=(1-3t^2)\,dt$$
Do i differentiate the differential dt?
$$d^2x=(-2)\,dt^2,~~d^2y=(-6)t\,dt^2$$
$$\frac{d^2y}{dx^2}=\frac{d^2y/dt^2}{d^2x/dt^2}=\frac{-6t}{-2}=...=3$$
The answer should be -2
 
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  • #2
You cannot do it like that. Use the chain and Leibniz rules.
 
  • #3
Karol said:

Homework Statement


View attachment 210022
Only the second part

Homework Equations


Second derivative:
$$\frac{d^2y}{dx^2}=\frac{d}{dx}\frac{dy}{dx}$$

The Attempt at a Solution


$$dx=(1-2t)\,dt,~~dy=(1-3t^2)\,dt$$
Do i differentiate the differential dt?
$$d^2x=(-2)\,dt^2,~~d^2y=(-6)t\,dt^2$$
$$\frac{d^2y}{dx^2}=\frac{d^2y/dt^2}{d^2x/dt^2}=\frac{-6t}{-2}=...=3$$
The answer should be -2
Set ##y' = dy/dx = y'(t)## and then comput ##d^2 y/dx^2 = dy'/dx## paremetrically
 
  • #4
$$\frac{d^2y}{dx^2}=\frac{d}{dx}\frac{dy}{dx}=\frac{d}{dx}\left( \frac{1-3t^2}{1-2t} \right)$$
$$=\frac{d \left( \frac{1-3t^2}{1-2t} \right)/dt}{dx/dt}=\frac{d \left( \frac{1-3t^2}{1-2t} \right)/dt}{1-2t}$$
It comes out right, -2
Thank you Orodruin and Ray
 
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Related to Second derivative in parametric equations

1. What is the second derivative in parametric equations?

The second derivative in parametric equations is the rate of change of the first derivative. It measures how quickly the slope of the curve is changing at a particular point.

2. Why is the second derivative important in parametric equations?

The second derivative helps determine the concavity of a curve and identify points of inflection. It also helps find the maximum and minimum points of a curve.

3. How do you find the second derivative in parametric equations?

To find the second derivative in parametric equations, you first find the first derivative using the chain rule. Then, you differentiate the first derivative with respect to the parameter again.

4. What is the geometric interpretation of the second derivative in parametric equations?

The second derivative in parametric equations represents the curvature of a curve. A positive second derivative indicates a curve is concave up, while a negative second derivative indicates a curve is concave down.

5. Can the second derivative be negative in parametric equations?

Yes, the second derivative can be negative in parametric equations. This indicates that the curve is concave down and has a decreasing slope. It may also indicate a point of maximum curvature on the curve.

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