Find the curvature of x = e^(t)

  • Thread starter brendan
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Sorry, one last thing. The curvature at t = 0 is 2 and not sqrt(2)/3.In summary, the curvature of the parametric curve x = e^(t), y = e^(-t), z = t at t = 0 is 2. The formula used was |r'(t) x r''(t)| / |r'(t)|^3. After substituting the values, the curvature was found to be 2.
  • #1
brendan
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Homework Statement



Find the curvature of x = e^(t) y = e^(-t) z = t t = 0

Homework Equations



I've used the equation of

k(t) = |r'(t) x r''(t) |/ |r'(t)|^3

The Attempt at a Solution



k(t) = |r'(t) x r''(t) |/ |r'(t)|^3


= |e^t i + -e^(-t)j + 1k| x |e^t i + e^(-t)j + 0k| / |e^t i + -e^(-t)j + 1k|^3

= |-e^(-t)i + e^(t)j +2k| / |e^t i + -e^(-t)j + 1k|^3

Using t = 0


= |-e^(0)i + e^(0)j +2k| / |e^0 i + -e^(0)j + 1k|^3


= 2/1

= 2

Is this right ?

regards
Brendan
 
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  • #2


I agree with everything to your penultimate line. The modulus of (-1,1,2) is not 2.
 
  • #3


How about now?

= |-e^(0)i + e^(0)j +2k| / |e^0 i + -e^(0)j + 1k|^3


= sqrt(2)/3


Brendan
 
  • #4


Looks good to me.
 
  • #5


Thanks mate!

Brendan
 

FAQ: Find the curvature of x = e^(t)

What is the equation for finding the curvature of x = e^(t)?

The equation for finding the curvature of x = e^(t) is k = |e^(t)| / (1+e^(2t))^(3/2).

How do you calculate the curvature of x = e^(t)?

The curvature of x = e^(t) can be calculated by using the formula k = |e^(t)| / (1+e^(2t))^(3/2), where t is the parameter value at a given point on the curve.

What does the value of the curvature represent?

The curvature at a given point on x = e^(t) represents the rate of change of the direction of the curve at that point. A higher curvature value indicates a sharper change in the direction of the curve, while a lower value indicates a more gradual change.

How is the curvature affected by the parameter value t?

The curvature of x = e^(t) is affected by the parameter value t, as seen in the formula k = |e^(t)| / (1+e^(2t))^(3/2). As t increases, the numerator (|e^(t)|) also increases, leading to a higher curvature value. Similarly, as t decreases, the curvature value decreases.

Can the curvature of x = e^(t) be negative?

Yes, the curvature of x = e^(t) can be negative. The absolute value in the formula k = |e^(t)| / (1+e^(2t))^(3/2) ensures that the curvature value is always positive. However, depending on the value of t, the numerator (|e^(t)|) can be negative, resulting in a negative curvature value.

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