What is the arclength of #f(t) = (sin^2t/sin(2t),tant-sec^2t)# on #t in [pi/12,pi/4]#?
We have
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To find the arc length of the curve described by the parametric equations (f(t) = \left(\frac{\sin^2 t}{\sin(2t)}, \tan t - \sec^2 t\right)) on the interval (\left[\frac{\pi}{12}, \frac{\pi}{4}\right]), you use the formula for arc length:
[L = \int_{a}^{b} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} , dt]
where (a) and (b) are the limits of integration, and (x(t)) and (y(t)) are the parametric equations for the curve.
In this case, (x(t) = \frac{\sin^2 t}{\sin(2t)}) and (y(t) = \tan t - \sec^2 t).
- Compute (\frac{dx}{dt}) and (\frac{dy}{dt}).
- Plug these derivatives into the formula for arc length.
- Integrate the resulting expression over the given interval (\left[\frac{\pi}{12}, \frac{\pi}{4}\right]) to find the arc length (L).
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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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