# What is the arclength of the polar curve #f(theta) = 2thetasin(5theta)-cottheta # over #theta in [pi/12,pi/6] #?

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To find the arc length of the polar curve ( f(\theta) = 2\theta \sin(5\theta) - \cot(\theta) ) over the interval ( \theta \in [\frac{\pi}{12}, \frac{\pi}{6}] ), you can use the following formula:

[ L = \int_{\theta_1}^{\theta_2} \sqrt{r^2 + \left(\frac{dr}{d\theta}\right)^2} , d\theta ]

Where ( r(\theta) = f(\theta) ) and ( \frac{dr}{d\theta} ) is the derivative of ( r(\theta) ) with respect to ( \theta ).

- Find ( r(\theta) = f(\theta) ).
- Compute ( \frac{dr}{d\theta} ).
- Plug ( r(\theta) ) and ( \frac{dr}{d\theta} ) into the arc length formula.
- Evaluate the integral over the given interval.

Once you have completed these steps, you'll have the arc length of the polar curve over the specified interval.

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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.

- What is the equation of the tangent line of #r=theta-sin(-theta+(pi)/3) # at #theta=(2pi)/3#?
- What is the slope of the polar curve #f(theta) = theta^2 - sec^3theta+tantheta # at #theta = (3pi)/4#?
- What is the slope of the polar curve #f(theta) = theta - sec^3theta+thetasin^3theta # at #theta = (5pi)/8#?
- What is the Cartesian form of #( 8 , (13pi)/6 ) #?
- What is the Cartesian form of #(12,(-7pi)/3))#?

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