What is the arc length of the polar curve #f(theta) = 2costhetatheta # over #theta in [pi/8, pi/3] #?
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To find the arc length of the polar curve ( f(\theta) = 2 \cos(\theta)  \theta ) over the interval ( \theta \in [\frac{\pi}{8}, \frac{\pi}{3}] ), you can use the formula for arc length in polar coordinates:
[ L = \int_{\alpha}^{\beta} \sqrt{r^2 + \left(\frac{dr}{d\theta}\right)^2} , d\theta ]
Where ( r = f(\theta) ) is the polar function, and ( \frac{dr}{d\theta} ) is the derivative of ( r ) with respect to ( \theta ).

Find ( \frac{dr}{d\theta} ) by taking the derivative of ( f(\theta) ) with respect to ( \theta ): [ \frac{dr}{d\theta} = \frac{d}{d\theta}(2\cos(\theta)  \theta) = 2\sin(\theta)  1 ]

Substitute ( r = f(\theta) ) and ( \frac{dr}{d\theta} ) into the arc length formula.

Integrate the expression from ( \frac{\pi}{8} ) to ( \frac{\pi}{3} ) to find the arc length:
[ L = \int_{\pi/8}^{\pi/3} \sqrt{(2\cos(\theta)  \theta)^2 + (2\sin(\theta)  1)^2} , d\theta ]
 Evaluate the integral to find the arc length ( L ).
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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