# How do you find the slope of the polar curve #r=cos(2theta)# at #theta=pi/2# ?

By converting into parametric equations,

#{(x(theta)=r(theta)cos theta=cos2theta cos theta), (y(theta)=r(theta)sin theta=cos2theta sin theta):}#

By Product Rule,

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To find the slope of the polar curve ( r = \cos(2\theta) ) at ( \theta = \frac{\pi}{2} ), first, differentiate ( r ) with respect to ( \theta ) to find ( \frac{dr}{d\theta} ). Then, plug in ( \theta = \frac{\pi}{2} ) into ( \frac{dr}{d\theta} ) to find the slope at that point.

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To find the slope of the polar curve (r = \cos(2\theta)) at (\theta = \frac{\pi}{2}), differentiate the equation (r = \cos(2\theta)) with respect to (\theta) and then evaluate it at (\theta = \frac{\pi}{2}).

(\frac{dr}{d\theta} = -2\sin(2\theta))

Evaluate at (\theta = \frac{\pi}{2}):

(\frac{dr}{d\theta}\Bigg|_{\theta = \frac{\pi}{2}} = -2\sin(\pi) = 0)

So, the slope of the polar curve (r = \cos(2\theta)) at (\theta = \frac{\pi}{2}) is (0).

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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 area enclosed by #r=theta # for #theta in [0,pi]#?
- What is the distance between the following polar coordinates?: # (2,(3pi)/2), (-2,pi/2) #
- What is the Cartesian form of #(100,(-17pi)/16))#?
- What is the Cartesian form of #(-10,(-17pi)/16))#?
- What is the equation of the line that is normal to the polar curve #f(theta)=-5theta- sin((3theta)/2-pi/3)+tan((theta)/2-pi/3) # at #theta = pi#?

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