# What is the slope of the polar curve #f(theta) = theta^2 - sec^3theta+tantheta # at #theta = (3pi)/4#?

Anytime a problem asks for the slope of a curve at a given point, it is equivalent to asking what the value of the derivative is at that point.

Finally,

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To find the slope of the polar curve ( f(\theta) = \theta^2 - \sec^3(\theta) + \tan(\theta) ) at ( \theta = \frac{3\pi}{4} ), we first find the derivative of ( f(\theta) ) with respect to ( \theta ), and then evaluate it at ( \theta = \frac{3\pi}{4} ).

The derivative of ( f(\theta) ) with respect to ( \theta ) is given by:

[ f'(\theta) = \frac{d}{d\theta}(\theta^2 - \sec^3(\theta) + \tan(\theta)) ]

[ f'(\theta) = 2\theta + 3\sec(\theta)\tan(\theta) - \sec^3(\theta) + \sec^2(\theta) ]

Now, to find the slope at ( \theta = \frac{3\pi}{4} ), substitute ( \frac{3\pi}{4} ) into ( f'(\theta) ):

[ f'\left(\frac{3\pi}{4}\right) = 2\left(\frac{3\pi}{4}\right) + 3\sec\left(\frac{3\pi}{4}\right)\tan\left(\frac{3\pi}{4}\right) - \sec^3\left(\frac{3\pi}{4}\right) + \sec^2\left(\frac{3\pi}{4}\right) ]

[ f'\left(\frac{3\pi}{4}\right) = \frac{3\pi}{2} + 3(-1)(1) - (-\sqrt{2})^3 + (-\sqrt{2})^2 ]

[ f'\left(\frac{3\pi}{4}\right) = \frac{3\pi}{2} - 3 + 2\sqrt{2} + 2 ]

[ f'\left(\frac{3\pi}{4}\right) = \frac{3\pi}{2} - 1 + 2\sqrt{2} ]

Therefore, the slope of the polar curve ( f(\theta) = \theta^2 - \sec^3(\theta) + \tan(\theta) ) at ( \theta = \frac{3\pi}{4} ) is ( \frac{3\pi}{2} - 1 + 2\sqrt{2} ).

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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.

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