# What is the slope of #r=tantheta^2-theta# at #theta=(3pi)/8#?

Using the product rule, we can say that:

So:

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The slope of ( r = \tan^2(\theta) - \theta ) at ( \theta = \frac{3\pi}{8} ) can be found by taking the derivative of ( r ) with respect to ( \theta ) and then evaluating it at ( \theta = \frac{3\pi}{8} ). The derivative of ( r ) with respect to ( \theta ) is:

[ \frac{dr}{d\theta} = 2\tan(\theta)\sec^2(\theta) - 1 ]

Evaluating this derivative at ( \theta = \frac{3\pi}{8} ), we get:

[ \frac{dr}{d\theta} \bigg|_{\theta = \frac{3\pi}{8}} = 2\tan\left(\frac{3\pi}{8}\right)\sec^2\left(\frac{3\pi}{8}\right) - 1 ]

Calculate the tangent and secant values at ( \frac{3\pi}{8} ), and then substitute into the expression. This will give you the slope of the curve at ( \theta = \frac{3\pi}{8} ).

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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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