What is the slope of the polar curve #f(theta) = theta^2 - sectheta # at #theta = (3pi)/4#?
The slope:
From the reference Tangents with Polar Coordinates we obtain the equation
Substitute this into equation [2]:
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To find the slope of the polar curve ( f(\theta) = \theta^2 - \sec(\theta) ) at ( \theta = \frac{3\pi}{4} ), you need to find the derivative of the function with respect to ( \theta ) and then evaluate it at ( \theta = \frac{3\pi}{4} ).
The derivative of the given function is:
[ \frac{df}{d\theta} = 2\theta - \sec(\theta)\tan(\theta) ]
Evaluating this derivative at ( \theta = \frac{3\pi}{4} ), we get:
[ \frac{df}{d\theta}\Bigg|_{\theta=\frac{3\pi}{4}} = 2\left(\frac{3\pi}{4}\right) - \sec\left(\frac{3\pi}{4}\right)\tan\left(\frac{3\pi}{4}\right) ]
Simplify this expression to find the slope of the polar curve at ( \theta = \frac{3\pi}{4} ).
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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.
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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- What is the polar form of #(16,-4#?
- What is the Cartesian form of #( 8, (23pi)/8 ) #?
- What is the arclength of #r=-2sin(theta/4+(7pi)/8) # on #theta in [(pi)/4,(7pi)/4]#?

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