What is the equation of the tangent line to the polar curve # f(theta)=theta^2cos(3theta)-thetasin(2theta)+tan(theta/6) # at #theta = pi#?
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To find the equation of the tangent line to the polar curve ( f(\theta) = \theta^2 \cos(3\theta) - \theta \sin(2\theta) + \tan(\frac{\theta}{6}) ) at ( \theta = \pi ), follow these steps:
- Calculate the value of ( f(\theta) ) at ( \theta = \pi ) to find the point of tangency.
- Find the derivative of ( f(\theta) ) with respect to ( \theta ).
- Evaluate the derivative at ( \theta = \pi ) to find the slope of the tangent line.
- Use the point of tangency and the slope to write the equation of the tangent line in polar form.
Step 1: ( f(\pi) = \pi^2 \cos(3\pi) - \pi \sin(2\pi) + \tan(\frac{\pi}{6}) )
Step 2: Find the derivative of ( f(\theta) ): [ f'(\theta) = (2\theta \cos(3\theta) - 3\theta^2 \sin(3\theta)) - (\sin(2\theta) - 2\theta \cos(2\theta)) + \frac{1}{6\cos^2(\theta/6)} ]
Step 3: Evaluate the derivative at ( \theta = \pi ): [ f'(\pi) = (2\pi \cos(3\pi) - 3\pi^2 \sin(3\pi)) - (\sin(2\pi) - 2\pi \cos(2\pi)) + \frac{1}{6\cos^2(\pi/6)} ]
Step 4: Use the point of tangency ( (\pi, f(\pi)) ) and the slope ( f'(\pi) ) to write the equation of the tangent line in polar form.
So, the equation of the tangent line is:
[ r = \frac{f(\pi)}{\cos(\theta - \pi)} ]
where ( r ) is the distance from the origin, and ( \theta ) is the angle.
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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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