How do you find the points of horizontal tangency of #r=asinthetacos^2theta#?

Answer 1
Horizontal tangents occur when #dy/dx=0#.
For polar equations, #dy/dx=(dy//d theta)/(dx//d theta)# where #x=rcostheta# and #r=sintheta#. Then, #dy/dx=(d/(d theta)rsintheta)/(d/(d theta)rcostheta)#.
So, horizontal tangents occur when #dy/dx=0#, which is the same as when #dy/(d theta)=0#, or when #d/(d theta)rsintheta=0#.
Here #r=asinthetacostheta#, so #y=rsintheta=asin^2thetacos^2theta#.
We can simplify this by noting that #y=1/4a(4sin^2thetacos^2theta)=1/4a(2sinthetacostheta)^2=1/4asin^2 2theta#.
Differentiating this with the chain rule, we see that #dy/(d theta)=1/4a(2sin^1 2theta)(cos2theta)(2)=asin2thetacos2theta#.
#dy/(d theta)=0# then when either #sin2theta=0# or #cos2theta=0#, which occur at #theta=(kpi)/4,kinZZ#.
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Answer 2

To find the points of horizontal tangency for the polar curve ( r = a \sin(\theta) \cos^2(\theta) ), you need to find where the derivative of ( r ) with respect to ( \theta ) is zero and the second derivative is negative.

  1. Find the first derivative of ( r ) with respect to ( \theta ).
  2. Set the first derivative equal to zero and solve for ( \theta ).
  3. Check the second derivative of ( r ) with respect to ( \theta ) to ensure concavity.
  4. Evaluate ( r ) at the found ( \theta ) values to get the corresponding points of horizontal tangency.
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Answer from HIX Tutor

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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