Question

Find the slope of the curve r=cos(2theta) at theta=pi/6.. . Must this be solved parametrically? If so, why?

Answer 1

By converting into parametric equations,

x(θ)=r(θ)cosθ=cos2θcosθ
y(θ)=r(θ)sinθ=cos2θsinθ

By Product Rule,

x'(θ)=−sin2θcosθ−cos2θsinθ

x'(π/2)=−sin(π)cos(π/2)−cos(π)sin(π/2)=1

y'(θ)=−sin2θsinθ+cos2θcosθ

y'(π/2)=−sin(π)sin(π/2)+cos(π)cos(π/2)=0

So, the slope m of the curve can be found by

m=dy/dx∣θ=π2
= y'(π/2)/x'(π/2)
=0/1

=0

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