Question

In: Advanced Math

Find the slope of the tangent line to the given polar curve at the point specified by the value of θ. r = 1 + 2 cos θ, θ = π/3

Find the slope of the tangent line to the given polar curve at the point specified by the value of θ. r = 1 + 2 cos θ, θ = π/3

Solutions

Expert Solution

Solution

step

Differentiate with respect to \theta by using the product rule

\dfrac{dy}{d\theta}=\left(0-2\sin\theta\right)\cdot\sin\theta+\left(1+2\cos\theta\right)\cdot\left(\cos\theta\right)

Substitute \theta=\pi/3 (which means \sin\theta=\sqrt{3}/2 and \cos\theta=1/2)

\dfrac{dy}{d\theta}\bigg|_{\theta=\pi/3}=\left(0-2\cdot\dfrac{\sqrt{3}}{2}\right)\cdot\dfrac{\sqrt{3}}{2}+\left(1+2\cdot\dfrac{1}{2}\right)\cdot\left(\dfrac{1}{2}\right)

\dfrac{dy}{d\theta}\bigg|_{\theta=\pi/3}=\left(-\sqrt{3}\right)\cdot\dfrac{\sqrt{3}}{2}+\left(1+1\right)\cdot\left(\dfrac{1}{2}\right)

\dfrac{dy}{d\theta}\bigg|_{\theta=\pi/3}=-\dfrac{3}{2}+1=-\dfrac{3}{2}+\dfrac{2}{2}=-\dfrac{1}{2}

Using the chain rule, we can write

 

\dfrac{dy}{dx}\bigg|_{\theta=\pi/3}=\dfrac{dy/d\theta}{dx/d\theta}\bigg|_{\theta=\pi/3}=\dfrac{-1/2}{-3\sqrt{3}/2}=\dfrac{1}{3\sqrt{3}}


Slope of the tangent is \dfrac{1}{3\sqrt{3}}

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