Question

Q.04: (11 points) Given the polar curve r = e θ , where 0 ≤ θ ≤ 2π. Find points on the curve in the form (r, θ) where there is a horizontal or vertical tangent line. g

Answer 1

I suppose the curve is
`r(\theta)=e^\theta`.

Tangent lines to the curve have slope
`(dy)/(dx)`; use the chain rule to get this in polar coordinates.

`(dy)/(dx)=(dy)/(d\theta)(d\theta)/(dx)=((dy)/(d\theta))/((dx)/(d\theta))`

We have

`y(\theta)=r(\theta)\sin\theta\implies(dy)/(d\theta)=(dr)/(d\theta)\sin\theta+r(\theta)\cos\theta`

`x(\theta)=r(\theta)\cos\theta\implies(dx)/(d\theta)=(dr)/(d\theta)\cos\theta-r(\theta)\sin\theta`

`r(\theta)=e^\theta\implies(dr)/(d\theta)=e^\theta`

`\implies(dy)/(dx)=(e^\theta\sin\theta+e^\theta\cos\theta)/(e^\theta\cos\theta-e^\theta\sin\theta)=(\sin\theta+\cos\theta)/(\cos\theta-\sin\theta)`

The tangent line is horizontal when the slope is 0, which happens wherever the numerator vanishes:

`\sin\theta+\cos\theta=0\implies\sin\theta=-\cos\theta\implies\tan\theta=-1`

`\implies\theta=\boxed{-\frac\pi4+n\pi}`

(where
`n` is any integer)

The tangent line is vertical when the slope is infinite or undefined, which happens when the denominator is 0:

`\cos\theta-\sin\theta=0\implies\sin\theta=\cos\theta\implies\tan\theta=1`

`\implies\theta=\boxed{\frac\pi4+n\pi}`