Find the points on the given curve where the tangent

Painevg 2021-12-17 Answered
Find the points on the given curve where the tangent line is horizontal or vertical.
r=eθ
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Expert Answer

encolatgehu
Answered 2021-12-18 Author has 27 answers

To find the points where the tangents are horizontal/vertical, we need to find dy/dx. Using the chain rule fore differentiation, we can write
dydx=dydθdθdx=dydθdxdθ
Substitute y=rsinθ=eθsinθ and x=rcosθ=eθ, to get
dydx=deθsinθdθdeθcosθdθ
Use product rule for differentiation
dydx=d(eθ)sinθ+eθ(sinθ)(eθ)cosθ+eθ(cosθ)
dydx=eθsinθ+eθcosθeθcosθeθsinθ
Divide both the numerator and the denominator by eθ
dydx=sinθ+cosθcosθsinθ
The tangents will be horizontal when the numerator is 0
sinθ+cosθ=0
Subtract cosθ from both sides
sinθ=cosθ
Divide both sides by cosθ
sinθcosθ=1
In the right-hand side, we will rewrite -1 as tan(π4)
tamθ=tan(π4)
Remember that: the general solution of the equation tanx=tanα is
x=α+nπ
Therefore, the tangents are horizontal when
θ=π4+nπ
The tangents will be vertical when the denomiantor is 0
sinθ=cosθ
Divides both sides by cosθ
sinθcosθ=1
In the right-hand side, we will rewrite 1 as tan(π4)
tanθ=tan(π4)
Therefore, the tangents are vertical when

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