   Chapter 10.2, Problem 20E

Chapter
Section
Textbook Problem

Find the points on the curve where the tangent is horizontal or vertical. If you have a graphing device, graph the curve to check your work.20. x = esin θ, y = ecos θ

To determine

To find: The tangent is horizontal or vertical and plot the curve for the parametric equations x=esinθ and y=ecosθ .

Explanation

Given:

The parametric equation for the variable x is as follows.

x=esinθ (1)

The parametric equation for the variable y is as follows.

y=ecosθ (2)

Calculation:

Differentiate the parametric equation x with respect to θ .

x=esinθdxdθ=esinθcosθ

Differentiate the parametric equation y with respect to θ .

y=ecosθdydθ=ecosθ(sinθ)

The tangent is horizontal when the expression dydt=0 is true.

dydθ=0ecosθ(sinθ)=0sinθ=0θ=0,π

Substitute (0) for θ in equation (1).

x=esinθ=esin(0)=1

Substitute (0) for θ in equation (2).

y=ecosθ=ecos(0)=e

Substitute (π) for θ in equation (1).

x=esinθ=esin(π)=1

Substitute (π) for θ in equation (2).

y=ecosθ=ecos(π)=1e

The curve has a horizontal tangents passing through points (1,e) and (1,1e)

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