   Chapter 10.3, Problem 33E

Chapter
Section
Textbook Problem

# Horizontal and Vertical Tangency In Exercises 29–38, find all points (if any) of horizontal and vertical tangency to the curve. Use a graphing utility to confirm your results.x = 3 cos 0, y = 3 sin θ

To determine

To calculate: The points of horizontal and vertical tangent to curve x=3cosθ,y=3sinθ and confirm by the use of a graphical utility.

Explanation

Given:

The function x=3cosθ,y=3sinθ.

Formula used:

The condition for horizontal tangent is dydθ=0 and the condition for vertical tangent is dxdθ=0.

Calculation:

As given in the function:

x=3cosθ,y=3sinθ

Differentiate these both with respect to θ,

dxdθ=3sinθ

And,

dydθ=3cosθ

The condition for horizontal tangent is dydθ=0.

Put the value of dydθ,

3cosθ=0cosθ=0θ=π2,3π2

Put the value of θ in the function,

x=3cosθ=3cos(π2),3cos(3π2)=0,0

And,

y=3sinθ=3sin(π2),3sin(3π2)=3,3

Hence, the points for horizontal tangent are (0,3) and (0,3).

Similarly, the condition for horizontal tangent is dxdθ=0

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