   Chapter 10.3, Problem 34E

Chapter
Section
Textbook Problem

# Horizontal and Vertical Tangency In Exercises 35-42, find all points (if any) of horizontal and vertical tangency to the curse. Use a graphing utility to confirm your results. x = cos θ ,     y = 2 sin 2 θ

To determine

All points of horizontal & vertical tangency to the curve. Verify the result by the use of graphing utility.

Explanation

a)Given:

x=cosθ......(1)y=2sin2θ......(2)

b) Calculation:

Differentiate both the functions with respect to θ.

dxdθ=sinθ & dydθ=4cos2θ

For vertical tangency equate dxdθ=0,

sinθ=0θ=0,π,2π

Now, for horizontal tangency equate dydθ=0,

dydθ=04cos2θ=0=cos(±π2)2θ=±π2.

θ=±π4

Substitute π4 for θ in equation (2) and obtain the value of y .

y=2sinπ2=2

Substitute π4 for θ in equation (1) and obtain the value of x.

x=cosπ4=12

Hence, the first horizontal tangent point is

(12,2)

Substitute π4 for θ in equation (2) and obtain the value of y

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