   Chapter 10.3, Problem 58E

Chapter
Section
Textbook Problem

# 55–60 Find the slope of the tangent line to the given polar curve at the point specified by the value of θ . r = cos ( θ / 3 ) ,     θ = π

To determine

To find:

The slope of the tangent line to the given polar curve at the point specified by the value of θ.

Explanation

1) Concept:

i) To find a tangent line to a polar curve r=fθ, regard θ as a parameter and write its parametric equations as x=rcosθ=fθcosθ,      y=rsinθ=fθsinθ

ii) To find slope of the tangent line use

dydx=dy/dθdx/dθ=drdθsinθ+rcosθdrdθcosθ-rsinθ

2) Given:

Polar curve:

r= cos θ3,    θ=π

3) Calculation:

Consider the given polar curve

r=cosθ3

By using concept i),

The parametric equations are

x=rcosθ=cosθ3·cosθ

y=rsinθ=cosθ3·sinθ

Now, by using concept ii),

The slope of the tangent line to the given polar curve is

dydx=dydθdxdθ

Substitute x and y

=ddθcosθ3·sinθddθcosθ3·cosθ

Use the product rule

dydx=cosθ3cosθ+sinθ-13

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