   Chapter 10.2, Problem 27E

Chapter
Section
Textbook Problem

(a) Find the slope of the tangent line to the trochoid x = rθ − d sin θ, y = r − d cos θ in terms of θ. (See Exercise 10.1.40.) (b) Show that if d < r, then the trochoid does not have a vertical tangent.

(a)

To determine

To find: The slope of the tangent line to trochoid and points for the parametric equations x=rθdsinθ and y=rdcosθ .

Explanation

Given:

The parametric equation for the variable x is as follows.

x=rθdsinθ (1)

The parametric equation for the variable y is as follows.

y=rdcosθ (2)

Calculation:

Differentiate the parametric equation x with respect to θ .

x=rθdsinθdxdθ=rdcosθ

Differentiate the parametric equation y with respect to θ .

y=rdcosθdydθ=dsinθ

Write the chain rule for dydx .

dydx=dydtdxdt

Substitute (dsinθ) for dydθ and (rdcosθ) for dxdθ in the above equation

(b)

To determine

To Show: The trochoid does not have vertical tangent for the parametric equations x=rθdsinθ and y=rdcosθ , when d<r .

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