Chapter 10.3, Problem 91E

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347

Chapter
Section

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347
Textbook Problem

# Cycloid Use the parametric equations x = a ( θ − sin )   and   y = a ( 1 − cos θ ) ,     a > 0 to answer the following.(a) Find d y / d x and d 2 y / d x 2 .(b) Find the equation of the tangent line at the point where θ = π / 6 (c) Find all points of horizontal tangency.(d) Determine where the curve is concave upward or concave downward.(e) Find the length of one arc of the curve.

(a)

To determine

To calculate: dydx and d2ydx2 for x=a(θsinθ) and y=a(1cosθ) and a>0

Explanation

Given: x=a(Î¸âˆ’sinÎ¸) and y=a(1âˆ’cosÎ¸) and a>0

Calculation:

Differentiation above equation with respect to Î¸

dxdÎ¸=a(1âˆ’cosÎ¸) and dydÎ¸=asinÎ¸

dydx=dydÎ¸dxdÎ¸=asinÎ¸a(1âˆ’cosÎ¸)=sinÎ¸(1âˆ’cos

(b)

To determine

To calculate: Equation of tangent line at the point where θ=π6

(c)

To determine

To calculate: All point of horizontal tangency

(d)

To determine
Where curve is concave upward or downward

(e)

To determine

To calculate: Length of one arc of curve

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