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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
BuyFindarrow_forward

Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
Textbook Problem

At what points does the curve

x = 2 a cos t a cos 2 t y = 2 a sin t a sin 2 t

have vertical or horizontal tangents? Use this information to help sketch the curve.

To determine

To find: The point at which the curve has vertical and horizontal tangents and also plots the curve.

Explanation

Given:

The curve equations are as below.

x=2acostacos2t (1)

y=2asintasin2t (2)

Calculation:

Differentiate the equation (1) with respect to t .

x=2acostacos2tdxdt=2asint+2asin2t

Assume dxdt is equal to zero.

dxdt=2asint+2asin2t0=2asint+2asin2t2asint+4asintcost=02asint(2cost1)=0

Calculate the value of sint and cost from the above equation.

2asint(2cost1)=02cost1=0cost=12

Take the values of t to get cost=12 . Therefore, the value of t is to be taken as π3 or 5π3 .

Then, apply the condition as below.

2asint(2cost1)=02asint=0sint=0

Take the values of t to get sint=0

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P-52RECh-10 P-53RECh-10 P-54RECh-10 P-55RECh-10 P-56RECh-10 P-57RECh-10 P-58RECh-10 P-1PCh-10 P-2PCh-10 P-3PCh-10 P-4PCh-10 P-5PCh-10 P-6P

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