   Chapter 10.2, Problem 16E

Chapter
Section
Textbook Problem

Find dy/dx and d2y/dx2. For which values of t is the curve concave upward?16. x = cos t, y = sin 2t, 0 < t < π

To determine

To find: The value of dydx and d2ydx2 for the parametric equations x=cost and y=sin2t , and check for value of t , whether the curve is concave upward.

Explanation

Given:

The parametric equation for the variable x is as follows.

x=cost (1)

The parametric equation for the variable y is as follows.

y=sin2t (2)

Calculation:

Differentiate the parametric equation x with respect to t .

x=costdxdt=sint

Differentiate the parametric equation y with respect to t .

y=sin2tdydt=2cos2t

Write the chain rule for dydx .

dydx=dydtdxdt

Substitute (2cos2t) for dydt and (sint) for dxdt in the above equation.

dydx=(2cos2t)(sint)=2cos2tsint

Calculate the second derivative to determine concavity.

d2ydx2=ddt(dydx)dxdt=(sint)(4sin2t)(2cos2t)(cost)(sint)2sint=(sint)(8sintcost)+2(12sin2t)(sint)sin2t=(cost)(8sin2t+24sin2t)(sint)sin2t=costsint

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