   Chapter 10.3, Problem 44E

Chapter
Section
Textbook Problem

# Determining Concavity In Exercises 43-48, determine the open t-intervals on which the curve is concave downward or concave upward. x = 4 cos t ,   y = 2 sin t ,   0 < t < 2 π

To determine

To calculate: The open t-interval on which the curve of the parametric equations, x=4cost,y=2sint is concave downward or concave upward.

Explanation

Given:

The parametric equation,

x=4costy=2sint

And, the interval is 0<t<2π.

Formula used:

If d2ydx2>0, then the curve is concave up and if d2ydx2<0, then the curve is concave downward.

Calculation:

Consider the equations,

x=4costy=2sint

Differentiate x=4cost with respect to t,

dxdt=4sint...... (1)

Differentiate y=2sint with respect to t,

dydt=2cost...... (2)

Divide equation (2) by (1),

dydx=dydtdxdt=2cost4sint=12cott

Now, differentiate again with respect to x,

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