Question

Find dy/dx and d^2y/dx^2.  For which values of t is the curve concave upward?

x = 2 sin t,     y = 3 cos t,          0 < t < 2π

Answer 1

Step 1:

The curve is and .

Consider .

Differentiate with respect to .

Consider

Differentiate with respect to .

Find .

Find .

Step 2:

Consider .

For concave upwards

Multiplying by negative sign will reverse the inequality.

Secant value is negative in Second and third quadrant that is .

Hence, the curve is concave upwards in .

Solution: 

, and

The curve is concave upwards in .