Determine the concavity of the curve with parametric equations x = 4 + t2, y = t2 + t3 at t = 2.

Question

Determine the concavity of the curve with parametric equations x = 4 + t², y = t² + t³ at t = 2. Justify your answer.

Answer 1

so we will need to find the 2nd derivative of dy/dx of x=4+t² and y=t²+t³, lets start with the first derivative:

dy/dt=2t+3t² and dx/dt=2t so (dy/dt)/(dx/dt)=2t+3t²/2t.

Now we must use the quotient rule to find the 2nd derivative:

(2t(2+6t)-2(2t+3t²))/(4t²), now we plug in t=2

(2*2(2+6*2)-2(2*2+3*2²))/4*2²=(4(14)-2(16))/16=(56-32)/16=24/16=3/2

Since the 2nd derivative at t=2 is positive (3/2), the curve is concave up