AP Calculus BC / Unit 9: Parametric Equations, Polar Coordinates, and Vector-Valued Functions
9.2 Second Derivatives of Parametric Equations
Notes
$$\frac{d^2y}{dx^2}=\frac{\frac{d}{dt}\left(\frac{dy}{dx}\right)}{dx/dt}$$
Practice
$x=t^2$, $y=t^3$, so $\displaystyle \frac{dy}{dx}=\frac{3t}{2}$. Then $\displaystyle \frac{d^2y}{dx^2}=$
$\displaystyle \frac{3/2}{2t}$.
$x=t$, $y=t^2$. $\displaystyle \frac{d^2y}{dx^2}=$
$\displaystyle \frac{dy}{dx}=2t$; $\displaystyle \frac{2}{1}=2$.
The second derivative of a parametric curve tells you:
Same meaning as for $y=f(x)$.
Ready for more? Take the Unit 9 test.