Revision sheet

Parametric slopes

• $\dfrac{dy}{dx} = \dfrac{dy/dt}{dx/dt}$.
• Horizontal: $y' = 0$. Vertical: $x' = 0$.

Watch out: $y$'s rate on top.

Second derivative

$\dfrac{d^2y}{dx^2} = \dfrac{d}{dt}\!\left(\dfrac{dy}{dx}\right) \Big/ \dfrac{dx}{dt}$.

Watch out: the division by $dx/dt$ is needed again.

Arc length

$L = \displaystyle\int_a^b \sqrt{x'(t)^2 + y'(t)^2}\,dt$.

Watch out: Pythagoras, not a sum.

Vector motion

• $\vec v = \vec r\,'$, $\vec a = \vec v\,'$, speed $= |\vec v|$.
• Position: start + $\int \vec v$.

Watch out: speed is a length, not a sum of components.

Polar slopes

$\dfrac{dy}{dx} = \dfrac{r'\sin\theta + r\cos\theta}{r'\cos\theta - r\sin\theta}$.

Watch out: $dr/d\theta$ is not the slope.

Polar area

$A = \tfrac{1}{2}\displaystyle\int_{\alpha}^{\beta} r^2\,d\theta$.

Watch out: the ½ and the square.