(BC only) Parametric equations and vector-valued functions

Learning objectives

Differentiate a curve given parametrically using dy/dx = (dy/dt)/(dx/dt), and find the points where the tangent is horizontal, where dy/dt = 0 and dx/dt is not, or vertical, where dx/dt = 0 and dy/dt is not

Find the second derivative of a parametric curve as d²y/dx² = (d/dt of dy/dx) divided by dx/dt, and use its sign to determine concavity, taking care not to divide the two second derivatives

Calculate the arc length of a parametric curve as ∫√((dx/dt)² + (dy/dt)²) dt over the given parameter interval

Define a vector-valued function r(t) = ⟨x(t), y(t)⟩, differentiate and integrate it component by component, and apply an initial condition to determine the constant vector

Find the velocity vector ⟨x′(t), y′(t)⟩, the acceleration vector ⟨x″(t), y″(t)⟩ and the speed √(x′(t)² + y′(t)²) of a particle moving in the plane

Solve planar motion problems by integrating velocity to recover position, compute the total distance traveled as the integral of the speed over the time interval, and distinguish that distance from the magnitude of the displacement vector joining the initial and final positions