By Jordyn Kohl, Soumya Kamath, and Peter Ballentine Parametric and Vector By Jordyn Kohl, Soumya Kamath, and Peter Ballentine

Part 1: Parametric

Parametric Functions defined: a group of quantities as functions of one or more independent variables called parameters x = f(t) y = g(t) input: t x = sin(3πt) y = cos(5πt) -1 .5 1 1.5

Derivative of Parametric Equation dy dt 1st derivative: = dy dx dx dt 2nd derivative: ( ) d dt dy dx dx dt

√(f’(t))2 +(g’(t))2)dx Additional Terms B A Parametric Arc Length Equation: B √(f’(t))2 +(g’(t))2)dx A

Part Two: Vectors

Key Definitions Vector- directed line segment with a initial and terminal point. Component form: < x(t), y(t) > Magnitude: Length of the vector. = x2+y2 Speed: l v l (absolute value of the velocity vector) Average Speed: Distance divided by time elapsed. Total distance traveled: Integrate velocity function using the bounds given and take the absolute value of the value.

Motion with vectors Position: s(t)= <x(t), y(t)> Velocity: v(t)= <x’(t), y’(t)> Acceleration: a(t)= <x’’(t), y’’(t)> Derivative Integration Moving left: x’(t) is negative Moving right: x’(t) is positive. Moving down: y’(t) is negative Moving up: y’(t) is positive. At rest when x’(t)=0 and y’(t)=0