Introduction to Parametric Equations and Vectors Rizzi – Calc BC

Parametric Equations Involve both x and y expressed in terms of a third variable, t Review: Sketch the following graph. 𝑥= 𝑡 2 −4 and 𝑦= 𝑡 2 , −2≤𝑡≤3

What will this graph look like? 𝑥=3 sin 𝑡 and 𝑦=3 cos 𝑡 , 0≤𝑡<2𝜋

Let’s Do Some Calculus! Given 𝑥=2 𝑡 and 𝑦=3 𝑡 2 −2𝑡 Find 𝑑𝑦 𝑑𝑥 and 𝑑 2 𝑦 𝑑 𝑥 2 and evaluate at 𝑡=1

FYI Everything you know from chapters 2-5 about derivatives and integrals still applies in these problems Recall what you know about tangent lines, extrema, and concavity

Try This! Given 𝑥=4 cos 𝑡 and 𝑦=3 sin 𝑡 , write an equation of the tangent line to the curve at point where 𝑡= 3𝜋 4

Vectors A vector is a quantity that has both magnitude and direction Generally written like this: 𝑥 𝑡 , 𝑦 𝑡 Description of x and y both in terms of t

Particle Motion Like above, everything you know about particle motion still applies The only difference: x and y are defined independently Particle Motion Free Response Questions framed in terms of Vectors OR Parametrics (But they’re basically the same in how they operate)

Position, Speed, Velocity, and Acceleration A particle moves in the xy-plane so that at any time t, 𝑡≥0, the position of the particle is given by 𝑥 𝑡 = 𝑡 3 +4 𝑡 2 and 𝑦 𝑡 = 𝑡 4 − 𝑡 3 Find the velocity vector at 𝑡=1 Find the speed of the particle at 𝑡=1 Find the acceleration vector at 𝑡=1