Parametric Equations and Motion Free Online FlashCards

What defines a parametric curve?

A parametric curve is described by coordinate functions x=x(t) and y=y(t), where t is the parameter.

What curve is x=cos t, y=sin t?

For x=cos t and y=sin t, 0≤t≤2π, the curve is the unit circle, traced counterclockwise once.

What information can parameter elimination lose?

Eliminating t can produce a rectangular equation, but it may lose information about direction, timing, and repeated traversal.

How do you find slope parametrically?

The slope is dy/dx=(dy/dt)/(dx/dt)=y′(t)/x′(t), provided x′(t)≠0.

What conditions give horizontal and vertical tangents?

A horizontal tangent occurs when y′(t)=0 and x′(t)≠0. A vertical tangent occurs when x′(t)=0 and y′(t)≠0.

What is the tangent-line formula?

The tangent line at t=t₀ is y−y(t₀)=[y′(t₀)/x′(t₀)](x−x(t₀)), when the slope is defined.

How is the second derivative found parametrically?

The second derivative is d²y/dx²=[d/dt(dy/dx)]/(dx/dt), equivalently [x′y″−y′x″]/[x′]³.

How are position, velocity, and acceleration related?

Position is r(t)=⟨x(t),y(t)⟩. Velocity is r′(t)=⟨x′(t),y′(t)⟩, and acceleration is r″(t)=⟨x″(t),y″(t)⟩.

What is the speed of a parametric particle?

Speed is the magnitude of velocity: √([x′(t)]²+[y′(t)]²). It is nonnegative, while velocity also records direction.

What are velocity and acceleration for the projectile model?

For x=3t and y=10t−4.9t², velocity is ⟨3,10−9.8t⟩ and acceleration is ⟨0,−9.8⟩.

When does the projectile reach its highest point?

The projectile reaches its highest point when its vertical velocity is zero: 10−9.8t=0, so t=10/9.8≈1.02 seconds.

What is the parametric arc-length formula?

Arc length, and therefore distance traveled, is L=∫ₐᵇ√([x′(t)]²+[y′(t)]²) dt.