Formulas for Rectangular, Parametric, & Polar Equations

Flashcards covering formulas for rectangular, parametric, and polar equations, including derivatives, area, volume, arc length, speed, total distance, and position.

When do the signs of velocity and acceleration need to be the same?

Speed is increasing.

For parametrically defined curves, what is the velocity vector?

x'(t), y'(t)

For parametrically defined curves, what is the acceleration vector?

x''(t), y''(t)

What is the formula for the derivative (slope of curve, velocity of a particle, etc.) in rectangular coordinates?

dy / dx

What is the formula for the derivative (slope of curve, velocity of a particle, etc.) in polar coordinates?

Convert to parametric. x = rcos(theta) y = rsin(theta)

dx d/dx (dy/dx) = dt/dx d/dt (dy/dx)

2nd derivative

What is the formula for area in rectangular coordinates?

∫[a to b] f(x) dx

What is the formula for area in polar coordinates?

∫[theta1 to theta2] 1/2 r^2 d(theta)

What is the formula for volume in rectangular coordinates?

Disc: ∫[a to b] πR^2 dx; Washer: ∫[a to b] π(R^2 - r^2) dx

What is the formula for arc length in rectangular coordinates?

∫[a to b] √(1 + (f'(x))^2) dx

What is the formula for arc length in parametric coordinates?

∫[t1 to t2] √((dx/dt)^2 + (dy/dt)^2) dt

What is the formula for arc length in polar coordinates?

∫[a to b] √((r(θ))^2 + (r'(θ))^2) dθ

What is the formula for speed, v(t), in parametric coordinates?

√((dx/dt)^2 + (dy/dt)^2)

What is the formula for total distance in parametric coordinates?

∫[t1 to t2] v(t) dt

What is the formula for position in parametric coordinates?

(x(t), y(t)), where x(t) = x(t1) + ∫[t1 to t2] x'(t) dt and y(t) = y(t1) + ∫[t1 to t2] y'(t) dt