Derivatives and arc length for parametric curves, polar area, and vector-valued function motion.
40 cards · basic cards · AI-written, checked twice. Edit anything.
- What is a parametric curve?
Show answer
A curve traced by a point (x(t), y(t)) where both x and y are functions of a parameter t.- To eliminate the parameter from parametric equations, what is the general procedure?
Show answer
Solve one parametric equation for t, then substitute that expression into the other equation to get a single equation in x and y.- For parametric equations x = t² and y = t + 1, what is the rectangular form?
Show answer
y = √x + 1 (for x ≥ 0) or y = -√x + 1- What is the formula for dy/dx when x and y are both functions of t?
Show answer
dy/dx = (dy/dt) / (dx/dt)- What is the formula for d²y/dx² in parametric form?
Show answer
d²y/dx² = d/dx(dy/dx) = (d/dt(dy/dx)) / (dx/dt)- How do you find where a parametric curve has a horizontal tangent?
Show answer
Set dy/dt = 0 (while dx/dt ≠ 0) and solve for t.- How do you find where a parametric curve has a vertical tangent?
Show answer
Set dx/dt = 0 (while dy/dt ≠ 0) and solve for t.- For a parametric curve, how do you determine concavity?
Show answer
Use d²y/dx² > 0 for concave up, d²y/dx² < 0 for concave down.- What is the arc length formula for a parametric curve from t = a to t = b?
Show answer
L = ∫[a to b] √((dx/dt)² + (dy/dt)²) dt- When setting up an arc length integral for a parametric curve, what goes under the square root?
Show answer
(dx/dt)² + (dy/dt)², where dx/dt and dy/dt are derivatives with respect to the parameter.- In polar coordinates, what do r and θ represent?
Show answer
r is the distance from the origin, and θ is the angle measured counterclockwise from the positive x-axis.- What is the conversion formula from polar (r, θ) to rectangular x?
Show answer
x = r cos(θ)- What is the conversion formula from polar (r, θ) to rectangular y?
Show answer
y = r sin(θ)- What is the conversion formula from rectangular (x, y) to polar r?
Show answer
r = ±√(x² + y²)- What is the key relationship between rectangular coordinates and polar r?
Show answer
r² = x² + y²
FlashKeepers