Parametric & Polar (Extended): every key term you need (+ practice quiz)
31 flashcard terms for AP Calculus BC Unit 9, written to match the course framework. Study them here, then drill them as interactive flashcards, or test yourself with the 27-question quiz — free, no account needed.
L = ∫√[(dx/dt)² + (dy/dt)²] dt. Integrates speed over interval.
Polar Coordinates
(r, θ) where r = distance, θ = angle. x = r cos θ, y = r sin θ.
Area in Polar
A = (1/2)∫r² dθ from θ₁ to θ₂. Integrates sector areas.
Polar Derivatives
dy/dx = (r' sin θ + r cos θ) / (r' cos θ - r sin θ) where r' = dr/dθ.
Rose Curves
r = a sin(nθ) or r = a cos(nθ). n petals if n odd, 2n if n even. Symmetric patterns.
Circles & Limacon
r = a (circle), r = a + b sin θ (limacon). Various polar curves from equations.
Lemniscate
r² = a² cos(2θ) or r² = a² sin(2θ). Figure-eight shaped. Symmetric about axes.
Spiral
r = aθ or r = ae^(bθ). Logarithmic spiral common in nature (shells, galaxies).
Unit 9 Summary
Parametric curves: dx/dx via derivatives of components. Polar: r² dθ integration. Rose, limacon, spiral, lemniscate curves.
Vector Velocity and Position
Given velocity ⟨x'(t),y'(t)⟩ and an initial position, integrate each component separately and add the corresponding initial coordinate.
Displacement Vector
The displacement from t=a to t=b is ⟨∫x'dt, ∫y'dt⟩, which differs from the distance traveled whenever the path is not straight.
Acceleration Vector Interpretation
The component of acceleration along velocity changes speed; the perpendicular component changes direction. Zero tangential component means constant speed.
Total Distance for Vector Motion
Distance = ∫√((x')² + (y')²)dt, requiring a calculator on most exam problems; setting it up correctly is worth most of the credit.
Direction of Travel
The unit vector v/|v| gives the direction of motion; its angle is arctan(y'/x') adjusted for the quadrant of the components.
Motion Along a Line
When x'(t) and y'(t) are proportional with a constant ratio, the path is a straight line even though the speed may vary.
Polar Region Boundaries
Determine the θ-limits by finding where the curve starts and stops enclosing area, checking whether r passes through zero.
Areas shared by two curves usually require splitting the θ-range at the intersection angles and using a different outer curve on each piece.
Polar Curve Symmetry Shortcut
If the region is symmetric about the polar axis, compute half the area and double it, which halves the algebra and the chance of a limit error.
Rate of Change of Distance from the Origin
For a polar path, dr/dθ measures how quickly the radius grows; dr/dθ = 0 marks the maximum or minimum distance from the pole.
Parametric Motion with a Table
Given tabulated velocity values, approximate ∫x'dt with a Riemann or trapezoidal sum, then add the initial coordinate to estimate position.
Speed Increasing Test
Speed increases when the derivative of √((x')²+(y')²) is positive, equivalently when x'x'' + y'y'' > 0.
Arc Length vs Parameter Interval
Arc length depends on the geometric path; reparametrizing changes the integrand and limits but not the resulting length, provided the curve is traced once.
Tangent Line to a Parametric Curve
At t=t0 the tangent line is y - y(t0) = (dy/dx)(x - x(t0)) with the slope evaluated from the component derivatives at t0.
Converting Polar Equations
Multiply a polar equation by r to introduce r² = x²+y² and r cos θ = x, converting r = 2cos θ into the circle (x-1)²+y²=1.
Area Swept by a Radius
The polar area integral literally sums sector areas swept by the radius as θ advances, which is why the factor of one-half appears.
Multiple Representations of a Point
(r,θ) and (-r, θ+π) name the same point, so intersection problems must be checked graphically as well as algebraically.
Vertical Tangents in Polar Form
Set dx/dθ = 0 where x = r cos θ, and confirm dy/dθ ≠ 0, to locate vertical tangents on a polar curve.
Position from Speed Alone
Speed determines distance traveled but not position; direction information from the individual components is indispensable.
Interpreting Vector Answers in Context
State units and describe motion in words: for example 'the particle is moving left and down at 5 units per second' rather than reporting a bare vector.