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Calculus II (Integration & Series) Unit 7: Parametric, polar & vector-valued functions

Parametric, polar and vector-valued functions describe curves and motion that aren't simple y = f(x) graphs.

Covers: Derivatives & arc length of parametric curves, polar area, vector motion

Study guide

Parametric equations x(t), y(t) describe a curve traced over time. The slope of the curve is dy/dx = (dy/dt)/(dx/dt), and horizontal tangents occur where dy/dt = 0 (with dx/dt ≠ 0). The second derivative is d²y/dx² = [d/dt(dy/dx)]/(dx/dt). Arc length is ∫ √((dx/dt)² + (dy/dt)²) dt.

Vector-valued functions r(t) = ⟨x(t), y(t)⟩ treat parametric motion as a position vector. Velocity is v(t) = r′(t) = ⟨x′(t), y′(t)⟩, acceleration is a(t) = r″(t), and speed is the magnitude |v(t)| = √(x′(t)² + y′(t)²). The total distance traveled is the integral of speed. To find position from velocity, integrate each component and use the initial position.

Polar curves r = f(θ) can be treated as parametric curves with x = r cos θ and y = r sin θ, which gives their slopes. The area enclosed by a polar curve from θ = α to θ = β is A = ½∫ r² dθ, because thin sectors have area ½r²Δθ. The area between two polar curves is ½∫ (R² − r²) dθ, with limits found where the curves intersect. Watch for loops traced twice and choose limits that trace the region exactly once.

When θ changes with time, dr/dt = (dr/dθ)·(dθ/dt). The same chain rule applies to x = r cos θ and y = r sin θ. If r > 0 and dr/dθ > 0, points on the curve move away from the origin as θ increases. If r > 0 and dr/dθ < 0, they move toward the origin. The conclusions reverse when r < 0.

For a polar curve, dy/dx = (r′ sin θ + r cos θ)/(r′ cos θ − r sin θ), where r′ = dr/dθ. The arc length of r = f(θ) from θ = α to θ = β is ∫ √(r² + (dr/dθ)²) dθ.

A parametric curve has a vertical tangent where dx/dt = 0 and dy/dt ≠ 0. A particle moves left when dx/dt < 0 and down when dy/dt < 0.

Key ideas

• Parametric slope: dy/dx = (dy/dt)/(dx/dt).
• Second derivative: d²y/dx² = [d/dt(dy/dx)] / (dx/dt).
• Speed is the magnitude of the velocity vector.
• Polar area uses wedges: A = ½∫ r² dθ.
• Polar slope converts with x = r cos θ, y = r sin θ.

Formulas and rules

Common mistakes

• Dividing d/dt(dy/dx) by dt instead of dx/dt for the second derivative.
• Forgetting the ½ in polar area.

Words to know

Slope of a parametric curve

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

Speed of a particle with position ⟨x(t), y(t)⟩

√(x′(t)² + y′(t)²)

Area inside a polar curve

½∫ r² dθ

Second derivative for parametric curves

[d/dt(dy/dx)] / (dx/dt)

Arc length of a parametric curve

∫ √((dx/dt)² + (dy/dt)²) dt

Parametric curve

x = x(t), y = y(t), traced as t changes.

Horizontal tangent (parametric)

Where dy/dt = 0 and dx/dt ≠ 0.

Velocity vector

v(t) = ⟨x′(t), y′(t)⟩

Acceleration vector

a(t) = ⟨x″(t), y″(t)⟩

Total distance (vector motion)

∫ speed dt = ∫ √(x′² + y′²) dt

Area between polar curves

½∫ (R² − r²) dθ

Sample practice questions

29 more questions like these, from level 1 to 5, are in the practice set.