Short on time? This page is the recap. For the limit argument, the diagram and worked examples, use the full study guide. BC-only material. Prerequisites (parametric derivatives 9.1 and 9.2, product and chain rules) are on the Calculus BC roadmap.
Recap
- r(t) = ⟨x(t), y(t)⟩ = x(t) i + y(t) j takes a number and returns a vector.
- As a position vector, its tip traces the same curve as the parametric equations x = x(t), y = y(t).
- Domain: the values of t where both components are defined.
- The limit definition works component by component, so you differentiate each component separately.
Key relationships
| Quantity | Formula | Meaning |
|---|---|---|
| First derivative | r′(t) = ⟨x′(t), y′(t)⟩ | Tangent vector; velocity if r is position |
| Second derivative | r″(t) = ⟨x″(t), y″(t)⟩ | Acceleration if r is position |
| Tangent slope | dy/dx = y′(t)/x′(t), x′(t) ≠ 0 | Rise over run |
| Horizontal tangent | y′ = 0, x′ ≠ 0 | r′ = ⟨nonzero, 0⟩ |
| Vertical tangent | x′ = 0, y′ ≠ 0 | r′ = ⟨0, nonzero⟩ |
| Size of r′ | √(x′² + y′²) | Arc length integrand (Topic 9.3) |
| Product rule | (f r)′ = f′ r + f r′ | f is an ordinary function |
Reading the direction
| Sign of x′ | Sign of y′ | The point is moving |
|---|---|---|
| + | + | right and up |
| + | − | right and down |
| − | + | left and up |
| − | − | left and down |
Mistakes to avoid
- Writing x′ + y′ instead of the vector ⟨x′, y′⟩.
- Missing chain or product rules inside a component.
- Slope as x′/y′. It is y′/x′.
- r″(t) confused with d²y/dx². d²y/dx² = (d/dt[dy/dx]) ÷ (dx/dt), a single number.
- Domain from one component only.
- r′ = ⟨0, 0⟩ called a vertical tangent. No conclusion is possible from it alone.
Quick self-check
- Find r′(t) for r(t) = ⟨t e^t, cos(3t)⟩. (⟨e^t + t e^t, −3 sin(3t)⟩)
- r(t) = ⟨t² − 4t, t³⟩. Which way is the point moving at t = 1? (r′(1) = ⟨−2, 3⟩: left and up)
- For r(t) = ⟨2t, t³⟩, find r″(1) and d²y/dx² at t = 1. (r″(1) = ⟨0, 6⟩; d²y/dx² = 3t/2 = 1.5)
Next: practice questions.