Chapter 7 - Parametric, Polar, and Vector Functions

📚 Table of Contents

1. [Parametric Functions]
2. [Polar Coordinates]
3. [Vector Functions]
4. [Motion in Space]
5. [Applications]

1. Parametric Functions 📊

Understanding Parametric Functions

A way to describe curves using parameter t. Think of it as:

• Motion over time
• Coordinated x and y movement
• Path tracing
• Indirect curve description

Basic Form

x = f(t), y = g(t)

Key Concepts

1. Parameter t:

   • Independent variable
   • Often represents time
   • Controls both x and y
   • Defines curve position

2. Elimination of t:

   • Find rectangular form
   • Solve for relationship
   • Identify curve type
   • Check domain restrictions

Derivatives

First Derivative

$\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}$

Process

1. Find dy/dt and dx/dt:

   • Differentiate y = g(t)
   • Differentiate x = f(t)
   • Keep in terms of t
   • Watch chain rule

2. Form Quotient:

   • Write fraction
   • Simplify if possible
   • Consider domain
   • Check undefined points

Example Walkthrough

Find dy/dx for x = t², y = t³

1. Find derivatives:
   • dx/dt = 2t
   • dy/dt = 3t²
2. Form quotient:
   • dy/dx = 3t²/2t
3. Simplify:
   • = 3t/2
   • = (3/2)x^(1/2)

Second Derivative

$\frac{d^2y}{dx^2} = \frac{\frac{d}{dt}(\frac{dy}{dx})}{\frac{dx}{dt}}$

Process

1. Find first derivative
2. Differentiate with respect to t
3. Divide by dx/dt
4. Simplify

Common Curves

Circle

• x = r cos t
• y = r sin t
• 0 ≤ t ≤ 2π
• Radius r

Cycloid

• x = r(t - sin t)
• y = r(1 - cos t)
• Rolling circle
• Period 2π

2. Polar Coordinates 🎯

Understanding Polar Form

Points described by distance and angle. Think of it as:

• Distance from origin (r)
• Angle from x-axis (θ)
• Alternative to xy-coordinates
• Circular description

Conversion Formulas

Polar to Rectangular

• x = r cos θ
• y = r sin θ
• r² = x² + y²
• θ = tan⁻¹(y/x)

Process

1. Identify r and θ
2. Use conversion formulas
3. Simplify
4. Check quadrant

Common Polar Curves

Circle

• r = a (centered at origin)
• r = 2a cos θ (through origin)
• r = 2a sin θ (through origin)

Rose Curves

• r = a sin(nθ)
• n odd: n petals
• n even: 2n petals
• Symmetry about origin

Cardioid

• r = a(1 + cos θ)
• Heart-shaped
• One loop
• Size determined by a

Area in Polar Form

$A = \frac{1}{2}\int_α^β r^2 dθ$

Process

1. Identify bounds:

   • Find full rotation
   • Consider symmetry
   • Check overlap
   • Verify angles

2. Set up integral:

   • Square r
   • Include 1/2
   • Use correct bounds
   • Watch for negative r

Example Walkthrough

Find area inside r = 2 cos θ

1. Identify curve:
   • Circle through origin
   • Radius = 1
2. Find bounds:
   • -π/2 to π/2
3. Evaluate:
   • $A = \frac{1}{2}\int_{-π/2}^{π/2} 4\cos^2 θ dθ$
   • = π

3. Vector Functions 🔄

Understanding Vector Functions

Functions that output vectors. Think of it as:

• Position in space
• Motion path
• Component functions
• Parametric curve in space

Basic Form

$\vec{r}(t) = \langle f(t), g(t), h(t) \rangle$

Derivatives

Position Vector

$\vec{r}(t) = \langle x(t), y(t), z(t) \rangle$

Velocity Vector

$\vec{v}(t) = \vec{r}'(t) = \langle x'(t), y'(t), z'(t) \rangle$

Acceleration Vector

$\vec{a}(t) = \vec{r}''(t) = \langle x''(t), y''(t), z''(t) \rangle$

Example Walkthrough

For $\vec{r}(t) = \langle t^2, t^3, t \rangle$, find velocity

1. Differentiate components:
   • x'(t) = 2t
   • y'(t) = 3t²
   • z'(t) = 1
2. Write velocity:
   • $\vec{v}(t) = \langle 2t, 3t^2, 1 \rangle$

4. Motion in Space 📈

Understanding Motion

Describing position, velocity, and acceleration. Think of it as:

• Path through space
• Rate of position change
• Speed and direction
• Force and movement

Key Concepts

Speed vs. Velocity

• Speed: $|\vec{v}(t)|$
• Velocity: $\vec{v}(t)$
• Direction matters
• Scalar vs. vector

TNB Frame

• Tangent vector
• Normal vector
• Binormal vector
• Orthogonal system

Arc Length

$L = \int_a^b |\vec{r}'(t)|dt$

Process

1. Find r'(t)
2. Calculate magnitude
3. Set up integral
4. Evaluate

5. Applications 🎯

Area Calculations

Polar Form

$A = \frac{1}{2}\int_α^β r^2dθ$

Parametric Form

$A = \int_a^b y\frac{dx}{dt}dt$

Example Walkthrough

Find area inside r = 4 sin θ

1. Identify curve:
   • Circle through origin
   • Diameter = 4
2. Set up integral:
   • $A = \frac{1}{2}\int_0^π 16\sin^2 θ dθ$
3. Evaluate:
   • = 4π

📝 AP-Style Examples

Example 1: Parametric Derivatives

Find dy/dx at t = 1 for x = t², y = t³

Solution:

1. Find derivatives:
   • dx/dt = 2t
   • dy/dt = 3t²
2. Form quotient:
   • dy/dx = 3t²/2t = 3t/2
3. Evaluate at t = 1:
   • dy/dx = 3/2

Example 2: Polar Area

Find area inside r = 2 sin 2θ

Solution:

1. Identify curve:
   • Four-leaved rose
   • One leaf: 0 to π/4
2. Set up integral:
   • $A = 2\int_0^{π/4} 4\sin^2 2θ dθ$
3. Evaluate:
   • = 2

💡 Success Strategies

1. Curve Recognition

• Sketch curves
• Identify key points
• Consider symmetry
• Check period

2. Common Mistakes

• Wrong parameter range
• Sign errors
• Chain rule errors
• Domain restrictions

3. Calculator Tips

• Graph to verify
• Check endpoints
• Use parametric mode
• Confirm results

🔍 AP Exam Focus

Free Response Tips

1. Show work:

   • Parameter elimination
   • Derivative calculations
   • Area/length setup
   • Vector operations

2. Common Questions:

   • Slope calculations
   • Area/arc length
   • Motion analysis
   • Curve sketching

Multiple Choice Strategy

1. Consider:

   • Multiple approaches
   • Graphical insights
   • Parameter ranges
   • Symmetry

2. Check:

   • Units
   • Sign
   • Domain
   • Reasonableness

💡 Pro Tip: Practice converting between parametric, polar, and rectangular forms - it's crucial for solving complex problems!