Section 12.7

Calculus with Vector Functions

Extending limits, derivatives, and integrals to vector-valued functions.

Introduction

Calculus operations on vector functions represent physical quantities. The derivative represents the velocity vector, which is always tangent to the path of motion.

Interactive: The Tangent Vector

The red vector points in the direction of motion.

Key Formulas

Derivative

Differentiate each component:

Integral

Integrate each component:

Geometric Meaning

is the Tangent Vector (Velocity).
is the Speed.
If , the curve is smooth.

Worked Examples

Example 1: Finding the Derivative (Level 1)

Find for .

Step 1: Differentiate components

Step 2: Assemble Vector

Example 2: Tangent Line (Level 2)

Find the parametric equations of the tangent line to the helix at .

1. Find Point P

.
Point .

2. Find Tangent Vector v

.
At :
.

3. Write Line Equation

Example 3: Integration (Level 3)

Evaluate .

Integrate term by term:

Result: