1. Derivatives of Vectors
Lesson 10.4

2. Component Vectors Unit vectors often used to express vectors
P = Pxi + Py j
i and j are vectors with length 1, parallel to x and y axes, respectively
P = Pxi + Py j j i

3. Vector Functions and Parametric Equations
Consider a curve described by parametric equations
x = f(t) y = g(t)
The curve can be expressed as the vector-valued function, P(t)
P(t) = f(t)i + g(t)j
t = 1
t = 2
t = 3
t = 4
t = 5

4. Example Consider the curve represented by parametric equations
Then the vector-valued function is …

5. Derivatives of Vector-Valued Functions
Given the vector valued function p(t) = f(t)i + g(t)j
Given also that f(t) and g(t) are differentiable
Then the derivative of p is p'(t) = f '(t)i + g'(t)j
Recall that if p is a position function
p'(t) is the velocity function
p''(t) is the acceleration function

6. Example
Given parametric equations which describe a vector-valued position function
x = t3 – t
y = 4t – 3t2
What is the velocity vector?
What is the acceleration vector?

7. Example For the same vector-valued function
x = t3 – t and y = 4t – 3t2
What is the magnitude of v(t) when t = 1?
The direction?

8. Application The Easter Bunny is traveling by balloon
Position given by height y = 360t – 9t2 and x = 0.8t sin2t (positive direction west)
Determine the velocity of the balloon at any time t
For time t = 2.5, determine
Position
Speed
Direction

9. Assignment
Lesson 10.4
Page 426
Exercises 1 – 13 odd