1. Derivatives of Vectors Lesson 10.4

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

3. 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. 4 Example Consider the curve represented by parametric equations Then the vector-valued function is …

5. 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. 6 Example Given parametric equations which describe a vector-valued position function  x = t 3 – t  y = 4t – 3t 2 What is the velocity vector? What is the acceleration vector?

7. 7 Example For the same vector-valued function  x = t 3 – tand y = 4t – 3t 2 What is the magnitude of v(t) when t = 1? The direction?

8. 8 Application The Easter Bunny is traveling by balloon  Position given by height y = 360t – 9t 2 and x = 0.8t 2 + 0.9 sin 2 t (positive direction west) Determine the velocity of the balloon at any time t For time t = 2.5, determine  Position  Speed  Direction

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