Derivatives of Vectors Lesson 10 4 Component Vectors
- Slides: 9
Derivatives of Vectors Lesson 10. 4
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 = Px i + Py j j i 2
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 =t 1= 2 t=3 t = 4 t = 5 3
Example • Consider the curve represented by parametric equations • Then the vector-valued function is … 4
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 § 5
Example • Given parametric equations which describe a vector-valued position function x = t 3 – t § y = 4 t – 3 t 2 § • What is the velocity vector? • What is the acceleration vector? 6
Example • For the same vector-valued function § x = t 3 – t and y = 4 t – 3 t 2 • What is the magnitude of v(t) when t = 1? • The direction? 7
Application • The Easter Bunny is traveling by balloon § Position given by height y = 360 t – 9 t 2 and x = 0. 8 t 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 § 8
Assignment • Lesson 10. 4 • Page 426 • Exercises 1 – 13 odd 9
