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 = Pxi + Py j j i

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

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

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

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?

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?

Application The Easter Bunny is traveling by balloon Position given by height y = 360t – 9t2 and x = 0.8t2 + 0.9 sin2t (positive direction west) Determine the velocity of the balloon at any time t For time t = 2.5, determine Position Speed Direction

Assignment Lesson 10.4 Page 426 Exercises 1 – 13 odd