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Derivatives of Vectors

Derivatives of Vectors. Lesson 10.4. 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. P = P x i + P y j. j. i. Vector Functions and Parametric Equations.

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Derivatives of Vectors

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  1. Derivatives of Vectors Lesson 10.4

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

  3. Vector Functions andParametric Equations • Consider a curve described by parametric equations • x = f(t) y = g(t) • The curve can be expressed as thevector-valued function, P(t) • P(t) = f(t)i + g(t)j t = 2 t = 1 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 functionp(t) = f(t)i + g(t)j • Given also that f(t) and g(t) are differentiable • Then the derivative of p isp'(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 = 0.8t2 + 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

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

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