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3.4 Velocity And Other Rates Of Change

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    3.4 Velocity And Other Rates Of Change 3.4 Velocity And Other Rates Of Change Presentation Transcript

    • 3.4 Velocity and Other Rates of Change Derivatives give the rates at which things change in the world
    • Velocity • Average Velocity = Displacement Time Instantaneous Velocity = Derivative of Position x(t) = Position x’(t) = Velocity x”(t) = v’(t) = a(t) = Acceleration
    • Determining the velocity • An object moves along a linear path according to the equation • s = 2t2 −12t +10,where s is measured in feet and t in seconds. • Determine the velocity when t = 4 and when t = 2. When is the velocity zero? • v(t) = s'(t) = 4t −12 • v(4) = s'(4) = 4(4) −12 = 4 ft/sec. • v(2) = s'(2) = 4(2) −12 = −4 ft/sec. • Velocity = 0 when 4t −12 = 0, or when • t = 3 sec.
    • Finding position, velocity and acceleration • An object is moving along a linear path according to the equation • s(t) = 1 t4 − 2t3 + 4t2 Determine the position, velocity, and acceleration 4 when t = 0 and when t = 3 seconds. • s(0) = 1 (0)2 − 2(0)3` + 4(0)2 = 0 ft. • 4 • s(3) = 1 (3)4 − 2(3)3 + 4(3)2 = 9 • 4 4 ft. • v(t) = s'(t) = t3 − 6t2 + 8t • v(0) = (0)3 − 6(0)2 + 8(0) = 0 ft / sec • v(3) = (3)3 − 6(3)2 + 8(3) = −3 ft / sec • a(t) = v'(t) = 3t2 −12t + 8 • a(0) = 3(0)2 −12(0) + 8 = 8 ft / sec2 • a(3) = 3(3)2 −12(3) + 8 = −1 ft / sec2
    • Particle Movement • V(t) = 0, then the particle is at rest • V(t) > 0, then the particle is moving right • V(t) < 0, then the particle is moving left