find the max & min values of voltage given by the expression: v=5cos4θ + 5 between and including values of θ=0 and θ=2π radians?

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  • 8 months ago
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    0 ≤ θ ≤ 2π

    0 ≤ 4θ ≤ 8π

    v = 5 cos(4θ) + 5

    dv/dθ = -20 sin(4θ)

    d²v/dθ² = -80 cos(4θ)

    When dv/dθ = 0:

    -20 sin(4θ) = 0

    4θ = 0, π, 2π, 3π, 4π, 5π, 6π, 7π, 8π

    θ = 0, π/4, π/2, 3π/4, π, 5π/4, 3π/2, 7π/4, 2π

    When θ = 0, π/2, π, 3π/2, 2π:

    v = 5 cos(4θ) + 5 = 10

    dv/dθ = 0

    d²v/dθ² = -80 cos(4θ) = -80 < 0

    When θ = π/4, 3π/4, 5π/4, 7π/4

    v = 5 cos(4θ) + 5 = 0

    dv/dθ = 0

    d²v/dθ² = -80 cos(4θ) = 80 > 0

    Answers:

    Maximum voltage = 10 V when θ = 0, π/2, π, 3π/2, 2π

    Minimum voltage = 0 V when θ =π/4, 3π/4, 5π/4, 7π/4

  • Ian H
    Lv 7
    8 months ago

    V = 5cos(4θ) + 5, (θ from 0 to 2π radians) 

    Graph looks like this 

    https://www.wolframalpha.com/input/?i=V+%3D+5cos%2...

     

    Values of voltage at the 5 maxima are 10v. 

    Values of voltage at the 4 minima are zero v. 

  • 8 months ago

    The 4θ term means there are 4 cycles between 0 and 2π, so the max and min are +10 and 0

    Attachment image
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