Calculus

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f(theta)= 4sec(theta)+ 2tan(theta), 0 less than theta less than 2pi.
find all critical numbers

  • Calculus -

    where is f' zero or undefined?

    f' = 4secθtanθ + 2sec^2 θ
    = 2secθ(2tanθ+secθ)

    so, critical numbers are where

    secθ = 0 or is undefined,
    2tanθ+secθ = 0 or is undefined

    That help?

  • Calculus -

    yes,i got 7pi/6 and 11pi/6 is that correct?

  • Calculus -

    f(Ø) = 4secØ + 2tanØ
    for Ø intercepts, f(Ø) = 0
    4secØ + 2tanØ = 0
    2/cosØ + sinØ/cosØ = 0
    times cosØ
    2 + sinØ = 0
    sinØ = -2 , which is not possible

    so the curve does not cross or touch the Ø axis

    for y-intercept, let Ø = 0
    f(0) = 4sec 0 + 2 tan 0 = 4 + 0 = 4
    crosses the vertical axis at (0,4)

    f ' (x) = 4 secØtanØ + 2sec^2 Ø
    = 0 for max/min of f(Ø)

    4/cosØ (sinØ/cosØ) = -2/cos^2 Ø

    4 sinØ = -2
    sinØ = -1/2
    Ø = 7π/6 or Ø = 11π/6

    if Ø = 7π/6 , f(7π/6) = 4(-√3/2) + 2/√3 = ....
    if Ø = 11π/6, f(11π/6) = ...

    I will let you finish the arithmetic

    If you want the points of inflection, take the 2nd derivative, and set that equal to zero.
    That will be quite a challenge.

  • Calculus -

    Thank you!!

  • Calculus -

    Hmm. I was under the impression that critical numbers were numbers in the domain where f'=0 or f is not differentiable.

    I think your math is great, Reiny, but maybe covers too much ground.

  • Calculus -

    f (0)=sinø + cos ø of(0)=3 f'(0)=4

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