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Math Help - questions about cusps and tangent lines in Derivatives

  1. #1
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    questions about cusps and tangent lines in Derivatives

    these questions i have answeared uncorrectly and don't understand why i'm wrong, anyway here are the questions:

    Determine whether f has :
    (a)a vertical tangent line at (0,0)
    (b)a cusp at (0,0)

    (1)f(x) = x ^ (1/3)
    Correct Answear: (a)yes (b) no
    My Answear: (a)no (b)no

    (2)f(x) = x ^ (2/5)
    Correct Answear: (a)yes (b)yes
    My Answear: (a)no (b)no

    (3)f(x) = 5x ^ (3/2)
    Correct Answear: (a)no (b)no
    My Answear: (a)yes (b)no

    if there is a cusp at a point then the right hand limit = infinity and the left hand limit = infinity with opposite signs at that point ?
    and a vrtical tangent line means the dericative of f at this point = infinity?
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  2. #2
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    Hello, mHadad!

    Your understanding is correct, but your answers were incorrect.


    Determine whether f has:
    (a) a vertical tangent line at (0,0)
    (b) a cusp at (0,0)

    (1) f(x) = x^{1/3}

    Correct answer: (a) yes (b) no
    My answer: (a) no (b) no

    f'(x) .= .(1/3)x^{-2/3} .= .(1/3)[x^{-1/3}]^2

    (a) f'(0) is undefined; there is a vertical tangent.

    (b) The right-hand limit and the left-hand limit are both +∞
    . . .There is no cusp there.



    (2) f(x) = x^{2/5}
    Correct answer: (a) yes (b) yes
    My answer: (a) no (b) no

    f'(x) .= .(2/5)x^{-3/5}

    (a) f(0) is undefined; there is a vertical tangent.

    (b) The left-hand limit is -∞ . . . The right-hand limit is +∞
    . . .There is a cusp.



    (3) f(x) = 5x^{3/2}

    Correct answer: (a) no (b) no
    My answer: (a) yes (b) no

    f'(x) .= .(15/2)x^{1/2}

    (a) f'(0) = 0 is defined . . . no vertical tangent.

    (b) There is no left-hand limit . . . no cusp.
    . . .(x cannot approach 0 from the left.)

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  3. #3
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    thanks Soroban for the reply, i have graphed those three functions and have some questions about ur reply.about the first answear isn't both the left hand limit and right hand limit for f prime is undefined? i mean how could u tell if it is +∞ or -∞?
    as for the second answear (The left-hand limit is -∞ . . . The right-hand limit is +∞
    ...There is a cusp.
    )both graphs of the derivative for the first and second questions look similiar how could u define one limit as +∞ and the second limit as -∞ (that's for the second question)??
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