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Math Help - differentiation and intergration of sinx/x and d/dt P(t)=P`(t)

  1. #1
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    differentiation and intergration of sinx/x and d/dt P(t)=P`(t)

    (1) problem

    differentiation [intergration of sint/t], limit from 0 to x^2

    let say f(t)=t, F(t)=t^2/2, if f(t)=sint/t, F(t)=intergration of sint/t

    d/dx [intergration of sint/t] limit from 0 to x^2
    d/dx [F(t)] limit from 0 to x^2
    {d/dx [F(t)} = differentiation of F(t)

    then {d/dx [F(t)} = [f(t)] limit from 0 to x^2

    f(x^2)-f(0)
    (sin x^2/x)-(sin0/0)

    (sin x^2/x)-1

    in this correct?

    (2) problem

    d/dt P(t)=P`(t)

    t=0, P`(0)=0

    t->infinite, then P`(infinite)=f(h)/h

    in this right?
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by toumah12 View Post
    (1) problem

    differentiation [intergration of sint/t], limit from 0 to x^2

    let say f(t)=t, F(t)=t^2/2, if f(t)=sint/t, F(t)=intergration of sint/t

    d/dx [intergration of sint/t] limit from 0 to x^2
    d/dx [F(t)] limit from 0 to x^2
    {d/dx [F(t)} = differentiation of F(t)

    then {d/dx [F(t)} = [f(t)] limit from 0 to x^2

    f(x^2)-f(0)
    (sin x^2/x)-(sin0/0)

    (sin x^2/x)-1

    in this correct?
    Given that F^{\prime}(t)=f(t), \frac{\,d}{\,dx}\int_a^{u(x)}f(t)\,dt=\frac{\,d}{\  ,dx}\left[F(u(x))-F(a)\right]=F^{\prime}(u(x))\cdot u^{\prime}(x)-0=f(u(x))\cdot u^{\prime}(x)

    So it follows that \frac{\,d}{\,dx}\int_0^{x^2}\frac{\sin t}{t}\,dt=\frac{\sin(x^2)}{x^2}\cdot 2x=\frac{2\sin(x^2)}{x}

    (2) problem

    d/dt P(t)=P`(t)

    t=0, P`(0)=0

    t->infinite, then P`(infinite)=f(h)/h

    in this right?
    I'm not quite sure what you're trying to do here...
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  3. #3
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    thanks a lot.

    (2) problem is a part of question.

    d/dt P(t)=P`(t)

    t=0, P`(0)=0

    t->infinite, then P`(infinite)=f(h)/h

    in this right?
    ---End Quote---

    I'm not quite sure what you're trying to do here...

    (2) problem is only sub part of the full question that is to find the P(t) or Q(t) when t=0 and t->infinite

    d/dt P(t) = -aP(t)+bQ(t)--1
    d/dt Q(t) = aP(t)-bQ(t)--2
    P(t)+Q(t) = 1

    a and b are real number.

    let d/dt P(t) = P`(t), d/dt Q(t) = Q`(t)

    from the eq 1 and 2

    Q(t)=[a-Q`(t)]/[a+b]---3
    P(t)=[b-P`(t)]/[a+b]---4

    from the eq 3 and 4

    when t=0, P`(t)=0 or Q`(t)=0
    Q(t)=[a-0]/[a+b]
    P(t)=[b-0]/[a+b]

    P(t)+Q(t) = 1, (a/[a+b])+(b/[a+b])=1 {prove}

    thus when t=0, Q(t)=a/(a+b) or P(t)=b/(a+b)

    but, i was wondering if t->infinite, what would be the value of P`(t)?

    Could this [f(h+a)-f(a)]/h to be used?

    lim t->infinite [P(h+infinte)-P(infinite)]/h

    P(h)/h in this correct?

    recall eq 4
    P(t)=[b-P(h)/h ]/[a+b] will this be answer?
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