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Math Help - Please check my answers

  1. #1
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    Please check my answers

    1>Lim (ln(ln(x))/ln(x)
    x->inf
    (a)-1 (b) 0 (c) 1 (d) inf
    Answer: infinity

    2>An ideal gas satisfies the equation PV = RT, where P is the pressure, V is the volume, R is a constant, T is the temperature, and n is the number of moles of the gas. State the rate of change of the temperature with respect to time in terms of P, V, R, and n.
    (a) Vdp/dt +pDv/dt )/ndR/dt (b)0 (c) (vdp/dt +p DV/dt)/nR (d) none
    Answer: 0

    3>An ideal gas satisfies the equation PV = RT, where P is the pressure, V is the volume, R is a constant, and T is the temperature. At a certain time, the temperature is maintained constant, the pressure P = 100 lb/in2 and is increasing at 4 lb/in2•sec. At what rate is the volume changing when it is 60 in3?
    (a) -2.4 (b) 2.4 (c)0 (d) none
    Answer: 2.4


    4>Lim sqrt(x+1)/ln(x+1)
    x->0
    (a) limit does not exist (b)0 (c)1 (d) 1/2

    Answer: (d) 1/2

    5>Lim 2x ln(x^2)
    x->0
    (a)inf (b) -1(c) 0 (d) -1/2
    Answer : 0


    6>Lim (1+5/x) ^x
    x->inf
    (a)5 (b)1 (c) e (d) e^5
    Answer: 5
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by bobby77
    1>Lim (ln(ln(x))/ln(x)
    x->inf
    (a)-1 (b) 0 (c) 1 (d) inf
    Answer: infinity
    The answer can't be \infty, as this is the same as:

    <br />
\lim_{x\rightarrow \infty} \frac{\ln(x)}{x}<br />

    which is 0, as L'Hopitals rule will verify (and numerical
    experiment confirm).

    RonL
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  3. #3
    Grand Panjandrum
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    [QUOTE=bobby77]
    2>An ideal gas satisfies the equation PV = RT, where P is the pressure, V is the volume, R is a constant, T is the temperature, and n is the number of moles of the gas. State the rate of change of the temperature with respect to time in terms of P, V, R, and n.
    (a) Vdp/dt +pDv/dt )/ndR/dt (b)0 (c) (vdp/dt +p DV/dt)/nR (d) none
    Answer: 0

    First the ideal gas equation is:

    <br />
PV=nRT<br />

    where n is the number of moles, and R is the universal gas
    constant, and P,\ V and T are the Pressure Volume
    and Temprature respectivly.

    Now:

    <br />
\frac{d}{dt}PV=P\frac{dV}{dt}+\frac{dP}{dt}V=nR \frac{dT}{dt}<br />

    So:

    <br />
\frac{dT}{dt}=\left( \frac{P\frac{dV}{dt}+\frac{dP}{dt}V}{nR} \right)<br />

    Which is answer (c)

    RonL
    Last edited by CaptainBlack; March 2nd 2006 at 03:13 AM.
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  4. #4
    Grand Panjandrum
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    Quote Originally Posted by bobby77
    3>An ideal gas satisfies the equation PV = RT, where P is the pressure, V is the volume, R is a constant, and T is the temperature. At a certain time, the temperature is maintained constant, the pressure P = 100 lb/in2 and is increasing at 4 lb/in2•sec. At what rate is the volume changing when it is 60 in3?
    (a) -2.4 (b) 2.4 (c)0 (d) none
    Answer: 2.4
    From (2) we get:

    <br />
\frac{d}{dt}PV=P\frac{dV}{dt}+\frac{dP}{dt}V=R \frac{dT}{dt}<br />

    ( n had been absorbed into [math[R[/tex] in this case)

    Now we are told that:

    <br />
\frac{dT}{dt}=0,\ \ \frac{dP}{dt}=4,\ \ P=100,\  \mbox{and}\  V=60<br />

    so:

    <br />
P\frac{dV}{dt}=-V\frac{dP}{dt}<br />
,

    form which it follows that

    <br />
\frac{dV}{dt}=-\ \frac{4 \times 60}{100}=-2.4<br />

    RonL
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  5. #5
    Grand Panjandrum
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    Quote Originally Posted by bobby77
    4>Lim sqrt(x+1)/ln(x+1)
    x->0
    (a) limit does not exist (b)0 (c)1 (d) 1/2

    Answer: (d) 1/2

    5>Lim 2x ln(x^2)
    x->0
    (a)inf (b) -1(c) 0 (d) -1/2
    Answer : 0
    get your calculator out and just try these for x=10, 100, 1000. This should
    be sufficient to identify what these limits are.

    RonL
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  6. #6
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    Quote Originally Posted by bobby77
    6>Lim (1+5/x) ^x
    x->inf
    (a)5 (b)1 (c) e (d) e^5
    Answer: 5
    The advice given for problems (4) and (5) also applies here. But this
    is usefull to know:

    <br />
\lim_{x \rightarrow \infty}(1+1/x)^x=e <br />

    So putting y=zx gives for fixed z:

    <br />
\lim_{y \rightarrow \infty}(1+z/y)^{y/z}=e <br />

    and so:

    <br />
\lim_{y \rightarrow \infty}(1+z/y)^{y}=e^z <br />

    So here the answer is (d)

    RonL
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