Alan

3.  

When x = 1 both x² - 1 and k(x - 1) are zero, so the function is continuous for all values of k.

d(x² - 1)/dx = 2x, which is 2 when x = 1

dk(x-1)/dx = k, which is 2 when k = 2, so the function is differentiable at x = 1 only when k = 2.

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Non-recursive form:

F(n) = 1+(n-1)n(n+1)

F(1) = 1

F(2) = 1+1*2*3 = 7

F(3) = 1 + 2*3*4 = 25

...etc.

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What was the ambient temperature?

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At constant pressure the volume will be proportional to the absolute temperature.

V2/(273+76) = 1690/(273+43)    (i've assumed your 1.69x103L means 1.69x10³L)

Can you take it from here?

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This might help:

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The solutions to this quadratic may be expressed as:

$$\\x_+=\frac{1-2m+\sqrt{6m^2+m+4}}{2m+3}\\\\x_-=\frac{1-2m-\sqrt{6m^2+m+4}}{2m+3}$$

There are an infinite number of values of m that have one root negative and one root positive.  However, if you are looking for the value of m that makes one root the negative of the other, then that is clearly m = 1/2.

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Rough explanation of where the expression in b comes from:

I should have noted that w_k goes to w(x)dx in passing from the discrete to the continuous (so "mass" goes to "mass per unit length"*"(infinitesimal) length"; with "mass per unit length" being "density").

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boy boy girl   0.51*0.51*0.49

boy girl boy   0.51*0.49*0.51

girl boy boy  0.49*0.51*0.51

So probability is

$${\mathtt{p}} = {\mathtt{3}}{\mathtt{\,\times\,}}{{\mathtt{0.51}}}^{{\mathtt{2}}}{\mathtt{\,\times\,}}{\mathtt{0.49}} \Rightarrow {\mathtt{p}} = {\mathtt{0.382\: \!347}}$$

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