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5
2010 Calculus #5: Two Defined Functions
2010 Calculus #5: Two Defined Functions
Source:
July 15, 2012
calculus
function
algebra
polynomial
Problem Statement
Let the functions
f
(
α
,
x
)
f(\alpha,x)
f
(
α
,
x
)
and
g
(
α
)
g(\alpha)
g
(
α
)
be defined as
f
(
α
,
x
)
=
(
x
2
)
α
x
−
1
g
(
α
)
=
d
4
f
d
x
4
∣
x
=
2
f(\alpha,x)=\dfrac{(\frac{x}{2})^\alpha}{x-1}\qquad\qquad\qquad g(\alpha)=\,\dfrac{d^4f}{dx^4}|_{x=2}
f
(
α
,
x
)
=
x
−
1
(
2
x
)
α
g
(
α
)
=
d
x
4
d
4
f
∣
x
=
2
Then
g
(
α
)
g(\alpha)
g
(
α
)
is a polynomial is
α
\alpha
α
. Find the leading coefficient of
g
(
α
)
g(\alpha)
g
(
α
)
.
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