Question

How does one do an asymptotic power expansion of an implicit function? Say the function y = y(x) is given implicitly as y = 1+x*y^2. If that function is too difficult, then how about x^2+y^2 = 1? I just don't know how to get started. I've thought about using the Taylor series expansion, but I don't know if that is a correct approach.

I was asked to do the power expansion of y = 1+x*y^2 to the power 8. If someone can help doing the power expansion of the first few terms, that would be helpful in getting started. Thank you.

Answer #1

1-(Partial Fraction Decomposition Revisited) Consider the
rational function 1/(1-x)(1-2x)
(a) Find power series expansions separately for 1/(1 − x) and
1/(1 − 2x).
(b) Multiply these two power series expansions together to get a
power series ex-pansion for
1 (1−x)(1−2x)
(This involves doing an infinite amount of distributing and
combining coeffi-cients, but you should be able to figure out the
pattern here.)
c) Separate the power series in terms of power series for A/(1 −
x) and B/(1 − 2x)...

Determine wether the given function has any local or
absolute extreme values, and find those values if
possible.
f(x) = 1/(x^2 + 1)
Can you please explain how to know if the point is abs max or
abs min if I don't think about how the graph of this function looks
like.
So I've found out that
f ' (x) = (-2x)/((x^2 + 1))^2 = 0, so x=0 is a critical
point.
f(0) = 1.
I know the answer is...

How do I express this recursive function?
an = 10n
I have found some of the values
a0 = 100 = 1
a1 = 101 =
10
a2 = 102 =
100
a3 = 103 =
1000
And this is the answer from the
textbook
an + 1 = 10an,
for ≥ 1 and a1 = 5
I'm confused because it says a1 = 5
but I got 10
Another problem
an = 5
I don't know how to find...

If prices increase at a monthly rate of 18,000?%, by what
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per? month.)
The annual inflation rate is about _______ x 10 _____ to what
power %? I don't know how to show that behind the 10 is a raised
number.
..when I looked at an example it shows steps like (I am going to
try and do this)
=Q(sligtly lower 0) x (1 + r) raised...

A potential energy function is given by U(x) = x^(−8)e^x^2 .
Let’s only focus on the region where x > 0.
a) Find the position where the potential energy is a minimum
b) For small oscillations around this minimum, what is the
angular frequency ω?
c) At what distance (either to the left or right) from the
equilibrium point is the exact value of the force (derived from the
full potential) more than 10% different from the force
corresponding to...

A subset of a power set.
(a)
Let X = {a, b, c, d}. What is { A: A ∈ P(X) and |A| = 2 }?
comment: Please give a clear explanation to what this
set builder notation translate to? Because I've checked the answer
for a) and it is A= {{a,b}, {a,c}, {a,d}, {b,c}, {b,d},
{c,d}}.
I don't understand because the
cardinality of A has to be 2 right? Meanwhile, the answer is
basically saying there's 6 elements. So...

How do I answer and solve this question?
Does the problem specify a function with independent variable x?
If so, find the domain of the function. If not, find a value of x
to which there corresponds more than one value of y.
x2 + y2 = 9

(1) (5 pts) Consider the function f : + ×+ → given by f(x, y) =
x! y!(x−y)! . Where and x and y are positive integers h Hint: this
is the combination formula, x y i (a) What types of relationships
are generated by this function, please justify your answers using
examples or counter examples. (b) How many combinations of 2 pairs
can be generated from a power of R, assuming there are 4 element in
set R .

Assume one input, one output production function which we know
very little about. But over a period of three years we see that (1)
with p = 1,w = 1 the firm choose x = 2,y = 5 (2) with p = 3,w =
0.5, the firm will choose x = 4 and y = 6 and (3)
withp=4,w=2,thefirmwillchoosex=3,y=11. Graph the isoprofit lines
associated with each pair of prices and quantity choices. Show on
your graph what we have learned...

2.At the bundle (x, y) = (2, 4), what is the MRS of the utility
function U = 3x + 4y?
1.You currently have (x, y) = (4, 1) and your utility function
is U = min{x, 4y}. How many units of x are you willing to give up
to get another unit of y?
3. You are currently at the bundle (x, y) = (4, 2) with a
utility function U = (3/4) ln(x) + (1/4)ln(y). I offer you...

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