Show all your work. Let f(x) = x 5 e x 3 (i) Use the Taylor...
Show all your work. Let f(x) = x 5 e x 3 (i) Use the Taylor
series for e x around 0 to find the Taylor series for f(x) (ii) Use
(i) to find f (20)(0), f (21)(0), f (22)(0), f (23)(0)
5. (a) Let f : R \ {−1} → R, f(x) = x+1. Show that f is
injective, but not surjective.
(b) Suppose g : R\{−1} → R\{a} is a function such that g(x) =
x−1, where a ∈ R. Determine x+1
a, show that g is bijective and determine its inverse
function.
Problem 3. Let F ⊆ E be a field extension.
(i) Suppose α ∈ E is algebraic of odd degree over F. Prove that
F(α) = F(α^2 ). Hints: look at the tower of extensions F ⊆ F(α^2 )
⊆ F(α) and their degrees.
(ii) Let S be a (possibly infinite) subset of E. Assume that
every element of S is algebraic over F. Prove that F(S) = F[S]
Let be the following probability density function f (x) = (1/3)[
e ^ {- x / 3}] for 0 <x <1 and f (x) = 0 in any other
case
a) Determine the cumulative probability distribution F (X)
b) Determine the probability for P (0 <X <0.5)
f(x)=e^x/(3+e^x)
Find the first derivative of f.
Use interval notation to indicate where f(x) is increasing.
List the x coordinates of all local minima of f. If there are no
local maxima, enter 'NONE'.
List the x coordinates of all local maxima of f. If there are no
local maxima, enter 'NONE'.
Find the second derivative of f:
Use interval notation to indicate the interval(s) of upward
concavity of f(x).
Use interval notation to indicate the interval(s) of downward
concavity...
Please use excel and show all work and formulas. (I will give
your work a like if you do this)
Size (1000s sq. ft)
Selling Price ($1000s)
1.26
117.5
3.02
299.9
1.99
139.0
0.91
45.6
1.87
129.9
2.63
274.9
2.60
259.9
2.27
177.0
2.30
175.0
2.08
189.9
1.12
95.0
1.38
82.1
1.80
169.0
1.57
96.5
1.45
114.9
What are the p-values of the t test (for the
slope estimate) and F test?
What is the coefficient of determination?
What is...
Let E and F be independent events. Show that the events E and F
care also independent.
Hint: Start with P(E∩Fc) =P(E)−P(E∩F) and use the independence
of E and F.
(a) Let <X, d> be a metric space and E ⊆ X. Show
that E is connected iff for all p, q ∈ E, there is a connected A ⊆
E with p, q ∈ E.
b) Prove that every line segment between two points in R^k
is connected, that is Ep,q = {tp + (1 − t)q |
t ∈ [0, 1]} for any p not equal to q in R^k.
C). Prove that every convex subset of R^k...
3. Let X = {1, 2, 3, 4}. Let F be the set of all functions from
X to X. For any relation R on X, define a relation S on F by: for
all f, g ∈ F, f S g if and only if there exists x ∈ X so that
f(x)Rg(x).
For each of the following statements, prove or disprove the
statement.
(a) For all relations R on X, if R is reflexive then S is
reflexive....