In: Advanced Math

Prove that if f, g are integrable, then the function (f(x) + cos(x)g(x))2is integrable.

Let f: X→Y and g: Y→Z be both onto. Prove that g◦f is an onto
function
Let f: X→Y and g: Y→Z be both onto. Prove that f◦g is an onto
function
Let f: X→Y and g: Y→Z be both one to one. Prove that g◦f is an
one to one function
Let f: X→Y and g: Y→Z be both one to one. Prove that f◦g is an
one to one function

prove using the definition of derivative that if f(x) and g(x)
is differentiable than (f'(x)g(x) - f(x)g'(x))/g^2(x)

Prove that a continuous function for sections(subrectangles) is
integrable in R^N

Consider a continuous, integrable, twice-differentiable function
f with input variable x.
In terms of the units of f and the units of x, choose the units of
each function or expression below:
(a) The units of f ' are
the units of
f
the units of
x
(the units of f)(the units of x
)
the units of f
the units of x
the units of f
(the units of x)2
the units of f
(the units...

True and False (No need to solve).
1. Every bounded continuous function is
integrable.
2. f(x)=|x| is not integrable in [-1, 1] because the function f
is not differentiable at x=0.
3. You can always use a bisection algorithm to find a root of a
continuous function.
4. Bisection algorithm is based on the fact that If f is a
continuous function and f(x1) and f(x2) have
opposite signs, then the function f has a root in the interval
(x1,...

For the given function determine the following: f (x) = (sin x +
cos x) 2 ; [−π,π] a) Find the intervals where f(x) is increasing,
and decreasing b) Find the intervals where f(x) is concave up, and
concave down c) Find the x-coordinate of all inflection points

The function F(x) = x2 - cos(π x) is defined on the
interval 0 ≤ x ≤ 1 radians. Explain how the Intermediate Value
Theorem shows that F(x) = 0 has a solution on the interval 0 < x
< .

TOPOLOGY
Let f : X → Y be a function.
Prove that f is one-to-one and onto if and only if f[A^c] =
(f[A])^c for every subset A of X. (prove both directions)

(a) State the definition of the derivative of f.
(b) Using (a), prove the following:d/dx(f(x) +g(x)) =d/dx(f(x))
+d/dx(g(x))

2. Let f(x) ≥ 0 on [1, 2] and suppose that f is integrable on
[1, 2] with R 2 1 f(x)dx = 2 3 . Prove that f(x 2 ) is integrable
on [1, √ 2] and √ 2 6 ≤ Z √ 2 1 f(x 2 )dx ≤ 1 3 .

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